Approximating Constraint Satisfaction Problems on High-Dimensional Expanders
aa r X i v : . [ c s . D S ] J u l Approximating Constraint Satisfaction Problems onHigh-Dimensional Expanders
Vedat Levi Alev ∗ Fernando Granha Jeronimo † Madhur Tulsiani ‡ We consider the problem of approximately solving constraint satisfaction problemswith arity k > k -CSPs) on instances satisfying certain expansion properties, when viewedas hypergraphs. Random instances of k -CSPs, which are also highly expanding, are well-known to be hard to approximate using known algorithmic techniques (and are widelybelieved to be hard to approximate in polynomial time). However, we show that this isnot necessarily the case for instances where the hypergraph is a high-dimensional expander .We consider the spectral definition of high-dimensional expansion used by Dinur andKaufman [FOCS 2017] to construct certain primitives related to PCPs. They measure theexpansion in terms of a parameter γ which is the analogue of the second singular valuefor expanding graphs. Extending the results by Barak, Raghavendra and Steurer [FOCS2011] for 2-CSPs, we show that if an instance of MAX k-CSP over alphabet [ q ] is a high-dimensional expander with parameter γ , then it is possible to approximate the maximumfraction of satisfiable constraints up to an additive error ε using q O ( k ) · ( k / ε ) O ( ) levels ofthe sum-of-squares SDP hierarchy, provided γ ≤ ε O ( ) · ( ( kq )) O ( k ) .Based on our analysis, we also suggest a notion of threshold-rank for hypergraphs,which can be used to extend the results for approximating 2-CSPs on low threshold-rankgraphs. We show that if an instance of MAX k-CSP has threshold rank r for a threshold τ = ( ε / k ) O ( ) · ( q ) O ( k ) , then it is possible to approximately solve the instance up toadditive error ε , using r · q O ( k ) · ( k / ε ) O ( ) levels of the sum-of-squares hierarchy. As inthe case of graphs, high-dimensional expanders (with sufficiently small γ ) have thresholdrank 1 according to our definition. ∗ Supported by NSERC Discovery Grant 2950-120715, NSERC Accelerator Supplement 2950-120719, andpartially supported by NSF awards CCF-1254044 and CCF-1718820. [email protected] . † Supported in part by NSF grants CCF-1254044 and CCF-1816372. [email protected] . ‡ Supported by NSF grants CCF-1254044 and CCF-1816372. [email protected] ontents t -local PSD Ensembles . . . . . . . . . . . . 7 MAX 4-XOR
84 Walks 11 S k , k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.2 Expanding Posets and Balanced Operators . . . . . . . . . . . . . . . . . . . 215.3 Quadratic Forms over Balanced Operators . . . . . . . . . . . . . . . . . . . 235.4 Rectangular Swap Walks S k , l . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.5 Bipartite Kneser Graphs - Complete Complex . . . . . . . . . . . . . . . . . . 32 k -CSP 34 i Introduction
We consider the problem of approximately solving constraint satisfaction problems (CSPs)on instances satisfying certain expansion properties. The role of expansion in understand-ing the approximability of CSPs with two variables in each constraint (2-CSPs) has beenextensively studied and has led to several results, which can also be viewed as no-go re-sults for PCP constructions (since PCPs are hard instances of CSPs). It was shown byArora et al. [AKK +
08] (and strengthened by Makarychev and Makarychev [MM11]) thatthe
Unique Games problem is easily approximable on expanding instances, thus provingthat the
Unique Games Conjecture of Khot [Kho02] cannot be true for expanding instances.Their results were extended to all 2-CSPs and several partitioning problems in works byBarak, Raghavendra and Steurer [BRS11], Guruswami and Sinop [GS11], and Oveis Gha-ran and Trevisan [OGT15] under much weaker notions of expansion.We consider the following question:
When are expanding instances of k-CSPs easy for k > ? At first glance, the question does not make much sense, since random instances of k -CSPs(which are also highly expanding) are known to be hard for various models of compu-tation (see [KMOW17] for an excellent survey). However, while the kind of expansionexhibited by random instances of CSPs is useful for constructing codes, it is not sufficientfor constructing primitives for PCPs, such as locally testable codes [BSHR05]. On the otherhand, objects such as high-dimensional expanders, which possess a form of “structuredmulti-scale expansion” have been useful in constructing derandomized direct-product anddirect-sum tests (which can be viewed as locally testable distance amplification codes)[DK17], lattices with large distance [KM18], list-decodable direct product codes [DHK + Connections to coding theory.
Algorithmic results related to expanding CSPs are alsorelevant for the problem of decoding locally testable codes. Consider a code C constructedvia k -local operations (such as k -fold direct-sum) on a base code C with smaller distance.Then, a codeword in C is simply an instance of a CSP, where each bit places a constraint on k bits (which is k -XOR in case of direct sum) of the relevant codeword in C . The task of de-coding a noisy codeword is then equivalent to finding an assignment in C , satisfying themaximum number of constraints for the above instance. Thus, algorithms for solving CSPson expanding instances may lead to new decoding algorithms for codes obtained by apply-ing local operations to a base code. In fact, the list decoding algorithm for direct-productcodes by Dinur et al. [DHK +
18] also relied on algorithmic results for expanding uniquegames. Since all constructions of locally testable codes need to have at least some weakexpansion [DK12], it is interesting to understand what notions of expansion are amenableto algorithmic techniques.
High-dimensional expanders and our results. A d -dimensional expander is a downward-closed hypergraph (simplicial complex), say X , with edges of size at most d +
1, such thatfor every hyperedge a ∈ X (with | a | ≤ d − G ( X a ) is a1pectral expander . Here, the graph G ( X a ) is defined to have the vertex set { i | a ∪ { i } ∈ X } and edge-set { i , j | a ∪ { i , j } ∈ X } . If the (normalized) second singular value of each of theneighborhood graphs is bounded by γ , X is said to be a γ -high-dimensional expander( γ -HDX).Note that (the downward closure of) a random sparse ( d + ) -uniform hypergraph,say with n vertices and c · n edges, is very unlikely to be a d -dimensional expander. Withhigh probability, no two hyperedges share more than one vertex and thus for any i ∈ [ n ] , the neighborhood graph G i is simply a disjoint union of cliques of size d , which isvery far from an expander. While random hypergraphs do not yield high-dimensionalexpanders, such objects are indeed known to exists via (surprising) algebraic constructions[LSV05b, LSV05a, KO18a, CTZ18] and are known to have several interesting propertiesand applications [KKL16, DHK +
18, KM17, KO18b, DDFH18, DK17, PRT16].Expander graphs can simply be thought of as the one-dimensional case of the abovedefinition. The results of Barak, Raghavendra and Steurer [BRS11] for 2-CSPs yield that ifthe constraint graph of a 2-CSP instance (with size n and alphabet size q ) is a sufficientlygood (one dimensional) spectral expander, then one can efficiently find solutions satisfying OPT − ε fraction of constraints, where OPT denotes the maximum fraction of constraintssatisfiable by any assignment. Their algorithm is based on ( q / ε ) O ( ) levels of the Sum-of-Squares (SoS) SDP hierarchy, and the expansion requirement on the constraint graph isthat the (normalized) second singular value should be at most ( ε / q ) O ( ) . We show a similarresult for k -CSPs when the corresponding simplicial complex X I , which is obtained by in-cluding one hyperedge for each constraint and taking a downward closure, is a sufficientlygood ( k − ) -dimensional expander. Theorem 1.1 (Informal) . Let I be an instance of MAX k-CSP on n variables taking values overan alphabet of size q, and let ε > . Let the simplicial complex X I be a γ -HDX with γ = ε O ( ) · ( ( kq )) O ( k ) . Then, there is an algorithm based on ( k / ε ) O ( ) · q O ( k ) levels of the Sum-of-Squareshierarchy, which produces an assignment satisfying OPT − ε fraction of the constraints. Remark 1.2.
While the level-t relaxation for
MAX k-CSP can be solved in time ( nq ) O ( t ) [RW17],the rounding algorithms used by [BRS11] and our work do not need the full power of this relaxation.Instead, they are captured by the “local rounding” framework of Guruswami and Sinop [GS12] whoshow how to implement a local rounding algorithm based on t levels of the SoS hierarchy, in timeq O ( t ) · n O ( ) (where q denotes the alphabet size). Our techniques.
We start by using essentially the same argument for analyzing the SoShierarchy as was used by [BRS11] (specialized to the case of expanders). They viewed theSoS solution as giving a joint distribution on each pair of variables forming a constraint,and proved that for sufficiently expanding graphs, these distributions can be made closeto product distributions, by conditioning on a small number of variables (which governsthe number of levels required). Similarly, we consider the conditions under which jointdistributions on k -tuples corresponding to constraints can be made close to product dis-tributions. Since the [BRS11] argument shows how to split a joint distribution into twomarginals, we can use it to recursively split a set of size k into two smaller ones (one canthink of all splitting operations as forming a binary tree with k leaves). While there are several definitions of high-dimensional expanders, we consider the one by Dinur andKaufman [DK17], which is most closely related to spectral expansion, and was also the one shown to berelated to PCP applications. Our results also work for a weaker but more technical definition by Dikstein etal. [DDFH18], which we defer till later. k -CSPs, we may split a set of size ( ℓ + ℓ ) intodisjoint sets of size ℓ and ℓ . This requires understanding the expansion of the followingfamily of (weighted) bipartite graphs arising from the complex X I : The vertices in thegraph are sets of variables of size ℓ and ℓ that occur in some constraint, and the weight ofan edge { a , a } for a ∩ a = ∅ , is proportional to the probability that a random constraintcontains a ⊔ a . Note that this graph may be weighted even if the k -CSP instance I isunweighted.We view the above graphs as random walks, which we call “swap walks” on the hy-peredges (faces) in the complex X . While several random walks on high-dimensional ex-panders have been shown to have rapid mixing [KM17, KO18b, DK17, LLP17], we need astronger condition. To apply the argument from [BRS11], we not only need that the secondsingular value is bounded away from one, but require it to be an arbitrarily small constant(as a function of ε , k and q ). We show that this is indeed ensured by the condition that a ∩ a = ∅ , and obtain a bound of k O ( k ) · γ on the second singular value. This bound,which constitutes much of the technical work in the paper, is obtained by first expressingthese walks in terms of more canonical walks, and then using the beautiful machinery ofharmonic analysis on expanding posets by Dikstein et al. [DDFH18] to understand theirspectra.The swap walks analyzed above represent natural random walks on simplicial com-plexes, and their properties may be of independent interest for other applications. Just asthe high-dimensional expanders are viewed as “derandomized” versions of the completecomplex (containing all sets of size at most k ), one can view the swap walks as derandom-ized versions of (bipartite) Kneser graphs, which have vertex sets ( [ n ] ℓ ) and ( [ n ] ℓ ) , and edges ( a , b ) iff a ∩ b = ∅ . We provide a more detailed and technical overview in Section 3 afterdiscussing the relevant preliminaries in Section 2. High-dimensional threshold rank.
The correlation breaking method in [BRS11] can beapplied as long as the graph has low threshold rank i.e., the number of singular valuesabove a threshold τ = ( ε / q ) O ( ) is bounded. Similarly, the analysis described above canbe applied, as long as all the swap walks which arise when splitting the k -tuples havebounded threshold rank. This suggests a notion of high-dimensional threshold rank forhypergraphs (discussed in Section 7), which can be defined in terms of the threshold ranksof the relevant swap walks. We remark that it is easy to show that dense hypergraphs(with Ω ( n k ) hyperedges) have small-threshold rank according to this notion, and thus itcan be used to recover known algorithms for approximating k -CSPs on dense instances[FK96] (as was true for threshold rank in graphs). Other related work.
While we extend the approach taken by [BRS11] for 2-CSPs, some-what different approaches were considered by Guruswami and Sinop [GS11], and Oveis-Gharan and Trevisan [OGT15]. The work by Guruswami and Sinop relied on the expan-sion of the label extended graph, and used an analysis based on low-dimensional approxi-mations of the SDP solution. Oveis-Gharan and Trevisan used low-threshold rank assump-tions to obtain a regularity lemma, which was then used to approximate the CSP. For thecase of k -CSPs, the Sherali-Adams hierarchy can be used to solve instances with bounded3reewidth [WJ04] and approximately dense instances [YZ14, MR17]. Brandao and Harrow[BH13] also extended the results by [BRS11] for 2-CSPs to the case of 2-local Hamiltonians.We show that their ideas can also be used to prove a similar extension of our results to k -local Hamiltonians on high-dimensional expanders.In case of high-dimensional expanders, in addition to canonical walks described here, a“non-lazy” version of these walks (moving from s to t only if s = t ) was also considered byKaufman and Oppenheim [KO18b], Anari et al. [ALGV18] and Dikstein et al. [DDFH18].The swap walks studied in this paper were also considered independently in a very recentwork of Dikstein and Dinur [DD19] (under the name "complement walks").In a recent follow-up work [AJQ + C via k -local operations to amplify distance. This workalso showed that the hypergraphs obtained by considering collections of length- k walkson an expanding graph also satisfy (a slight variant of) splittability, and admit similaralgorithms. Recall that for an operator A : V → W between two finite-dimensional inner productspaces V and W , the operator norm can be written as k A k op = sup f , g = h A f , g ik f k k g k .Also, for such an A the adjoint A † : W → V is defined as the (unique) operator satisfying h A f , g i = h f , A † g i for all f ∈ V , g ∈ W . For A : V → W , we take k A k op = σ ( A ) ≥ σ ( A ) ≥· · · ≥ σ r ( A ) > A : V → V , σ ( A ) denotes its second largest eigenvalue in absolute value. A high-dimensional expander (HDX) is a particular kind of downward-closed hypergraph(simplicial complex) satisfying an expansion requirement. We elaborate on these proper-ties and define well known natural walks on HDXs below. A simplicial complex X with ground set [ n ] is a downward-closed collection ofsubsets of [ n ] i.e., for all sets s ∈ X and t ⊆ s , we also have t ∈ X. The sets in X are also referredto as faces of X.We use the notation X ( i ) to denote the collection of all faces s in X with | s | = i. When facesare of cardinality at most d, we also use the notation X ( ≤ d ) to denote all the faces of X. Byconvention, we take X ( ) : = { ∅ } . simplicial complex X ( ≤ d ) is said to be a pure simplicial complex if every face of X iscontained in some face of size d. Note that in a pure simplicial complex X ( ≤ d ) , the top slice X ( d ) completely determines the complex. Note that it is more common to associate a geometric representation to simplicial com-plexes, with faces of cardinality i being referred to as faces of dimension i − X ( i − ) instead of X ( i ) ). However, since we will only be treatingthese as hypergraphs, we prefer to index faces by their cardinality, to improve readabilityof related expressions.An important simplicial complex is the complete complex. Definition 2.2 (Complete Complex ∆ d ( n ) ) . We denote by ∆ d ( n ) the complete complex withfaces of size at most d i.e., ∆ d ( n ) : = { s ⊆ [ n ] | | s | ≤ d } . Let C k denote the space of real valued functions on X ( k ) i.e., C k : = { f | f : X ( k ) → R } ∼ = R X ( k ) .We describe natural walks on simplicial complexes considered in [DK17, DDFH18, KO18b],as stochastic operators, which map functions in C i to C i + and vice-versa.To define the stochastic operators associated with the walks, we first need to describea set of probability measures which serve as the stationary measures for these randomwalks. For a pure simplicial complex X ( ≤ d ) , we define a collection of probability mea-sures ( Π , . . . Π d ) , with Π i giving a distribution on faces in the slice X ( i ) . Definition 2.3 (Probability measures ( Π , . . . , Π d ) ) . Let X ( ≤ d ) be a pure simplicial complexand let Π d be an arbitrary probability measure on X ( d ) . We define a coupled array of randomvariables ( s ( d ) , . . . , s ( ) ) as follows: sample s ( d ) ∼ Π d and (recursively) for each i ∈ [ d ] , take s ( i − ) to be a uniformly random subset of s ( i ) , of size i − .The distributions Π d − , . . . , Π are then defined to be the marginal distributions of the randomvariables s ( d − ) , . . . , s ( ) as defined above. The following is immediate from the definition above.
Proposition 2.4.
Let a ∈ X ( ℓ ) be an arbitrary face. For all j ≥ , one has ∑ b ∈ X ( ℓ + j ) : b ⊇ a Π ℓ + j ( b ) = (cid:18) ℓ + jj (cid:19) · Π ℓ ( a ) .For all k , we define the inner product of functions f , g ∈ C k , according to associatedmeasure Π k h f , g i = E s ∼ Π k [ f ( s ) g ( s )] = ∑ s ∈ X ( k ) f ( s ) g ( s ) · Π k ( s ) .We now define the up and down operators U i : C i → C i + and D i + : C i + → C i as [ U i g ]( s ) = E s ′ ∈ X ( i ) , s ′ ⊆ s (cid:2) g ( s ′ ) (cid:3) = i + · ∑ x ∈ s g ( s \{ x } )[ D i + g ]( s ) = E s ′ ∼ Π i + | s ′ ⊃ s (cid:2) g ( s ′ ) (cid:3) = i + · ∑ x / ∈ s g ( s ⊔ { x } ) · Π i + ( s ⊔ { x } ) Π i ( s )
5n important consequence of the above definition is that U i and D i + are adjoints withrespect to the inner products of C i and C i + . Fact 2.5. U i = D † i + , i.e., h U i f , g i = h f , D i + g i for every f ∈ C i and g ∈ C i + . Note that the operators can be thought of as defining random walks in a simplicialcomplex X ( ≤ d ) . The operator U i moves down from a face s ∈ X ( i + ) to a face s ′ ∈ X ( i ) ,but lifts a function g ∈ C i up to a function U g ∈ C i + . Similarly, the operator D i + can bethought of as defining a random walk which moves up from s ∈ X ( i ) to s ′ ∈ X ( i + ) . It iseasy to verify that these walks respectively map the measure Π i + to Π i , and Π i to Π i + . We recall the notion of high-dimensional expansion (defined via local spectral expansion)considered by [DK17]. We first need a few pieces of notation.For a complex X ( ≤ d ) and s ∈ X ( i ) for some i ∈ [ d ] , we denote by X s the link complex X s : = { t \ s | s ⊆ t ∈ X } .When | s | ≤ d −
2, we also associate a natural weighted graph G ( X s ) to a link X s , withvertex set X s ( ) and edge-set X s ( ) . The edge-weights are taken to be proportional to themeasure Π on the complex X s , which is in turn proportional to the measure Π | s | + on X .The graph G ( X s ) is referred to as the skeleton of X s . Dinur and Kaufman [DK17] definehigh-dimensional expansion in terms of spectral expansion of the skeletons of the links. Definition 2.6 ( γ -HDX from [DK17]) . A simplicial complex X ( ≤ d ) is said to be γ -High Di-mensional Expander ( γ -HDX) if for every ≤ i ≤ d − and for every s ∈ X ( i ) , the graph G ( X s ) satisfies σ ( G ( X s )) ≤ γ , where σ ( G ( X s )) denotes the second singular value of the (normalized)adjacency matrix of G ( X s ) . A k -CSP instance I = ( H , C , w ) with alphabet size q consists of a k -uniform hypergraph, aset of constraints C = {C a ⊆ [ q ] a : a ∈ H } ,and a non-negative weight function w ∈ R H + on the constraints, satisfying ∑ a ∈ H w ( a ) = C a is said to be satisfied by an assignment η if we have η | a ∈ C a i.e., therestriction of η on a is contained in C a . We write, SAT I ( η ) for the (weighted fraction of theconstraints) satisfied by the assignment η i.e., SAT I ( η ) = ∑ a ∈ H w ( a ) · [ η | a ∈ C a ] = E a ∼ w [ [ η | a ∈ C a ]] .We denote by OPT ( I ) the maximum of SAT I ( η ) over all η ∈ [ q ] V ( H ) .Any k -uniform hypergraph H can be associated with a pure simplicial complex ina canonical way by just setting X I = { b : ∃ a ∈ H and a ⊇ b } – notice that X I ( k ) = H .We will refer to this complex as the constraint complex of the instance I . The probabilitydistribution Π k on X I will be derived from the weights function w of the constraint, i.e Π k ( a ) = w ( a ) ∀ a ∈ X I ( k ) = H .6 .4 Sum-of-Squares Relaxations and t -local PSD Ensembles The Sum-of-Squares (SoS) hierarchy gives a sequence of increasingly tight semidefiniteprogramming relaxations for several optimization problems, including CSPs. Since wewill use relatively few facts about the SoS hierarchy, already developed in the analysis ofBarak, Raghavendra and Steurer [BRS11], we will adapt their notation of t -local distributions to describe the relaxations. For a k -CSP instance I = ( H , C , w ) on n variables, we considerthe following semidefinite relaxation given by t -levels of the SoS hierarchy, with vectors v ( S , α ) for all S ⊆ [ n ] with | S | ≤ t , and all α ∈ [ q ] S . Here, for α ∈ [ q ] S and α ∈ [ q ] S , α ◦ α ∈ [ q ] S ∪ S denotes the partial assignment obtained by concatenating α and α . maximize E a ∼ w " ∑ α ∈ C a k v ( a , α ) k = : SDP ( I ) subject to D v ( S , α ) , v ( S , α ) E = ∀ α | S ∩ S = α | S ∩ S D v ( S , α ) , v ( S , α ) E = D v ( S , α ) , v ( S , α ) E ∀ S ∪ S = S ∪ S , α ◦ α = α ◦ α ∑ j ∈ [ q ] k v ( { i } , j ) k = ∀ i ∈ [ n ] k v ( ∅ , ∅ ) k = For any set S with | S | ≤ t , the vectors v ( S , α ) induce a probability distribution µ S over [ q ] S such that the assignment α ∈ [ q ] S appears with probability k v ( S , α ) k . Moreover, thesedistributions are consistent on intersections i.e., for T ⊆ S ⊆ [ n ] , we have µ S | T = µ T , where µ S | T denotes the restriction of the distribution µ S to the set T . We use these distributions todefine a collection of random variables Y , . . . , Y n taking values in [ q ] , such that for any set S with | S | ≤ t , the collection of variables { Y i } i ∈ S have a joint distribution µ S . Note that theentire collection ( Y , . . . , Y n ) may not have a joint distribution: this property is only true forsub-collections of size t . We will refer to the collection ( Y , . . . , Y n ) as a t -local ensemble ofrandom variables.We also have that that for any T ⊆ [ n ] with | T | ≤ t −
2, and any β ∈ [ q ] T , we candefine a ( t − | T | ) -local ensemble ( Y ′ , . . . , Y ′ n ) by “conditioning” the local distributions onthe event Y T = β , where Y T is shorthand for the collection { Y i } i ∈ T . For any S with | S | ≤ t − | T | , we define the distribution of Y ′ S as µ ′ S : = µ S ∪ T |{ Y T = β } . Finally, the semidefiniteprogram also ensures that for any such conditioning, the conditional covariance matrix M ( S , α )( S , α ) = Cov (cid:0) [ Y ′ S = α ] , [ Y ′ S = α ] (cid:1) is positive semidefinite, where | S | , | S | ≤ ( t − | T | ) /2. Here, for each pair S , S the co-variance is computed using the joint distribution µ ′ S ∪ S . The PSD-ness be easily verifiedby noticing that the above matrix can be written as the Gram matrix of the vectors w ( S , α ) : = k v ( T , β ) k · v ( T ∪ S , β ◦ α ) − k v ( T ∪ S , β ◦ α ) k k v ( T , β ) k · v ( T , β ) In this paper, we will only consider t -local ensembles such that for every conditioning ona set of size at most t −
2, the conditional covariance matrix is PSD. We will refer to theseas t -local PSD ensembles . We will also need a simple corollary of the above definitions.7 act 2.7. Let ( Y , . . . , Y n ) be a t-local PSD ensemble, and let X be any simplicial complex withX ( ) = [ n ] . Then, for all s ≤ t /2 , the collection { Y a } a ∈ X ( ≤ s ) is a ( t / s ) -local PSD ensemble,where X ( ≤ s ) = S si = X ( i ) . For random variables Y S in a t -local PSD ensemble, we use the notation { Y S } to denotethe distribution of Y S (which exists when | S | ≤ t ). We also define Var [ Y S ] as ∑ α ∈ [ q ] S Var [ [ Y S = α ]] . MAX 4-XOR
We consider a simple example of a specific k -CSP, which captures most of the key ideas inour proof. Let I be an unweighted instance of 4-XOR on n Boolean variables. Let H be a 4-uniform hypergraph on vertex set [ n ] , with a hyperedge corresponding to each constrainti.e., each a = { i , i , i , i } ∈ H corresponds to a constraint in I of the form x i + x i + x i + x i = b a ( mod 2 ) ,for some b a ∈ {
0, 1 } . Let X denote the constraint complex for the instance I such that X ( ) = [ n ] , X ( ) = H and let Π , . . . , Π be the associated distributions (with Π beinguniform on H ). Local vs global correlation: the BRS strategy.
We first recall the strategy used by [BRS11],which also suggests a natural first step for our proof. Given a 2-CSP instance with an as-sociated graph G , and a t -local PSD ensemble Y , . . . , Y n obtained from the SoS relaxation,they consider if the “local correlation" of the ensemble is small across the edges of G (whichcorrespond to constraints) i.e., E { i , j }∼ G h(cid:13)(cid:13)(cid:8) Y i Y j (cid:9) − { Y i } (cid:8) Y j (cid:9)(cid:13)(cid:13) i ≤ ε .If the local correlation is indeed small, we easily produce an assignment achieving a value SDP − ε in expectation, simply by rounding each variable x i independently according tothe distribution { Y i } . On the other hand, if this is not satisfied, they show (as a specialcase of their proof) that if G is an expander with second eigenvalue λ ≤ c · ( ε / q ) , thenvariables also have a high “global correlation", between a typical pair ( i , j ) ∈ [ n ] . Here, q is the alphabet size and c is a fixed constant. They use this to show that for ( Y ′ , . . . , Y ′ n ) obtained by conditioning on the value of a randomly chosen Y i , we have E i [ Var [ Y i ]] − E i , Y i E i (cid:2) Var (cid:2) Y ′ i (cid:3)(cid:3) ≥ Ω ( ε / q ) ,where the expectations over i and i are both according to the stationary distribution on thevertices of G . Since the variance is bounded between 0 and 1, this essentially shows thatthe local correlation must be at most ε after conditioning on a set of size O ( q / ε ) (althoughthe actual argument requires a bit more care and needs to condition on a somewhat largerset). Extension to 4-XOR.
As in [BRS11], we check if the t -local PSD ensemble ( Y , . . . , Y n ) obtained from the SDP solution satisfies E { i , i , i , i }∈ H (cid:2) k{ Y i Y i Y i Y i } − { Y i } { Y i } { Y i } { Y i }k (cid:3) ≤ ε .8s before, independently sampling each x i from { Y i } gives an expected value at least SDP − ε in this case. If the above inequality is not satisfied, an application of triangleinequality gives E { i , i , i , i }∈ H " k{ Y i Y i Y i Y i } − { Y i Y i } { Y i Y i }k + k{ Y i Y i } − { Y i } { Y i }k + k{ Y i Y i } − { Y i } { Y i }k > ε .Symmetrizing over all orderings of { i , i , i , i } , we can write the above as ε + · ε > ε ,which gives max { ε , ε } ≥ ε /3. Here, ε : = E { i , i }∼ Π (cid:2) k{ Y i Y i } − { Y i } { Y i }k (cid:3) , and ε : = E { i , i , i , i }∼ Π (cid:2) k{ Y i Y i Y i Y i } − { Y i Y i } { Y i Y i }k (cid:3) = E { i , i , i , i }∼ Π h(cid:13)(cid:13)(cid:8) Y { i , i } Y { i , i } (cid:9) − (cid:8) Y { i , i } (cid:9) (cid:8) Y { i , i } (cid:9)(cid:13)(cid:13) i .As before, ε measures the local correlation across edges of a weighted graph G withvertex set X ( ) = [ n ] and edge-weights given by Π . Also, ε measures the analogousquantity for a graph G with vertex set X ( ) (pairs of variables occurring in constraints)and edge-weights given by Π .Recall that the result from [BRS11] can be applied to any graph G over variables in a2-local PSD ensemble, as long as the σ ( G ) is small. Since { Y i } i ∈ [ n ] and { Y s } s ∈ X ( ) are both ( t /2 ) -local PSD ensembles (by Fact 2.7), we will apply the result to the graph G on thefirst ensemble and G on the second ensemble. We consider the potential Φ ( Y , . . . , Y n ) : = E i ∼ Π [ Var [ Y i ]] + E s ∼ Π [ Var [ Y s ]] .Since local correlation is large along at least one of the graphs G and G , using the abovearguments (and the non-decreasing nature of variance under conditioning) it is easy toshow that in expectation over the choice of { i , j } ∼ Π and β ∈ [ q ] chosen from n Y { i , j } o , the conditional ensemble ( Y ′ , . . . , Y ′ n ) satisfies Φ ( Y , . . . , Y n ) − E i , j , β (cid:2) Φ ( Y ′ , . . . , Y ′ n ) (cid:3) = Ω ( ε ) , provided G and G satisfy σ ( G ) , σ ( G ) ≤ c · ε for an appropriate constant c .The bound on the eigenvalue of G follows simply from the fact that it is the skeletonof X , which is a γ -HDX. Obtaining bounds on the eigenvalues of G and similar higher-order graphs, constitutes much of the technical part of this paper. Note that for a randomsparse instance of MAX 4-XOR , the graph G will be a matching with high probability (since { i , i } in a constraint will only be connected to { i , i } in the same constraint). However,we show that in case of a γ -HDX, this graph has second eigenvalue O ( γ ) . We analyzethese graphs in terms of modified high-dimensional random walks, which we call “swapwalks”.We remark that our potential and choice of a “seed set” of variables to condition on, isslightly different from [BRS11]. To decrease the potential function above, we need that for9ach level X ( i ) ( i =
1, 2 in the example above) the seed set must contain sufficiently manyindependent samples from X ( i ) sampled according to Π i . This can be ensured by drawingindependent samples from the top level X ( k ) (though X ( ) suffices in the above example).In contrast, the seed set in [BRS11] consists of random samples from Π . Analyzing Swap Walks.
The graph G defined above can be thought of as a randomwalk on X ( ) , which starts at a face s ∈ X ( ) , moves up to a face (constraint) s ′ ∈ X ( ) containing it, and then descends to a face t ∈ X ( ) such that t ⊂ s ′ and s ∩ t = ∅ i.e.,the walk “swaps out” the elements in s for other elements in s ′ . Several walks consideredon simplicial complexes allow for the possibility of a non-trivial intersection, and hencehave second eigenvalue lower bounded by a constant. On the other hand, swap walkscompletely avoid any laziness and thus turn out to have eigenvalues which can be madearbitrarily small. To understand the eigenvalues for this walk, we will express it in termsof other canonical walks defined on simplicial complexes.Recall that the up and down operators can be used to define random walks on simpli-cial complexes. The up operator U i : C i → C i + defines a walk that moves down from aface s ∈ X ( i + ) to a random face t ∈ X ( i ) , t ⊂ s (the operator thus “lifts” a function in C i to a function in C i + ). Similarly, the down operator D i : C i → C i − moves up from a face s ∈ X ( i − ) to t ∈ X ( i ) , t ⊃ s , with probability Π i ( t ) / ( i · Π i − ( s )) . These can be used todefine a canonical random walk N ( u ) : = D · · · D u + U u + · · · U , N ( u ) : C → C ,which moves from up for u steps s ∈ X ( ) to s ′ ∈ X ( u + ) , and then descends back to t ∈ X ( ) . Such walks were analyzed optimally by Dinur and Kaufman [DK17], who provedthat λ (cid:16) N ( u ) (cid:17) = ( u + ) ± O u ( γ ) when X is a γ -HDX. Thus, while this walk gives anexpanding graph with vertex set X ( ) , the second eigenvalue cannot be made arbitrarilysmall for a fixed u (recall that we are interested in showing that σ ( G ) ≤ c · ε ). However,note that we are only interested in N ( ) conditioned on the event that the two elements from s are “swapped out” with new elements in the final set t i.e., s ∩ t = ∅ . We define S ( u , j ) ( s , t ) : = ( u + )( uj ) · ( − j ) · N ( u ) if | t \ s | = j G corresponds to the random-walk matrix S ( ) . We show that while σ ( N ( ) ) ≈ σ ( S ( ) ) = O ( γ ) . We first write the canonical walks in terms of the swap walks.Note that N ( ) = · I + · S ( ) + · S ( ) ,since the “descent” step from s ′ ∈ X ( ) containing s ∈ X ( ) , produces a t ∈ X ( ) which“swaps out” 0, 1 and 2 elements with probabilities / , / and / respectively. Similarly, N ( ) = · I + · S ( ) .Finally, we use the fact (proved in Section 4) that while the canonical walks do dependon the “height” u (i.e., N ( u ) = N ( u ′ ) ) the swap walks (for a fixed number of swaps j ) are10ndependent of the height to which they ascend! In particular, we have S ( ) = S ( ) .Using these, we can derive an expression for the swap walk S ( ) as S ( ) = I + · N ( ) − · N ( ) = I + · ( D D U U − D U ) To understand the spectrum of operators such as the ones given by the above expression,we use the beautiful machinery for harmonic analysis over HDXs (and more generallyover expanding posets) developed by Dikstein et al. [DDFH18]. They show how to decom-pose the spaces C k into approximate eigenfunctions for operators of the form DU . Usingthese decompositions and the properties of expanding posets, we can show that distincteigenvalues of the above operator are approximately the same (up to O ( γ ) errors) whenanalyzing the walks on the complete complex. Finally, we use the fact that swap walks ina complete complex correspond to Kneser graphs (for which the eigenvectors and eigen-values are well-known) to show that λ ( S ( ) ) = O ( γ ) . Splittable CSPs and high-dimensional threshold rank.
We note that the ideas usedabove can be generalized (at least) in two ways. In the analysis of distance from productdistribution for a 4-tuple of random variables forming a contraint, we split it in 2-tuples.In general, we can choose to split tuples in a k -CSP instance along any binary tree T with k leaves, with each parent node corresponding to a swap walk between tuples forming itschildren. Finally, the analysis from [BRS11] also works if the each of the swap walks insome T have a bounded number (say r ) of eigenvalues above some threshold τ , whichprovide a notion of high-dimensional threshold rank for hypergraphs. We refer to such aninstance as a ( T , τ , r ) -splittable .The arguments sketched above show that high-dimensional expanders are ( T , O ( γ ) , 1 ) -splittable for all T . Since the knowledge of T is only required in our analysis and notin the algorithm, we say that rank τ ( I ) ≤ r (or that I is ( τ , r ) -splittable) if I is ( T , τ , r ) -splittable for any T . We defer the precise statement of results for ( τ , r ) -splittable instancesto Section 7. It is important to note that both U i and D i + can be thought of as row-stochastic matrices,i.e. we can think of them as the probability matrices describing the movement of a walkfrom X ( i + ) to X ( i ) ; and from X ( i ) to X ( i + ) respectively. More concretely, we willthink [ D ⊤ i + e s ]( t ) = P h the walk moves up from s ∈ X ( i ) to t ∈ X ( i + ) i and similarly [ U ⊤ i e t ]( s ) = P [ the walk moves down from t ∈ X ( i + ) to s ∈ X ( i )] .By referring to the definition of the up and down operators in Section 2, it is easy toverify that [ D ⊤ i + e s ]( t ) = [ t ⊇ s ] · i + Π i + ( t ) Π i ( s ) and [ U ⊤ i e t ]( s ) = [ s ⊆ t ] · i + Π j ,i.e. we have U ⊤ i Π i + = Π i and D ⊤ i + Π i = Π i + ,i.e., randomly moving up from a sample of Π j gives a sample of Π j + and similarly, movingdown from a sample of Π j + results in a sample of Π j .Instead of going up and down by one dimension, one can try going up or down bymultiple dimensions since ( D i + · · · D i + ℓ ) and ( U i + ℓ · · · U i ) are still row-stochastic matrices.Further, the corresponding probability vectors still have intuitive explanations in terms ofthe distributions Π j . For a face s ∈ X ( k ) , we introduce the notation p ( u ) s = ( D k + · · · D k + u ) ⊤ e s where we take p ( ) s = e s . This notation will be used to denote the probability distributionof the up-walk starting from s ∈ X ( k ) and ending in a random face t ∈ X ( k + u ) satisfying t ⊇ s .Note that the following Lemma together with Proposition 2.4 implies that p ( u ) s is in-deed a probability distribution. Proposition 4.1.
For s ∈ X ( k ) and a ∈ X ( k + u ) one has,p ( u ) s ( a ) = [ a ⊇ s ] · ( k + uu ) · Π k + u ( a ) Π k ( s ) . Proof.
Notice that for u =
0, the statement holds trivially. We assume that there exists some u ≥ p ( u ) s ( a ) = [ a ⊇ s ] · ( k + uu ) · Π k + u ( a ) Π k ( s ) for all a ∈ X ( k + u ) .For b ∈ X ( k + ( u + )) one has, p ( u + ) s ( b ) = [ D ⊤ k + u + p ( u ) s ]( b ) = k + u + · ∑ x ∈ b Π k + u + ( b ) Π k + u ( b \{ x } ) · p ( u ) s ( b \{ x } ) .Plugging in the induction assumption, this implies p ( u + ) s ( b ) = ( k + u + ) · ∑ x ∈ b Π k + u + ( b ) Π k + u ( b \{ x } ) · [( b \{ x ) } ) ⊇ s ] · ( k + uu ) · Π k + u ( b \{ x } ) Π k ( s ) ! , = ( k + u + ) · ( k + uu ) · ∑ x ∈ b [ b \{ x } ⊇ s ] · Π k + u + ( b ) Π k ( s ) .First, note that the up-walk only hits the faces that contain s , otherwise [ b \{ x } ⊇ s ] = b ∈ X ( k + u + ) satisfies b ⊇ s . Since there are precisely ( u + ) indices whose deletion still preserves the containment of s , we can write p ( u + ) s ( b ) = [ b ⊇ s ] · u + k + u + · ( k + uu ) Π k + u + ( b ) Π k ( s ) , = [ b ⊇ s ] · ( k + u + u + ) · Π k + u + ( b ) Π k ( s ) .Thus, proving the proposition. 12imilarly, we introduce the notation q ( u ) a , as q ( u ) a ( s ) = ( U k + u − · · · U k ) ⊤ e s ,i.e. for the probability distribution of the down-walk starting from a ∈ X ( k + u ) and end-ing in a random face of X ( k ) contained in a . The following can be verified using Proposi-tion 4.1, and the fact that ( U k + u − · · · U k ) † = D k + u · · · D k + . Corollary 4.2.
Let X ( ≤ d ) be a simplicial complex, and k , u ≥ be parameters satisfying k + u ≤ d. For a ∈ X ( k + u ) and s ∈ X ( k ) , one hasq ( u ) a ( s ) = ( k + uu ) · [ s ⊆ a ] .In the remainder of this section, we will try to construct more intricate walks on X from X ( k ) to X ( l ) . Definition 4.3 (Canonical and Swap u -Walks) . Let d ≥ , X ( ≤ d ) be a simplicial complex,and k , l , u ≥ be parameters satisfying l ≤ k, u ≤ l and d ≥ k + u; where the constraints on theseparameters are to ensure well-definedness. We will define the following random walks,- canonical u -walk from X ( k ) to X ( l ) . Let N ( u ) k , l be the (row-stochastic) Markov operatorthat represents the following random walk: Starting from a face s ∈ X ( k ) , – (random ascent/up-walk) randomly move up a face s ′′ ∈ X ( k + u ) that contains s ,where s ′′ is picked with probabilityp ( u ) s ( s ′′ ) = [( D k + · · · D k + u ) ⊤ e s ]( s ′′ ) . – (random descent/down-walk) go to a face s ′ ∈ X ( l ) picked uniformly among all thel-dimensional faces that are contained in s ′′ , i.e., the set s ′ is picked with probabilityq s ′′ ( s ′ ) = [ s ′ ⊆ s ′′ ] · ( k + ul ) = [( U k + u − · · · U l ) ⊤ e s ′′ ]( s ′ ) . The operator N ( u ) k , l : C l → C k satisfies the following equation, N ( u ) k , l = D k + · · · D k + u U k + u − · U k · · · U l . Notice that we have N ( ) k , k = I , and N ( ) k , l = ( U k − . . . U l ) for l < k.- swapping walk from X ( k ) to X ( l ) . Let S k , l be the Markov operator that represents thefollowing random walk: Starting from a face s ∈ X ( k ) , – (random ascent/up-walk) randomly move up to a face s ′′ ∈ X ( k + l ) that contains s , where as before s ′′ is picked with probabilityp ( l ) s ( s ′′ ) = [( D k + · · · D k + l + ) ⊤ e s ]( s ′′ ) .13 (deterministic descent) deterministically go to s ′ = s ′′ \ s ∈ X ( l ) . For our applications, we will need to show that the walk S k , l has good spectral expan-sion whenever X is a d -dimensional γ -expander, for γ sufficiently small. To show this, wewill relate the swapping walk operator S k , l on X to the canonical random walk operators N ( u ) k , l (q.v. Lemma 4.4).By the machinery of expanding posets (q.v. Section 5) it is possible to argue that thespectral expansion of the random walk operator N ( u ) k , l on a high dimensional expander willbe close to that of the complete complex. This will allow us to conclude using the relationbetween the swapping walks and the canonical walks (q.v. Lemma 4.4) that the spectralexpansion of the swapping walk on X , will be comparable with the spectral expansion ofthe swap walk on the complete complex. More precisely, we will show Lemma 4.4 (Lemma 5.34) . For any d , k , l ≥ , and the complete simplicial simplicial complexX ( ≤ d ) , one has the following: If k ≥ l ≥ and d ≥ k + l, we have σ ( S k , l ) = O k , l (cid:18) n (cid:19) .Using these two, and the expanding poset machinery, we will conclude Theorem 4.5 (Theorem 5.2 simplified) . Let X be a d-dimensional γ expander. If k ≥ l ≥ satisfy d ≥ l + k we have, σ ( S k , l ) = O k , l ( γ ) where S k , l is the swapping walk on X from X ( k ) to X ( l ) . To prove Theorem 4.5 we will need to define an intermediate random walk that wewill call the j -swapping u -walk from X ( k ) to X ( l ) : Definition 4.6 ( j -swapping u -walk from X ( k ) to X ( l ) ) . Given d , u , j , k , l ≥ satisfying l ≤ k,j ≤ u, u ≤ l, and d ≥ k + u. Let S ( u , j ) k , l be the Markov operator that represents the followingrandom walk from X ( k ) to X ( l ) on a d-dimensional simplicial complex X: Starting from s ∈ X ( k ) - (random ascent/up-walk) randomly move up to a face s ′′ ∈ X ( k + u ) that contains s ,where s ′′ is picked with probabilityp ( u ) s ( s ′′ ) = [( D k + · · · D k + u ) ⊤ e s ]( s ′′ ) . - (conditioned descent) go to a face s ′ ∈ X ( l ) sampled uniformly among all the subsets of s ′′ ∈ X ( k + u ) that have intersection j with s ′′ \ s , i.e. | s ′ ∩ ( s ′′ \ s ) | = j.Notice that S k , l = S ( l , l ) k , l for any k and I = S ( u ,0 ) k , k for any u. Remark 4.7.
We will prove that the parameter u does not effect the swapping walk S ( u , j ) k , l so longas u ≥ j, i.e. for all u , u ′ ≥ j we have S ( u ′ , j ) k , l = S ( u , j ) k , l . Thus, we will often write S ( j ) k , l for S ( j , j ) k , l . .2 Swap Walks are Height Independent Recall that the swap walk S ( u , j ) k , l is the conditional walk defined in terms of N ( u ) k , l where s ∈ X ( k ) is connected to t ∈ X ( l ) only if | t \ s | = j . The parameter u is called the heightof the walk, namely the number of times it moves up. Since up and down operators havesecond singular value bounded away from 1, the second singular value of N ( u ) k , l shrinks as u increases. In other words, the operator N ( u ) k , l depends on the height u . Surprisingly, thewalk S ( u , j ) k , l which is defined in terms of N ( u ) k , l does not depend on the particular choice of u as long as it is well defined. More precisely, we have the following result. Lemma 4.8.
If X is a d-dimensional simplicial complex, ≤ l ≤ k, and u , u ′ ∈ [ j , d − k ] , then S ( u , j ) k , l = S ( u ′ , j ) k , l .In order to obtain Lemma 4.8, we will need a simple proposition: Proposition 4.9.
Let s ∈ X ( k ) , s ′ ⊆ s and | t ′ | = j. Suppose s ′ ⊔ t ′ ∈ X ( l ) . Then, we have S ( u , j ) k , l ( s , s ′ ⊔ t ′ ) = ( kl − j ) · ( uj ) · ∑ a ∈ X ( k + u ) : a ⊇ ( s ⊔ t ′ ) p ( u ) s ( a ) . Proof.
The only way of picking s ′ ⊔ t ′ at the descent step is picking some a ∈ X ( k + u ) thatcontains s ′ ⊔ t ′ in the ascent step. The probability of this happening is precisely, p = ∑ a ∈ X ( k + u ) : a ⊇ ( s ⊔ t ′ ) p ( u ) s ( a ) .Suppose we are at a set a = s ⊔ t , such that t ⊇ t ′ and s ∩ t = ∅ . Now, the probability of thedescent step ending at s ′ ⊔ t ′ is the probability of a randomly sampled ( l − j ) -elementedsubset of s being s ′ and the probability of a randomly sampled j -elemented subset of t being t ′ . It can be verified that this probability is p = ( kl − j ) · ( uj ) .By law of total probability we establish that S ( u , j ) k , l ( s , s ′ ⊔ t ′ ) = p · p = ( kl − j ) · ( uj ) · ∑ a ∈ X ( k + u ) : a ⊔ ( s ⊔ t ′ ) p ( u ) s ( a ) . Lemma 4.10 (Height Independence) . Let u ∈ [ j , d − k ] . For any s ∈ X ( k ) , s ′ ⊆ s and t ′ ∈ X ( j ) satisfying s ′ ⊔ t ′ ∈ X ( l ) we have the following, S ( u , j ) k , l ( s , s ′ ⊔ t ′ ) = ( kl − j )( k + jj ) · Π k + j ( s ⊔ t ′ ) Π k ( s ) . In particular, the choice of u ∈ [ j , d − k ] does not affect the swap walk. roof. We have, ∑ a ∈ X ( k + u ) : a ⊇ s ⊔ t ′ p ( k + u ) s ( a ) = ( k + uu ) · Π k ( s ) · ∑ a ∈ X ( k + u ) : a ⊇ s ⊔ t Π k + u ( a ) , = ( k + uu − j )( k + uu ) · Π k + j ( s ⊔ t ′ ) Π k ( s ) where the first equality is due to Proposition 4.1 and the second is due to Proposition 2.4together with the observation that s ⊔ t ′ ∈ X ( k + j ) .Thus, by Proposition 4.9 we get, S ( u , j ) k , l ( s , t ) = ( uj ) · ( kl − j ) ( k + uu − j )( k + uu ) · Π k + j ( s ⊔ t ′ ) Π k ( s ) .We complete the proof by noting that, ( k + uu − j )( k + uu ) = ( uj )( k + jj ) ,and thus S ( u , j ) k , l ( s , t ) = ( kl − j ) · ( k + jj ) · Π k + j ( s ⊔ t ′ ) Π k ( s ) which proves the formula.Since the choice of u does not affect the formula, we obtain Lemma 4.8. We show that the canonical walks are given by an average of swap walks with respect tothe hypergeometric distribution.
Lemma 4.11.
Let u , l , k , d ≥ be given satisfying l ≤ k and u ≤ l. Then, we have the followingformula for the canonical u-walk on any X ( ≤ d ) satisfying d ≥ k + u N ( u ) k , l = u ∑ j = ( uj )( kl − j )( k + ul ) · S ( j ) k , l . Proof.
Suppose the canonical u -walk starting from s ∈ X ( k ) picks s ′′ ∈ X ( k + u ) in thesecond step. Write E j ( s ′′ ) for the event that the random face s ′ the canonical u -walk picksin the descent step satisfies (cid:12)(cid:12) s ′ \ s (cid:12)(cid:12) = j .By elementary combinatorics, P s ′ ⊆ s ′′ (cid:2) E j ( s ′′ ) | s ′′ (cid:3) = ( uj )( kl − j )( k + ul ) s ′ ∈ X ( l ) of s ′′ . Further, let t ′ j bethe random variable that stands for the face picked in the descent step of the j -swapping u -walk from X ( k ) to X ( l ) .By the definition of the j -swapping walk from X ( k ) to X ( l ) , conditioning that the as-cent step picks the same s ′′ ∈ X ( k + u ) we have P h t ′ j = t | s ′′ i = P (cid:2) s ′ = t | s ′′ and E j ( s ′′ ) (cid:3) . (1)Now, by the law of total probability we have N ( u ) k , l ( s , t ) = P (cid:2) S ′ = t (cid:3) = u ∑ j = ∑ s ′′ ∈ X ( k + u ) P (cid:2) s ′′ (cid:3) · P (cid:2) E j ( s ′′ ) | s ′′ (cid:3) · P (cid:2) s ′ = t | s ′′ and E j ( s ′′ ) (cid:3) , = u ∑ j = ( uj )( kl − j )( k + ul ) · E s ′′ ⊇ s (cid:2) P (cid:2) s ′ = t | s ′′ and E j ( s ′′ ) (cid:3)(cid:3) , = u ∑ j = ( uj )( k + ul − j )( k + ul ) · E s ′′ ⊇ s h P h t ′ j = t | s ′′ ii where we used Equation (1) to get the last equality. Another application of the law of totalprobability gives us E s ′′ ⊇ s h P h t ′ j = t | s ′′ ii = P h t ′ j = t i .This allows us to write, N ( u ) k , l ( s , t ) = u ∑ j = ( uj )( kl − j )( k + ul ) · P h t ′ j = t i , = u ∑ j = ( uj )( kl − j )( k + ul ) · S ( u , j ) k , l ( s , t ) ,The statement follows using height independence, i.e. Lemma 4.8 We show how the swap walks can be obtained as a signed sum of canonical walks. Thisresult follows from binomial inversion which we recall next.
Fact 4.12 (Binomial Inversion, [BS02]) . Let ( a n ) n ≥ , ( b n ) n ≥ be arbitrary sequences. Supposefor all n ≥ we have, b n = n ∑ j = (cid:18) nj (cid:19) · ( − ) j · a j . Then, we also have a n = n ∑ j = (cid:18) nj (cid:19) · ( − ) j · b j .17 orollary 4.13. Let k , l , d ≥ be given parameters such that k + l ≤ d and k ≥ l. For anysimplicial complex X ( ≤ d ) , one has the following formula for the u-swapping walk from X ( k ) toX ( l ) in terms of the canonical j-walks: (cid:18) kl − u (cid:19) S ( u ) k , l = u ∑ j = ( − ) u − j · (cid:18) k + jl (cid:19) · (cid:18) uj (cid:19) · N ( j ) k , l . Proof.
Fix faces s ∈ X ( k ) and t ∈ X ( l ) and set for all j ∈ [ u ] a j : = (cid:18) kl − j (cid:19) · ( − ) j · S ( j ) k , l ( s , t ) .Notice that we have by Lemma 4.11 (cid:18) k + ul (cid:19) · N ( u ) k , l ( s , t ) = u ∑ j = (cid:18) uj (cid:19) · ( − ) j · a j = u ∑ j = (cid:18) uj (cid:19) · (cid:18) kl − j (cid:19) · · S ( j ) k , l ( s , t ) .i.e. if we set b u = (cid:18) k + ul (cid:19) · N ( u ) k , l ( s , t ) ,we can apply Fact 4.12 to obtain (cid:18) kl − u (cid:19) · ( − ) u · S ( u ) k , l ( s , t ) = a u = u ∑ j = (cid:18) uj (cid:19) · ( − ) j · b j = u ∑ j = (cid:18) uj (cid:19) · (cid:18) k + jl (cid:19) · ( − ) j · N ( j ) k , l ( s , t ) .Dividing both sides of this equation by ( − ) u yields the desired result. Swap walks arise naturally in our k -CSPs approximation scheme on HDXs where the run-ning time and the quality of approximation depend on the expansion of these walks. Forthis reason, we analyze the spectra of swap walks. We show that swap walks S k , k of γ -HDXs are indeed expanding for γ sufficiently small. More precisely, the first main resultof this section is the following. Theorem 5.1 (Swap Walk Spectral Bound) . Let X ( ≤ d ) be a γ -HDX with d ≥ k. Then thesecond largest singular value σ ( S k , k ) of the swap operator S k , k is σ ( S k , k ) ≤ γ · (cid:16) · k · k · k k (cid:17) .Theorem 5.1 is enough for the analysis of our k -CSP approximation scheme when k isa power of two. However, to analyze general k -CSPs on HDXs we need to understand thespectra of general swap walks S k , l where k may differ from l . Therefore, we generalize thespectral analysis of S k , k above to S k , l obtaining Theorem 5.2, our second main result of thissection. 18 heorem 5.2 (Rectangular Swap Walk Spectral Bound) . Suppose X ( ≤ d ) is a γ -HDX withd ≥ k + l and k ≤ l. Then the largest non-trivial singular value σ ( S k , l ) of the swap operator S k , l is σ ( S k , l ) ≤ q γ · ( · k ℓ · k + l · k k ) . S k , k We prove Theorem 5.1 by connecting the spectral structure of S k , k of general γ -HDXs tothe well behaved case of complete simplicial complexes. To distinguish these two caseswe denote by S ∆ k , k the swap S k , k of complete complexes . In fact, S ∆ k , k is the random walkoperator of the well known Kneser graph K ( n , k ) (see Definition 5.3). Definition 5.3 (Kneser Graph K ( n , k ) [GM15]) . The Kneser graph K ( n , k ) is the graph G =( V , E ) where V = ( [ n ] k ) and E = {{ s , t } | s ∩ t = ∅ } . Then at least for complete complexes we know that S ∆ k , k is expanding. This is a directconsequence of Fact 5.4. Fact 5.4 (Kneser Graph [GM15]) . The singular values of the Kneser graph K ( n , k ) are (cid:18) n − k − ik − i (cid:19) , for i =
0, . . . , k. This means that σ ( S ∆ k , k ) = O k ( n ) as shown in Claim 5.5. Claim 5.5.
Let d ≥ k and ∆ d ( n ) be the complete complex. The second largest singular value σ ( S ∆ k , k ) of the swap operator S ∆ k , k on ∆ d ( n ) is σ ( S ∆ k , k ) = kn − k , provided n ≥ M k where M k ∈ N only depends on k.Proof. First note that for the complete complex ∆ d ( n ) , the operator S ∆ k , k is the walk ma-trix of the Kneser graph K ( n , k ) . Since the degree of K ( n , k ) is ( n − kk ) , the result followsfrom Fact 5.4.Therefore, if we could claim that σ ( S k , k ) of an arbitrary γ -HDX is close to σ ( S ∆ k , k ) (pro-vided γ is sufficiently small), we would conclude that general S k , k walks are also expand-ing. A priori there is no reason why this claim should hold since a general d -sized γ -HDXmay have much fewer hyperedges ( O d ( n ) versus ( nd ) in the complete ∆ d ( n ) ). Fortunately,it turns out that this claim is indeed true (up to O k ( γ ) errors).To prove Theorem 5.1 we employ the beautiful expanding poset (EPoset) machineryof Dikstein et al. [DDFH18]. Before we delve into the full technical analysis, it might be The precise parameters of the complete complex ∆ d ( n ) where S ∆ k , k lives will not be important. We onlyrequire that S ∆ k , k is well defined in the sense that d ≥ k and n > d . The precise eigenvalues are also well known, but singular values are enough in our analysis. h S k , k f , f i where f ∈ C k .First we informally recall the decomposition C k = ∑ ki = C ki from the EPoset machinerywhere C ki can be thought of as the space of approximate eigenfunctions of degree i of C k (theprecise definitions are deferred to 5.2). In this decomposition, C k is defined as the space ofconstant functions of C k .We prove the stronger result that the S k , k operators of any γ -HDX has an an approx-imate spectrum that only depends on k provided γ is small enough. More precisely, weprove Lemma 5.6. Lemma 5.6 (Swap Quadratic Form) . Let f = ∑ ki = f i with f i ∈ C ki . Suppose X ( ≤ d ) is a γ -HDX with d ≥ k. If γ ≤ ε (cid:0) k k + k + (cid:1) − , then h S k , k f , f i = k ∑ i = λ k ( i ) · h f i , f i i ± ε , where λ k ( i ) depends only on k and i, i.e., λ k ( i ) is an approximate eigenvalue of S k , k associated tospace C ki . Remark 5.7.
From Lemma 5.6, it might seem that we are done since there exist approximate eigen-values λ k ( i ) that only depend on k and i. However, giving an explicit expression for these approx-imate eigenvalues is tricky. For this reason, we rely on the expansion of Kneser graphs as will beclear later. Towards showing Lemma 5.6, we introduce the notion of balanced operators whichin particular captures canonical and swap walks and we show that the quadratic formexpression of Lemma 5.6 is a particular case of a general result for h B f , f i where B is ageneral balanced operator. A balanced operator in C k is any operator that can be obtainedas linear combination of pure balanced operators, the later being operators that are a formalproduct of an equal number of up and down operators. Lemma 5.8 (General Quadratic Form) . Let ε ∈ (
0, 1 ) and let Y ⊆ { Y | Y : C k → C k } be acollection of formal operators that are product of an equal number of up and down walks (i.e., purebalanced operators) not exceeding ℓ walks. Let B = ∑ Y ∈Y α Y Y where α Y ∈ R and let f = ∑ ki = f i with f i ∈ C ki . If γ ≤ ε (cid:0) k k + ℓ ∑ Y ∈Y | α Y | (cid:1) − , then h B f , f i = k ∑ i = ∑ Y ∈Y α Y λ Y k ( i ) ! · h f i , f i i ± ε , where λ Y k ( i ) depends only on the operators appearing in the formal expression of Y , i and k, i.e., λ Y k ( i ) is the approximate eigenvalue of Y associated to C ki . Remark 5.9.
Note that our result generalizes the analysis of [DDFH18] for expanding posets ofHDXs which considered the particular case B = D k + U k . Moreover, their error term analysistreated all the parameters not depending on the number of vertices n as constants. In this workwe make the dependence on the parameters explicit since this dependence is important in under-standing the limits of our k-CSPs approximation scheme on HDXs. The beautiful EPoset machin-ery [DDFH18] is instrumental in our analysis. Theorem 5.10 (Swap Walk Spectral Bound (restatement of Theorem 5.1)) . Let X ( ≤ d ) be a γ -HDX with d ≥ k. For every σ ∈ (
0, 1 ) , if γ ≤ σ · (cid:0) k k + k + (cid:1) − , then the second largestsingular value σ ( S k , k ) of the swap operator S k , k is σ ( S k , k ) ≤ σ . Proof.
First we show that for i ∈ [ k ] the i -th approximate eigenvalue λ k ( i ) of the swapoperator S k , k is actually zero. Note that for i ∈ [ k ] the space C ki is a non-trivial eigenspace(i.e., C ki is not the space of constant functions). Let S ∆ k , k be the swap operator of the completecomplex ∆ d ( n ) . On one hand Claim 5.5 gives σ ( S ∆ k , k ) = max f ∈ C k : f ⊥ k f k = (cid:12)(cid:12)(cid:12) h S ∆ k , k f , f i (cid:12)(cid:12)(cid:12) = O k (cid:18) n (cid:19) .On the other hand since ∆ d ( n ) is a γ ∆ -HDX where γ ∆ = O k ( n ) , if n is sufficiently largewe have γ ∆ ≤ γ and thus Lemma 5.8 can be applied to give σ ( S ∆ k , k ) ≥ max f i ∈ C ki : i ∈ [ k ] , k f i k = (cid:12)(cid:12)(cid:12) h S ∆ k , k f i , f i i (cid:12)(cid:12)(cid:12) = | λ k ( i ) | · h f i , f i i ± O k (cid:18) n (cid:19) .Since n is arbitrary and λ k ( i ) depends only on k and i , we obtain λ k ( i ) = S k , k of the γ -HDX X ( ≤ d ) yields σ ( S k , k ) = max f ∈ C k : f ⊥ k f k = |h S k , k f , f i| ≤ max i ∈ [ k ] | λ k ( i ) | + σ = σ ,concluding the proof. We state the definitions used in our technical proofs starting with γ -EPoset from [DDFH18]. Definition 5.11 ( γ -EPoset adapted from [DDFH18]) . A complex X ( ≤ d ) with operators U , . . . , U d − , D , . . . , D d is said to be a γ -EPoset provided (cid:13)(cid:13) M + i − U i − D i (cid:13)(cid:13) op ≤ γ , (2) for every i =
1, . . . , d − , where M + i : = i + i (cid:18) D i + U i − i + I (cid:19) , i.e., M + i is the non-lazy version of the random walk N ( ) i , i = D i + U i . Definition 5.11 can be directly used as an operational definition of high-dimension ex-pansion as done in [DDFH18]. To us it is important that γ -HDXs are also γ -EPosets asestablished in Lemma 5.12. In fact, these two notions are known to be closely related. We tailor their general EPoset definition to HDXs. In fact, what they call γ -HDX we call γ -EPoset. More-over, what they call γ -HD expander we call γ -HDX. emma 5.12 (From [DDFH18]) . Let X be a d-sized simplicial complex.- If X is a γ -HDX, then X is a γ -EPoset.- If X is a γ -EPoset, then X is a d γ -HDX. Naturally the complete complex ∆ d ( n ) is a γ -EPoset since it is a γ -HDX. Moreover, inthis particular case γ vanishes as n grows. Lemma 5.13 (From [DDFH18]) . The complete complex ∆ d ( n ) is a γ -EPoset with γ = O d ( n ) . Harmonic Analysis on Simplicial Complexes
The space C k defined in Section 2.2.2 can be decomposed into subspaces C ki of functions of degree i for 0 ≤ i ≤ k where C ki : = { U k − i h i | h i ∈ H i } ,with H i : = ker ( D i ) , and C k : = { f : X ( k ) → R | f is constant } .More precisely, we have the following. Lemma 5.14 (From [DDFH18]) . C k = k ∑ i = C ki .Lemma 5.14 is proven in Appendix B as Lemma B.3.For convenience set ~ δ ∈ R d − such that δ i = ( i + ) for i ∈ [ d − ] . It will also beconvenient to work with the following equivalent version of Eq. (2) k D i + U i − ( − δ i ) U i − D i − δ i I k op ≤ ii + γ . (3)Towards our goal of understanding quadratic forms of swap operators we study theapproximate spectrum of operators of the form Y = Y ℓ . . . Y where each Y i is either anup or down operator, namely, Y is a generalized random walk of ℓ steps. We regard theexpression Y ℓ . . . Y defining Y as a formal product. Definition 5.15 (Pure Balanced Operator) . We call Y : C k → C k a pure balanced operator if Y can be defined as product Y ℓ . . . Y where each Y i is either an up or down operator. When we saythat the spectrum of Y depends on Y we mean that it depends on k and on the formal expression Y ℓ . . . Y (i.e., pattern of up and down operators). Remark 5.16.
By definition canonical walks N ( u ) k , k are pure balanced operators. Taking linear combinations of pure balanced operators leads to the notion of balanced operators. For the analysis it is convenient to order the indices appearing in Y ℓ . . . Y in decreasing order from left toright. efinition 5.17 (Balanced Operator) . We call B : C k → C k a balanced operator provided thereexists a set of pure balanced operators Y such that B = ∑ Y ∈Y α Y · Y , where α Y ∈ R . Remark 5.18.
Corollary 4.13 establishes that S ( u ) k , k are balanced operators. In particular, S k , k is abalanced operator. It turns out that at a more crude level the behavior of Y is governed by how the numberof up operators compares to the number of down operators. For this reason, it is conve-nient to define U ( Y ) = { Y i | Y i is an up operator } and D ( Y ) = { Y i | Y i is a down operator } where Y is a pure balanced operator. When Y is clear in the context we use U = U ( Y ) and D = D ( Y ) .Henceforth we assume h i ∈ H i = ker ( D i ) , f i ∈ C ki and g ∈ C k . This convention willmake the statements of the technical results of Section 5.3 cleaner. Now we establish all the technical results leading to and including the analysis of quadraticforms over balanced operators . By considering this general class of operators our analysisgeneralizes the analysis given in [DDFH18]. At the same time we refine their error termsanalysis by making the dependence on the EPoset parameters explicit. Recall that an ex-plicit dependence on these parameters is important in understanding the limits of our k -CSP approximation scheme. Lemma 5.19 (General Quadratic Form (restatement of Lemma 5.8)) . Let ε ∈ (
0, 1 ) and let Y ⊆ { Y | Y : C k → C k } be a collection of formal operators that are product of an equal numberof up and down walks (i.e., pure balanced operators) not exceeding ℓ walks. Let B = ∑ Y ∈Y α Y Y where α Y ∈ R and let f = ∑ ki = f i with f i ∈ C ki . If γ ≤ ε (cid:0) k k + ℓ ∑ Y ∈Y | α Y | (cid:1) − , then h B f , f i = k ∑ i = ∑ Y ∈Y α Y λ Y k ( i ) ! · h f i , f i i ± ε , where λ Y k ( i ) depends only on the operators appearing in the formal expression of Y , i and k, i.e., λ Y k ( i ) is the approximate eigenvalue of Y associated to C ki . Since swap walks are balanced operators , we will deduced the following (as provenlater).
Lemma 5.20 (Swap Quadratic Form (restatement of Lemma 5.6)) . Let f = ∑ ki = f i withf i ∈ C ki . Suppose X ( ≤ d ) is a γ -HDX with d ≥ k. If γ ≤ ε (cid:0) k k + k + (cid:1) − , then h S k , k f , f i = k ∑ i = λ k ( i ) · h f i , f i i ± ε , where λ k ( i ) depends on only on k an i, i.e., λ k ( i ) is an approximate eigenvalue of S k , k associated tospace C ki . γ -EPosets is fully determined by the parameters in ~ δ provided γ issmall enough. Note that the Eposet Definition 5.11 provides a “calculus” for rearranging asingle pair of up and down DU . The next result treats the more general case of DU · · · U . Lemma 5.21 (Structure Lemma) . Suppose |D| = . Let Y c ∈ D be the unique down operator in Y ℓ . . . Y . If k A k op ≤ , then h AY ℓ . . . Y h i , g i = ( ℓ = c = Q c , i ( ~ δ ) · h AU ℓ − h i , g i ± ( c − ) · γ k h i k k g k otherwise, where Q c , i is a polynomial in the variables ~ δ depending on c , i such that Q c , i ( ~ δ ) ≤ .Proof. We induct on ( ℓ , c ) . If ℓ = c =
1, we have Y h i = D i h i = Y c Y c − = D j + U j where j = i + c −
2. Then h AY ℓ . . . Y c + ( Y c Y c − ) Y c − . . . Y h i , g i ,becomes ( − δ j ) · h AY ℓ . . . Y c + U j − D j Y c − . . . Y h i , g i + δ j · h AY ℓ . . . Y c + Y c − . . . Y h i , g i ± γ k h i k k g k (Eq. (2)) = ( − δ j ) · h AY . . . Y c − U j − D j Y c + . . . Y ℓ h i , g i + δ j · h AU ℓ − h i , g i ± γ k h i k k g k = ( − δ j ) · Q c − i ( ~ δ ) · h AU ℓ − h i , g i ± ( − δ j ) · ( c − ) γ k h i k k g k + δ j · h AU ℓ − h i , g i ± γ k h i k k g k (I.H.) = Q c , i ( ~ δ ) · h AU ℓ − h i , g i ± ( c − ) · γ k h i k k g k . With Lemma 5.21 we are close to recover the approximate spectrum of D k + U k from [DDFH18].However, in our application we will need to analyze more general operators, namely, purebalanced and balanced operators. Lemma 5.22 (Refinement of [DDFH18]) . If k A k op ≤ , then h AD k + U k f i , g i = λ i · h A f i , g i ± ( k − i + ) · γ k h i k k g k , where λ i = Q k − i + i ( ~ δ ) .Proof. Recall that f i = U k − i h i where h i ∈ ker ( D i ) . Set Y = D k + U k U k − i . Lemma 5.21 yields h AD k + U k f i , g i = λ i · h A f i , g i ± ( k − i + ) · γ k h i k k g k ,where λ i = Q k − i + i ( ~ δ ) .Then powers of the operator D k + U k behave as expected. Lemma 5.23 (Exponentiation Lemma) . h ( D k + U k ) s f i , f i i = λ si · k f i k ± s · ( k − i + ) · γ k h i k k f i k , where λ i is given in Lemma 5.22. roof. Follows immediately from the foregoing and the fact that k D k + U k k op = |D| > |U | , Y : C i → C j is an operator whose kernel approximately containsker ( D i ) as the following lemma makes precise. Lemma 5.24 (Refinement of [DDFH18]) . If |D| > |U | and h i ∈ ker ( D i ) , then h AY ℓ . . . Y h i , g i = ± ℓ · γ k h i k k g k , provided k A k op ≤ .Proof. Let c ∈ [ ℓ ] be the smallest index for which Y c is a down operator. Observe that c < ℓ /2 since |D| > |U | . We induct on m = |D| . If c =
1, then h AD i h i , g i =
0. Henceassume c , m > Y c Y c − = D i + c U i + c − . Applying Lemma 5.21 we obtain h AY ℓ . . . Y h i , g i = h ( AY ℓ . . . Y c + ) DUU c − h i , g i = Q c , i ( ~ δ ) · h ( AY ℓ . . . Y c + ) U c − h i , g i ± ℓ · γ k h i k k g k = ± Q c , i ( ~ δ ) · ( ℓ − ) · γ k h i k k g k ± ℓ · γ k h i k k g k (Induction) = ± ℓ · γ k h i k k g k ,where in the last derivation we used Q c , i ( ~ δ ) ≤ |D| = |U | , namely, the canonical walks.We show that N ( u ) k , k is approximately a polynomial in the operator D k + U k . As a warm upconsider the case N ( ) k , k = D k + D k + U k + U k . Using the Eq. (3), we get N ( ) k , k ≈ ( − δ k + ) · D k + U k D k + U k + δ k + · D k + U k = ( − δ k + ) · ( D k + U k ) + δ k + · D k + U k .Inspecting this polynomial more carefully we see that that its coefficients form a probabil-ity distribution. This property holds in general as the following Lemma 5.25 shows. Thisgives an alternative (approximate) random walk interpretation of N ( u ) k , k as the walk thatfirst selects the power s according to the distribution encoded in the polynomial and thenmoves according to ( D k + U k ) s . Lemma 5.25 (Canonical Polynomials) . For k , u ≥ there exists a degree u univariate polyno-mial F Nu , k , ~ δ depending only on u , k , ~ δ such that (cid:13)(cid:13)(cid:13) N ( u ) k , k − F Nu , k , ~ δ ( D k + U k ) (cid:13)(cid:13)(cid:13) op ≤ ( u − ) · γ . Moreover, the coefficients of this polynomial form a probability distribution, i.e., F Nu , k , ~ δ ( x ) = ∑ ui = c i x i where ∑ ui = c i = and c i ≥ for i =
0, . . . , u. roof. For u = N ( ) k , k = I and the lemma trivially follows. Similarly, if u = N ( ) k , k = D k + U k . Now suppose u ≥
2. Set Y = N ( u ) k , k , i.e., Y = D k + . . . ( D k + u U k + u − ) . . . U k .For convenience let j = k + u −
1. Using the Eq. (3) we can replace D j + U j in Y by ( − δ j ) U j − D j + δ j I incurring an error of γ (in spectral norm) and yielding Y ≈ ( − δ j ) · Y ′ + δ j · N ( u − ) k , k ,where Y ′ was obtained from Y by moving the rightmost occurence of a down operator (inthis case D j + ) one position to right. We continue this process of moving the rightmostoccurrence of a down operator until the resulting operator is up to ( u − ) · γ error α · N ( u − ) k , k ( D k + U k ) + β · N ( u − ) k , k ,where α = ∏ ji = k + ( − δ i ) and β = ∑ ji = k + δ i ∏ ji = k + ( − δ i ) . Since δ i = δ i > α , β are nonnegative and form a probability distribution. Now the result follows from the inductionhypothesis applied to N ( u − ) k , k . Remark 5.26.
Having a polynomial expression F Nu , k , ~ δ ( D k + U k ) ≈ N ( u ) k , k and knowing that S k , k canbe written as linear combination of canonical walks, we could deduce that S k , k is also approximatelya polynomial in D k + U k . Using an error refined version of the Lemma 5.23 (showing that expo-nentiation of D k + U k behaves naturally), we could deduce the approximate spectrum of S k , k . Weavoid this approach since it analysis introduces unnecessary error terms and we can understandquadratic forms of pure balanced operators directly. Remark 5.27.
The canonical polynomial F Nu , k , ~ δ ( D k + U k ) is used later in the error analysis thatrelates the norms k h i k and k f i k (Lemma 5.30). Now we consider Y where |D| = |U | in full generality. We show how the quadraticform of Y behaves in terms of the approximate eigenspace decomposition C k = ∑ ki = C ki . Lemma 5.28 (Pure Balanced Walks) . Suppose Y = Y ℓ . . . Y is a product of an equal number ofup and down operators, i.e., |D| = |U | . Then for f i ∈ C ki h Y f i , f i i = λ Y k , i · h f i , f i i ± γ · ( ℓ + ℓ ( k − i − )) k h i k k f i k , where λ Y k , i is an approximate eigenvalue depending only on Y , k and i.Proof. We induct on even ℓ . For ℓ =
0, the result trivially follows so assume ℓ ≥
2. Let c ∈ [ ℓ ] be the smallest index of a down operator. Set A = Y ℓ . . . Y c + and let Y ′ = Y c . . . Y = DU . . . U . Observe that h AY ′ f i , f i i = h ADU c − + k − i h i , f i i .Applying Lemma 5.21 to the RHS above gives h ADU c − + k − i h i , f i i = Q c − + k − i , i ( ~ δ ) · h AU c − f i , f i i ± ( c + k − i − ) · γ k h i k k f i k .26pplying the induction hypothesis to Y ′′ = A U c − in the above RHS yields Q c − + k − i , i ( ~ δ ) · λ Y ′′ k , i h f i , f i i± Q c − + k − i , i ( ~ δ ) · γ · (( ℓ − ) + ( ℓ − )( k − i − )) k h i k k f i k± ( c + k − i − ) · γ k h i k k f i k = λ Y k , i · h f i , f i i ± γ · ( ℓ + ℓ ( k − i − )) k h i k k f i k ,where λ Y k , i = Q c − + k − i , i ( ~ δ ) · λ Y ′′ k , i and the last equality follows from Q c − + k − i , i ( ~ δ ) ≤ c ≤ ℓ .To understand all errors in the analysis in Lemma 5.28 we need to derive the approxi-mate orthogonality of f i and f j for i = j from [DDFH18] in more detail. We start with thefollowing bound in terms of h i , h j . Lemma 5.29 (Refinement of [DDFH18]) . For i = j, h f i , f j i = ± ( k − i − j ) · γ k h i k (cid:13)(cid:13) h j (cid:13)(cid:13) . Proof.
Recall that f i = U k − i h i , f j = U k − j h j where h i ∈ ker ( D i ) , h j ∈ ker (cid:0) D j (cid:1) . Without lossof generality suppose i > j . We have h U k − i h i , U k − j h j i = h D k − j U k − i h i , h j i .Since k − j > k − i , the result follows from Lemma 5.24.To give a bound for Lemma 5.29 only in terms of the eigenfunction norms k f i k and notin terms of k h i k , we need to understand how the norms of h i and f i are related. Lemma 5.30 (Refinement of [DDFH18]) . Let η k , i = ( k − i ) + and let β i = r(cid:12)(cid:12)(cid:12) F Nk − i , i , ~ δ ( δ i ) ± γ · η k , i (cid:12)(cid:12)(cid:12) where F Nk − i , k , ~ δ is a canonical polynomial of degree k − i from Lemma 5.25. Then h f i , f i i = β i · h h i , h i i . Let θ k , i = ( i + ) k − i . Furthermore, if γ ≤ ( · η k , i · θ k , i ) , then β i ≥ θ k , i .Proof. Recall that f i = U k − i h i where h i ∈ ker ( D i ) . For i = k the result trivially follows soassume k > i . First consider the case k = i +
1. We have h U i h i , U i h i i = h D i + U i h i , h i i = δ i · h h i , h i i ± γ · h h i , h i i . (4)For general k > i we have h U k − i h i , U k − i h i i = h D k − i U k − i h i , h i i .Applying Lemma 5.25 to D k − i U k − i yields h D k − i U k − i h i , h i i = h F Nk − i , i , ~ δ ( D i + U i ) h i , h i i ± γ · ( k − i − ) .27ombining Eq. (4) and Lemma 5.23 gives h F Nk − i , i , ~ δ ( D i + U i ) h i , h i i ± γ · ( k − i − ) = h F Nk − i , i , ~ δ ( δ i ) h i , h i i ± γ · (( k − i ) + ) .Since F Nk − i , i , ~ δ ( x ) = ∑ k − ii = c i x i where the coefficients c i form a probability distribution, we get F Nk − i , i , ~ δ ( δ i ) ≥ δ k − ii = (cid:18) i + (cid:19) k − i .Now, we can state the approximate orthogonality Lemma 5.31 in terms of the eigen-function norms. Lemma 5.31 (Approximate Orthogonality (refinement of [DDFH18])) . Let η k , s , θ k , s , β s fors ∈ { i , j } be given as in Lemma 5.30. If i = j and β i , β j > , then h f i , f j i = ± γ · ( k − i − j ) β i β j k f i k (cid:13)(cid:13) f j (cid:13)(cid:13) . Furthermore, if γ ≤ min (cid:0) ( · η k , i · θ k , i ) , 1/ ( · η k , j · θ k , j ) (cid:1) , then β i , β j > and h f i , f j i = ± γ · θ k , i · θ k , j · ( k − i − j ) k f i k (cid:13)(cid:13) f j (cid:13)(cid:13) . Proof.
Follows directly from Lemma 5.30.We generalize the quadratic form of Lemma 5.28 to linear combinations of general purebalanced operators Y , namely, to balanced operators. Lemma 5.32 (General Quadratic Form (restatement of Lemma 5.8)) . Let ε ∈ (
0, 1 ) and let Y ⊆ { Y | Y : C k → C k } be a collection of formal operators that are product of an equal numberof up and down walks (i.e., pure balanced operators) not exceeding ℓ walks. Let B = ∑ Y ∈Y α Y Y where α Y ∈ R and let f = ∑ ki = f i with f i ∈ C ki . If γ ≤ ε (cid:0) k k + ℓ ∑ Y ∈Y | α Y | (cid:1) − , then h B f , f i = k ∑ i = ∑ Y ∈Y α Y λ Y k ( i ) ! · h f i , f i i ± ε , where λ Y k ( i ) depends only on the operators appearing in the formal expression of Y , i and k, i.e., λ Y k ( i ) is the approximate eigenvalue of Y associated to C ki .Proof. Using Lemma 5.28 and the assumption on γ gives h B f , f i = k ∑ i = ∑ Y ∈Y α Y λ Y k ( i ) · h f i , f i i + ∑ i = j ∑ Y ∈Y (cid:16) α Y λ Y k ( i ) · h f i , f j i ± γ · α Y ( ℓ + ℓ ( k − i − )) h h i , f j i (cid:17) = k ∑ i = ∑ Y ∈Y α Y λ Y k ( i ) · h f i , f i i + ∑ i = j ∑ Y ∈Y α Y λ Y k ( i ) · h f i , f j i ± ε S k , k = k ∑ j = ( − ) k − j · (cid:18) k + jk (cid:19) · (cid:18) kj (cid:19) · N ( j ) k , k = k ∑ j = α j · N ( j ) k , k ,where α j = ( − ) k − j · ( k + jk ) · ( kj ) .Finally, we have all the pieces to prove Lemma 5.6 restated below. Lemma 5.33 (Swap Quadratic Form (restatement of Lemma 5.6)) . Let f = ∑ ki = f i withf i ∈ C ki . Suppose X ( ≤ d ) is a γ -HDX with d ≥ k. If γ ≤ ε (cid:0) k k + k + (cid:1) − , then h S k , k f , f i = k ∑ i = λ k ( i ) · h f i , f i i ± ε , where λ k ( i ) depends on only on k an i, i.e., λ k ( i ) is an approximate eigenvalue of S k , k associated tospace C ki .Proof. First note that Lemma 5.28 establishes the existence of approximate eigenvalues λ k , j ( i ) of N ( j ) k , k corresponding to space C ki for i =
0, . . . , k such that λ k , j ( i ) depends onlyon k , i and j . To apply Lemma 5.8 we need to bound ∑ kj = | α j | . Since k ∑ j = | α j | = k ∑ j = (cid:18) k + jk (cid:19) · (cid:18) kj (cid:19) ≤ k · k ∑ j = (cid:18) k + jk (cid:19) ≤ k + ,we are done. S k , l We turn to the spectral analysis of rectangular swap walks, i.e., S k , l where k = l . Recallthat to bound σ ( S k , k ) in Section 5.1 we proved that the spectrum of S k , k for a γ -HDX isclose to the spectrum of S ∆ k , k using the analysis of quadratic forms over balanced operatorsfrom Section 5.3. Then we appealed to the fact that S ∆ k , k is expanding since it is the walkoperator of the well known Kneser graph. In this rectangular case, we do not have a clas-sical result establishing that S ∆ k , l is expanding, but we were able to establish it Lemma 5.34. Lemma 5.34.
Let d ≥ k + l and ∆ d ( n ) be the complete complex. The second largest singular value σ ( S ∆ k , l ) of the swap operator S ∆ k , l on ∆ d ( n ) is σ ( S ∆ k , l ) ≤ max (cid:18) kn − k , ln − l (cid:19) , provided n ≥ M k , l where M k , l ∈ N only depends on k and l. Towards proving Lemma 5.34 we first introduce a generalization of Kneser graphswhich we denote bipartite Kneser graphs defined as follows.29 efinition 5.35 (General Bipartite Kneser Graph) . Let X ( ≤ d ) where d ≥ k + l. We denoteby K X ( n , k , l ) the bipartite graph on (vertex) partition ( X ( k ) , X ( l )) where s ∈ X ( k ) is adjacent to t ∈ X ( l ) if and only if s ∩ t is empty. We also refer to graphs of the form K X ( n , k , l ) as bipartiteKneser graphs. It will be convenient to distinguish bipartite Kneser graphs coming from general γ -HDXand the complete complex ∆ d ( n ) . Definition 5.36 (Complete Bipartite Kneser Graph) . Let X ( ≤ d ) where d ≥ k + l. If X is thecomplete complex, i.e., X = ∆ d ( n ) , then we denote K X ( n , k , l ) as simply as K ( n , k , l ) and we referto it as complete bipartite Kneser. We obtain the spectra of bipartite Kneser graphs generalizing the classical result of Fact 5.4.More precisely, we prove Lemma 5.37. Lemma 5.37 (Bipartite Kneser Spectrum) . The non-zero eigenvalues of the (normalized) walkoperator of K ( n , k , l ) are ± λ i where λ i = ( n − k − il − i )( n − l − ik − i )( n − kl )( n − lk ) , for i =
0, . . . , min ( k , l ) . Now the proof follows a similar strategy to the S k , k , namely, we analyze quadraticforms over S k , k using the results from Section 5.3Let X ( ≤ d ) where d ≥ k + l . Let A k , l be the (normalized) walk operator of K X ( n , k , l ) ,i.e., A k , l = S ( l ) k , l (cid:16) S ( l ) k , l (cid:17) † .To determine the spectrum of A k , l it is enough to consider the spectrum of B = S ( l ) k , l (cid:16) S ( l ) k , l (cid:17) † .Using Corollary 4.13, we have B = l ∑ j = ( − ) l − j (cid:18) k + jl (cid:19) · (cid:18) lj (cid:19) · N ( j ) k , l ! l ∑ j ′ = ( − ) l − j ′ (cid:18) k + j ′ l (cid:19) · (cid:18) lj ′ (cid:19) · (cid:16) N ( j ′ ) k , l (cid:17) † = l ∑ j , j ′ = α k , l , j , j ′ N ( j ) k , l N ( j ′ + k − l ) l , k , for some coefficients α k , l , j , j ′ depending only on k , l , i , j and j ′ . Since we have not yet usedany specific property of HDXs, these coefficients are the same for the complete complexand general HDXs. Lemma 5.38.
Let X ( ≤ d ) be a γ -HDX with d ≥ k + l. Let f = ∑ ki = f i with f i ∈ C ki . For ε ∈ (
0, 1 ) , if γ ≤ ε (cid:0) k k + ℓ k + l + (cid:1) − , then h B f , f i = k ∑ i = l ∑ j , j ′ = α k , l , j , j ′ λ k , l , j , j ′ ( i ) ! · h f i , f i i + ε , Note that the singular values of K ( n , k ) can be deduced from the bipartite case. here λ k , l , j , j ′ ( i ) is the approximate eigenvalues of N ( j ) k , l N ( j ′ + k − l ) l , k corresponding to space C ki . Fur-thermore, λ k , l , j , j ′ ( i ) depends only on k, l, i, j and j ′ .Proof. First observe that each N ( j ) k , l N ( j ′ + k − l ) l , k maps C k to itself, so it is a product of the samenumber of up and down operators. Now to apply Lemma 5.8 it only remains to bound ∑ lj , j ′ = | α k , l , j , j ′ | . Since l ∑ j , j ′ = | α k , l , j , j ′ | = l ∑ j , j ′ = (cid:18) k + jl (cid:19) · (cid:18) lj (cid:19)(cid:18) k + j ′ l (cid:19) · (cid:18) lj ′ (cid:19) ≤ l l ∑ j = (cid:18) k + jl (cid:19)! · l ∑ j ′ = (cid:18) k + j ′ l (cid:19)! ≤ k + l + ,we are done.Let B and B ∆ stand for the B operator for general γ -HDX and the complete complex,respectively. Lemma 5.39.
Suppose X ( ≤ d ) is a γ -HDX with d ≥ k + l. For ε ∈ (
0, 1 ) , if γ ≤ ε (cid:0) k k + ℓ k + l + (cid:1) − ,then the second largest singular value σ ( B ) of B is σ ( B ) ≤ ε . Furthermore, the second largest non-trivial eigenvalue λ ( A k , l ) of the walk matrix of K ( n , k , l ) is λ ( A k , l ) ≤ ε . Proof.
The proof follows the same strategy of Theorem 5.1, namely, we first consider B ∆ and show that ∑ lj , j ′ = α k , l , j , j ′ λ k , l , j , j ′ ( i ) =
0. Using Lemma 5.34, we deduce that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l ∑ j , j ′ = α k , l , j , j ′ λ k , l , j , j ′ ( i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O k , l (cid:18) n (cid:19) for i ∈ [ k ] where in this range each C ki is not the trivial approximate eigenspace (associatedwith eigenvalue 1). Since α k , l , j , j ′ and λ k , l , j , j ′ ( i ) do not depend on n and n is arbitrary, theLHS above is actually zero. Then our choice of γ Lemma 5.8 givesmax f ∈ C k : f ⊥ k f k = |h B f , f i| ≤ max i ∈ [ k ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l ∑ j , j ′ = α k , l , j , j ′ λ k , l , j , j ′ ( i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ε = ε .Now the proof of Theorem 5.2 follows. For convenience, we restate it. Theorem 5.40 (Rectangular Swap Walk Spectral Bound (restatement of Theorem 5.2)) . Sup-pose X ( ≤ d ) is a γ -HDX with d ≥ k + l and k ≤ l. For σ ∈ (
0, 1 ) , if γ ≤ σ · (cid:0) k k + ℓ k + l + (cid:1) − ,then the largest non-trivial singular value σ ( S k , l ) of the swap operator S k , l is σ ( S k , l ) ≤ σ . Proof.
Follows directly from Lemma 5.39. 31 .5 Bipartite Kneser Graphs - Complete Complex
Now we determine the spectrum of the complete bipartite Kneser graph K ( n , k , l ) . Moreprecisely, we prove the following. Lemma 5.41 (Bipartite Kneser Spectrum (restatement of Lemma 5.37)) . The non-zero eigen-values of the normalized walk operator of K ( n , k , l ) are ± λ i where λ i = ( n − k − il − i )( n − l − ik − i )( n − kl )( n − lk ) , for i =
0, . . . , min ( k , l ) . Henceforth, set X = ∆ d ( n ) . To prove Lemma 5.37 we work with the natural rectangu-lar matrix associated with K ( n , k , l ) , namely, the matrix W ∈ R X ( k ) × X ( l ) such that W ( s , t ) = [ s ∩ t = ∅ ] for every s ∈ X ( k ) and t ∈ X ( l ) .Observe that the entries of WW ⊤ and W ⊤ W only depend on the size of the intersectionof the sets indexing the row and columns. Hence, these matrices belong to the Johnsonscheme [GM15] J ( n , k ) and J ( n , l ) , respectively. Moreover, the left and right singular vec-tors of W are eigenvectors of these schemes.We adopt the eigenvectors used in Filmus’ work [Fil16], i.e., natural basis vectors com-ing from some irreducible representation of S n (see [Sag13]). First we introduce some no-tation. Let µ = ( n − i , i ) be a partition of n and let τ µ be a standard tableau of shape µ . Sup-pose the first row τ µ contains a < · · · < a n − i whereas the second contains b < · · · < b i .To τ µ we associate the function ϕ τ µ ∈ R ( [ n ] k ) as follows ϕ τ µ = ( a − b ) . . . ( a i − b i ) ,where a ∈ R ( nk ) is the containment indicator of element a , i.e., a ( s ) = a ∈ s . Filmus proved that (cid:8) ϕ τ µ | ≤ i ≤ k , µ ⊢ ( n − i , i ) , τ µ standard (cid:9) is an eigenbasis of J ( n , k ) . We abuse the notation by considering ϕ τ µ as both a function in R ( nk ) and R ( nl ) as long as these functions are well defined. Claim 5.42. If µ = ( n − i , i ) and k , l ≥ i, then W ϕ τ µ = ( − ) i · (cid:18) n − k − il − i (cid:19) · ϕ τ µ . Proof.
We follow a similar strategy of Filmus. For convenience suppose ϕ τ µ = ( − ) . . . ( i − − i ) . For i = i ≥
1. Consider (cid:0) W ϕ τ µ (cid:1) ( s ) where s ∈ ( [ n ] k ) . Note that (cid:0) W ϕ τ µ (cid:1) ( s ) = ∑ t ∈ Y : s ∩ t = ∅ ϕ τ µ ( t ) .32f 2 j −
1, 2 j ∈ s for some j ∈ [ i ] , then 2 j −
1, 2 j t so ϕ τ µ ( s ) = = (cid:0) W ϕ τ µ (cid:1) ( s ) . If2 j −
1, 2 j s for some j ∈ [ i ] , for each t adjacent to s there four cases: 2 j −
1, 2 j ∈ t ,2 j −
1, 2 j t , 2 j − ∈ t and 2 j t or vice-versa. The first two cases yield ϕ τ µ ( t ) = ϕ τ µ ( s ) = = (cid:0) W ϕ τ µ (cid:1) ( s ) .Now suppose that s contains exactly one element of each pair 2 j −
1, 2 j . For any adjacent t to yield ϕ τ µ ( t ) = t must contain [ i ] \ s . Since there are ( n − k − il − i ) such possibilities for t weobtain W ϕ τ µ = ( − ) i · (cid:18) n − k − il − i (cid:19) · ϕ τ µ ,where the sign ( − ) i follows from the product of the signs of each the i pairs and the factthat s and t partition the elements in each pair.Since we are working with singular vectors, we need to be careful with their normal-ization when deriving the singular values. We stress that the norm of ϕ τ µ depends on thespace where ϕ τ µ lies. Claim 5.43. If µ = ( n − i , i ) and ϕ τ µ ∈ R ( nk ) , then (cid:13)(cid:13) ϕ τ µ (cid:13)(cid:13) = s i (cid:18) n − ik − i (cid:19) . Proof.
Since ϕ τ µ assumes values in {−
1, 0, 1 } so its enough to count the number of sets s ∈ ( [ n ] k ) such that ϕ τ µ ( s ) =
0. To have ϕ τ µ ( s ) = s must contain exactly one elementin each pair and the remaining k − i elements of s can be chosen arbitrarily among theelements avoiding the 2 i elements appearing in the indicators defining ϕ τ µ .Now the singular values of W follow. Corollary 5.44 (Singular Values) . The singular values of W are σ i = (cid:18) n − k − il − i (cid:19) · (cid:13)(cid:13)(cid:13) ϕ k τ µ (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ϕ l τ µ (cid:13)(cid:13)(cid:13) , for i =
0, . . . , min ( k , l ) . Note that for k = l we recover the well know result of Fact 5.4.Finally we compute the eigenvalues of the bipartite graph K ( n , k , l ) . Let A n , k , l be itsnormalized adjacency matrix, i.e., A n , k , l = ( n − kl ) W ( n − lk ) W ⊤ . Lemma 5.45 (Bipartite Kneser Spectrum (restatement of Lemma 5.37)) . The non-zero eigen-values of the normalized walk operator of K ( n , k , l ) are ± λ i where λ i = ( n − k − il − i )( n − l − ik − i )( n − kl )( n − lk ) , for i =
0, . . . , min ( k , l ) . roof. Since the spectrum of a bipartite graph is symmetric around zero, it is enough tocompute the eigenvalues of A n , k , l . Set α = ( n − kl )( n − lk ) . Moreover, we consider α · WW ⊤ since α · W ⊤ W has the same non-zero eigenvalues. The non-zero eigenvalues of α · WW ⊤ are λ i = ( n − k − il − i )( n − l − ik − i )( n − kl )( n − lk ) ,for i =
0, . . . , min ( k , l ) . k -CSP In the following, we will show that k -CSP instances I whose constraint complex X I ( ≤ k ) is a suitable expander admit an efficient approximation algorithm. We will assumethroughout that X I ( ) = [ n ] , and drop the subscript I .This was shown for 2-CSPs in [BRS11]. In extending this result to k -CSPs we will relyon a central Lemma of their paper. Before, we explain our algorithm we give a basic outlineof our idea:We will work with the SDP relaxation for the k -CSP problem given by L -levels of SoShierarchy, as defined in Section 2.4 (for L to be specified later). This will give us an L -local PSD ensemble { Y , . . . , Y n } , which attains some value SDP ( I ) ≥ OPT ( I ) . Since { Y , . . . , Y n } , is a local PSD ensemble, and not necessarily a probability distribution, wecannot sample from it directly. Nevertheless, since (cid:8) Y j (cid:9) will be actual probability distri-butions for all j ∈ [ n ] , one can independently sample η j ∼ (cid:8) Y j (cid:9) and use η = ( η , . . . , η n ) as the assignment for the k -CSP instance I .Unfortunately, while we know that the local distributions { Y a } a ∈ X ( k ) induced by { Y , . . . , Y n } will satisfy the constraints of I with good probability, i.e., E a ∼ Π k E { Y a } [ Y a satisfies the constraint on a | {z } ⇐⇒ Y a ∈ C a ] = SDP ( I ) ≥ OPT ( I ) ,this might not be the case for the assignment η sampled as before. It might be that therandom variables Y a , . . . , Y a k are highly correlated for a ∈ X ( k ) , i.e., E a ∼ Π k k{ Y a } − { Y a } · · · { Y a k }k is large. One strategy employed by [BRS11] to ensure that the quantity above is small,is making the local PSD ensemble { Y , . . . , Y n } be consistent with a randomly sampledpartial assignment for a small subset of variables (q.v. Section 2.4). We will show that thisstrategy is succesful if X ( ≤ k ) is a γ -HDX (for γ sufficiently small). Our final algorithmwill be the following, 34 lgorithm 6.1 (Propagation Rounding Algorithm) . Intput
An L-local PSD ensemble { Y , . . . , Y n } and some distribution Π on X ( ≤ ℓ ) . Output
A random assignment η : [ n ] → [ q ] .1. Choose m ∈ {
1, . . . , L / ℓ } uniformly at random.2. Independently sample m ℓ -faces, s j ∼ Π for j =
1, . . . , m.3. Write S = S mj = s j , for the set of the seed vertices.4. Sample assignment η S : S → [ q ] according to the local distribution, { Y S } .5. Set Y ′ = { Y , . . . Y n | Y S = η S } , i.e. the local ensemble Y conditioned on agreeing with η S .6. For all j ∈ [ n ] , sample independently η j ∼ { Y ′ j } .7. Output η = ( η , . . . , η n ) . In our setting, we will apply Algorithm 6.1 with the distribution Π k and the L -local PSDensemble { Y , . . . , Y n } . Notice that in expectation, the marginals of Y ′ on faces a ∈ X ( k ) – which are actual distributions – will agree with the marginals of Y , i.e. E S , η S E Y ′ a = E Y a . In particular, the approximation quality of Algorithm 6.1 will depend on the averagecorrelation of Y ′ a , . . . , Y ′ a k on the constraints a ∈ X ( k ) , where Y ′ is the local PSD ensembleobtained at the end of the first phase of Algorithm 6.1.In the case where k =
2, the following is known
Theorem 6.2 (Theorem 5.6 from [BRS11]) . Suppose a weighted undirected graph G = ([ n ] , E , Π ) and an L-local PSD ensemble Y = { Y , . . . , Y n } are given. There exists absolute constantsc ≥ and C ≥ satisfying the following: If L ≥ c · q ε , Supp ( Y i ) ≤ q for all i ∈ V, and λ ( G ) ≤ C · ε / q then we have E { i , j }∼ Π (cid:13)(cid:13)(cid:13) { Y ′ i , Y ′ j } − { Y ′ i }{ Y ′ j } (cid:13)(cid:13)(cid:13) ≤ ε , where Y ′ is as defined in Algorithm 6.1 on the input of { Y , . . . , Y n } and Π . To approximate k -CSPs well, we will show the following generalization of Theorem 6.2for k -CSP instances I , whose constraint complex X ( ≤ k ) is γ -HDX, for γ sufficiently small. Theorem 6.3.
Suppose a simplicial complex X ( ≤ k ) with X ( ) = [ n ] and an L-local PSD ensem-ble Y = { Y , . . . , Y n } are given.There exists some universal constants c ′ ≥ and C ′ ≥ satisfying the following: If L ≥ c ′ · ( q k · k / ε ) , Supp ( Y j ) ≤ q for all j ∈ [ n ] , and X is a γ -HDX for γ ≤ C ′ · ε / ( k + k · k · q k ) .Then, we have E a ∼ Π k (cid:13)(cid:13) { Y ′ a } − (cid:8) Y ′ a (cid:9) · · · (cid:8) Y ′ a k (cid:9)(cid:13)(cid:13) ≤ ε , (5) where Y ′ is as defined in Algorithm 6.1 on the input of { Y , . . . , Y n } and Π k . Indeed, using Theorem 6.3, it will be straightforward to prove the following,
Corollary 6.4.
Suppose I is a q-ary k-CSP instance whose constraint complex X ( ≤ k ) is a γ -HDX. here exists absolute constants C ′ ≥ and c ′ ≥ , satisfying the following: If γ ≤ C ′ · ε / ( k + k · k · q k ) , there is an algorithm that runs in time n O ( k · q k · ε − ) based on ( c ′ · k · q k ε ) -levelsof SoS-hierarchy and Algorithm 6.1 that outputs a random assignment η : [ n ] → [ q ] that inexpectation ensures SAT I ( η ) = OPT ( I ) − ε .Proof of Corollary 6.4. The algorithm will just run Algorithm 6.1 on the local PSD-ensemble { Y , . . . , Y n } given by the SDP relaxation of I strengthened by L = c ′ · k · q k ε -levels of SoS-hierarchy and Π k – where c ′ ≥ Y satisfies, SDP ( I ) = E a ∼ Π k (cid:20) E { Y a } [ [ Y a ∈ C a ]] (cid:21) ≥ OPT ( I ) . (6)Let S , η S , and Y ′ be defined as in Algorithm 6.1 on the input of Y and Π k . Since the con-ditioning done on { Y ′ } is consistent with the local distribution, by law of total expectationand Eq. (6) one has E S E η S ∼{ Y S } E a ∼ Π k E { Y ′ a } (cid:2) [ Y ′ a ∈ C a ] (cid:3) = SDP ( I ) ≥ OPT ( I ) . (7)By Theorem 6.3 we know that E S E η S ∼{ Y S } E a ∼ Π k (cid:13)(cid:13) { Y ′ a } − { Y ′ a } · · · { Y ′ a k } (cid:13)(cid:13) ≤ ε (8)Now, the fraction of constraints satisfied by the algorithm in expectation is E η [ SAT I ( η )] = E S E η S ∼{ Y S } E a ∼ Π k E ( η ,..., η n ) ∼{ Y ′ }···{ Y ′ n } [ [ η | a ∈ C a ]] .By using Eq. (8), we can obtain E η [ SAT I ( η )] ≥ E S (cid:20) E η S ∼{ Y S } E { Y a } [ Y ′ a satisfies the constraint on a ] (cid:21) − ε .Using Eq. (7), we can conclude E η [ SAT I ( η )] ≥ SDP ( I ) − ε = OPT ( I ) − ε .Our proof of Theorem 6.3 will hinge on the fact that we can upper-bound the expectedcorrelation of a face of large cardinality ℓ , in terms of expected correlation over faces ofsmaller cardinality and expected correlations along the edges of a swap graph. The swapgraph here is defined as a weighted graph G ℓ , ℓ = ( X ( ℓ ) ⊔ X ( ℓ ) , E ( ℓ , ℓ ) , w ℓ , ℓ ) , where E ( ℓ , ℓ ) = {{ a , b } : a ∈ X ( ℓ ) , b ∈ X ( ℓ ) , and a ⊔ b ∈ X ( ℓ + ℓ ) } .We will assume ℓ ≥ ℓ , and if ℓ = ℓ we are going to identify the two copies of everyvertex. We will endow E ( ℓ , ℓ ) with the weight function, w ℓ , ℓ ( a , b ) = Π ℓ + ℓ ( a ⊔ b )( ℓ + ℓ ℓ ) ,36hich can easily be verified to be a probability distribution on E ( ℓ , ℓ ) Notice that in thecase where ℓ = ℓ the random walk matrix of G ℓ , ℓ is given by A ℓ , ℓ = (cid:18) S ℓ , ℓ S † ℓ , ℓ (cid:19) ,and if ℓ = ℓ we have A ℓ , ℓ = S ℓ , ℓ . The stationary distribution of A ℓ , ℓ is Π ℓ , ℓ definedby, Π ℓ , ℓ ( b ) = [ b ∈ X ( ℓ )] · · Π ℓ ( b ) + [ b ∈ X ( ℓ )] · · Π ℓ ( b ) . (9)When we write an expectation of f ( • , • ) over the edges in E ( ℓ , ℓ ) with respect to w ℓ , ℓ ,it is important to note, E { s , t }∼ w ℓ ℓ [ f ( s , t )] = ∑ { s , t }∈ E ( ℓ , ℓ ) ( ℓ + ℓ ℓ ) · f ( s , t ) · Π ℓ + ℓ ( s ⊔ t ) = ( ℓℓ ) E a ∼ Π k " ∑ { s , t }∼ a f ( s , t ) ,(10)where sum within the expectation in the RHS runs over the ( ℓ + ℓ ℓ ) possible ways of split-ting a into s ⊔ t such that s ∈ X ( ℓ ) and t ∈ X ( ℓ ) . When we are speaking about thespectral expansion of G ℓ , ℓ , we will be speaking with regards to λ ( G ℓ , ℓ ) and not withregards to σ ( G ℓ , ℓ ) . Remark 6.5.
By simple linear algebra, we have λ ( G ℓ , ℓ ) : = λ ( A ℓ , ℓ ) ≤ σ ( S ℓ , ℓ ) , where we employ the notation λ ( M ) to denote the second largest eigenvalue (signed) of the matrix M . With this, we will show
Lemma 6.6 (Glorified Triangle Inequality) . For a simplicial complex X ( ≤ k ) , ℓ ≥ ℓ ≥ , ℓ = ℓ + ℓ , ℓ ≤ k, and an ℓ -local ensemble { Y , . . . , Y n } , one has E a ∈ Π ℓ "(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) { Y a } − ℓ ∏ i = { Y a i } (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ E { s , t }∼ w ℓ ℓ [ k{ Y s , Y t } − { Y s }{ Y t }k ]+ E s ∼ Π ℓ "(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) { Y s } − ℓ ∏ i = { Y s i } (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + E t ∼ Π ℓ "(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) { Y t } − ℓ ∏ i = { Y t i } (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (11)One useful observation, is that by using Lemma 6.6 repeatedly, we can reduce the prob-lem of bounding E a ∈ Π ℓ (cid:13)(cid:13)(cid:13) { Y a } − ∏ ℓ i = { Y a i } (cid:13)(cid:13)(cid:13) to a problem of bounding E { s , t }∼ w ℓ ℓ k{ Y s , Y t } − { Y s }{ Y t }k ,for ℓ + ℓ ≤ k . Though it is not a direct implication, it is heavily suggested by Fact 2.7 andTheorem 6.2, that if G ℓ , ℓ is a good spectral expander, after an application of Algorithm 6.1with our chosen parameters, we should be able to bound these expressions. Using a keylemma used from [BRS11], we will prove that this is indeed the case. The only thing weneed to make sure after this point, is that the second eigenvalue λ ( G ℓ , ℓ ) of the swapgraphs G ℓ , ℓ we will be using are small enough for our purposes. Indeed, our choice of γ in Theorem 6.3 and Corollary 6.4 is to make sure that the bound we get on λ ( G ℓ , ℓ ) fromTheorem 5.2 (together with Remark 6.5) is good enough for our purposes.37 .1 Breaking Correlations for Expanding CSPs: Proof of Theorem 6.3 Throughout this section, we will use the somewhat non-standard definition of varianceintroduced in [BRS11], Var [ Y a ] = ∑ η ∈ [ q ] a Var [ [ Y a = η ]] .We will use the following central lemma from [BRS11] in our proof of Theorem 6.3: Lemma 6.7 (Lemma 5.4 from [BRS11]) . Let G = ( V , E , Π ) be a weighted graph, { Y , . . . , Y n } a local PSD ensemble, where we have Supp ( Y i ) ≤ q for every i ∈ V, and q ≥ . Suppose ε ≥ is a lower bound on the expected statistical difference between independent and correlated samplingalong the edges,i.e., ε ≤ E { i , j }∼ Π h(cid:13)(cid:13) { Y ij } − { Y i }{ Y j } (cid:13)(cid:13) i . There exists absolute constants c ≥ and c ≥ that satisfy the following: If λ ( G ) ≤ c · ε q .Then, conditioning on a random vertex decreases the variances, E i ∼ Π E j ∼ Π E { Y j } (cid:2) Var (cid:2) Y i | Y j (cid:3)(cid:3) ≤ E i ∼ Π [ Var [ Y i ]] − c · ε q .For our applications, we will be instantiating Lemma 6.7 with G ℓ , ℓ as G ; and withthe local PSD ensemble { Y a } a ∈ X that is obtained from { Y , . . . , Y n } (q.v. Fact 2.7). Forconvenience, we will write the concrete instance of the Lemma that we will use, Corollary 6.8.
Let ℓ ≥ ℓ ≥ satisfying ℓ + ℓ ≤ k be given parameters, and let G ℓ , ℓ bethe swap graph defined for a γ -HDX X ( ≤ k ) . Let { Y a } a ∈ X be a local PSD ensemble; satisfying Supp ( Y a ) ≤ q k for every a ∈ X ( ℓ ) ∪ X ( ℓ ) for some q ≥ . Suppose ε ≥ satisfies, ε k ≤ E { s , t }∈ w ℓ ℓ [ k{ Y s ⊔ t } − { Y s }{ Y t }k ] . There exists absolute constants c ≥ and c ≥ that satisfy the following: If λ ( G ) ≤ c · ( ε / ( k · q k )) . Then, conditioning on a random face a ∼ Π ℓ , ℓ decreases the variances, i.e. · E a , b ∼ Π ℓ ℓ (cid:20) E { Y a } [ Var [ Y b | Y a ]] (cid:21) = E a ∈ Π ℓ ℓ " E s ∼ Π ℓ [ Var [ Y s | Y a ]] + E t ∼ Π ℓ [ Var [ Y t | Y a ]] , ≤ E s ∼ Π ℓ [ Var [ Y s ]] + E t ∼ Π ℓ [ Var [ Y t ]] − c · ε · k · q k .Here, it can be verified that the expansion criterion presupposed by Lemma 6.7 is sat-isfied by Corollary 6.8 by Theorem 5.2. The constant c satisfies c = · c . Proof of Theorem 6.3.
We will follow the same proof strategy in [BRS11], and extend theirarguments for k -CSPs.Write Π mk for the distribution of the random set that is obtained in steps (2)-(3) ofAlgorithm 6.1 with Π = Π k , i.e. S ∼ Π mk is sampled by1. independently sampling m k -faces s j ∼ Π k for j =
1, . . . , m .38. outputting S = S mj = s j .First, for m ∈ [ L / k ] we will define ε m = E S ∼ Π mk E { Y S } E a ∼ Π k "(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) { Y a | Y S } − k ∏ j = { Y a j | Y S } (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ,which will measure the average correlation along X ( k ) after conditioning on m k -faces.Notice that our goal is ensuring, E m ∼ [ L / k ] ε m ≤ ε where m is sampled uniformly at random.To help us with this goal, we will define a potential function Φ m = E i ∼ [ k ] E S ∼ Π mk E { Y S } E a ∼ Π i Var [ Y a | Y S ] . (12)where i is sampled uniformly at random. Observe that Φ m always satisfies 0 ≤ Φ m ≤ m ∈ [ L / k ] such that ε m is large,i.e., say ε m ≥ ε /2. To this end assume ε m ≥ ε /2, i.e. we have E S ∼ Π mk E { Y S } E a ∼ Π k "(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) { Y a | Y S } − k ∏ i = { Y a | Y S } (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≥ ε T be any binary tree with k leaves.We will label each of the vertices v ∈ T with the number of leaves of the subtree rooted at v . Notice that this ensures that1. the root vertex of T has the label k ,2. for any vertex v ∈ T with label ℓ , the label ℓ of the left child of v and the label ℓ ofthe right child of v add up to k , i.e. ℓ + ℓ = k ,3. every vertex v ∈ T with the label 1 is a leaf.We write J ( T ) for the set of labels ℓ of the internal nodes of T , note | J ( T ) | ≤ k . We willuse the notation ℓ (resp. ℓ ) to refer to the label of the left (resp. right) of a vertex v ∈ T with the label ℓ .By applying Lemma 6.6, we obtain that for any local PSD ensemble Z one has E a ∼ Π k "(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) { Z a } − k ∏ i = { Z a i } (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ∑ ℓ ∈ J ( T ) E { t , t }∈ w ℓ ℓ [ k{ Z t ⊔ t } − { Z t }{ Z t }k ] .Now, by plugging this in Eq. (13), with Z a = { Y a | Y S } , we obtain E S ∼ Π mk E { Y S } " ∑ ℓ ∈ J ( T ) E { t , t }∼ w ℓ ℓ k{ Y t ⊔ t | Y S } − { Y t | Y S }{ Y t | Y S }k ≥ ε J ( T ) there should be some large term corresponding to some ℓ ∈ J ( T ) . i.e. we have, E S ∼ Π mk E { Y S } " E { t , t }∈ w ℓ ℓ k{ Y t ⊔ t | Y S } − { Y t | Y S }{ Y t | Y S }k ≥ ε · | J ( T ) | ≥ ε k .Now, we have P S ∼ Π mk { Y S } " E { s , t }∈ w ℓ ℓ k{ Y t ⊔ t | Y S } − { Y t | Y S }{ Y t | Y S }k ≥ ε k ≥ ε k .This together with Corollary 6.8 implies, P S ∼ Π mk { Y S } " E a ∈ Π ℓ ℓ " E t ∼ Π ℓ [ Var [ Y t | Y S ] − Var [ Y t | Y S , Y a ]]+ E t ∈ Π ℓ [ Var [ Y t | | Y S ] − Var [ Y t | Y S , Y a ]] ≥ c · ε · k · q k ≥ ε k , (15)provided that λ ( G ℓ , ℓ ) ≤ c ( ε / ( k · q k )) .Now, observe that a sample a ∼ Π ℓ , ℓ can be obtained from a sample s m + ∼ Π k in thefollowing way,1. with probability each, pick j = j = ℓ j elements from s m + .It is important to note that for the sample a ∼ Π ℓ , ℓ obtained this way, we have s m + ⊇ a . An application of Jensen’s inequality shows that the variance is non-increasing underconditioning, i.e. for random variables Z and W we have, E Z [ Var [ W | Z ]] = E Z (cid:20) E W (cid:2) W | Z (cid:3)(cid:21) − E Z (cid:20)(cid:18) E W [ W | Z ] (cid:19)(cid:21) , ≤ E (cid:2) W (cid:3) − (cid:18) E Z [ E [ W | Z ]] (cid:19) , = Var [ W ] .This means conditioning on s m + , the drop in variance can only be more, i.e., Eq. (15)implies P S ∼ Π mk { Y S } " E s m + ∈ Π k " E t ∼ Π ℓ [ Var [ Y t | Y S ] − Var [ Y t | Y S , Y s m + ]]+ E t ∈ Π ℓ [ Var [ Y t | | Y S ] − Var [ Y t | Y S , Y s m + ]] ≥ c · ε · k · q k ≥ ε k .By relabeling ℓ as ℓ if needed, we can obtain the following inequality from the above P S ∼ Π mk { Y S } " E s m + ∈ Π k " E t ∼ Π ℓ [ Var [ Y t | Y S ] − Var [ Y t | Y S , Y s m + ]] ≥ c · ε · k · q k ≥ ε k . (16)40his implies Φ m − Φ m + ≥ k · ε k · (cid:18) c · ε · k · q k (cid:19) = c · ε · k · q k ,where the k term in the RHS corresponds to ℓ ∈ [ k ] being chosen in Eq. (12), the ε k termin the RHS corresponds to the probability of the variances in X ( ℓ ) drop by (cid:16) c · ε · k · q k (cid:17) .Since, the variance is non-increasing under conditioning1 ≥ Φ ≥ · · · ≥ Φ m ≥ k · q k / ( c · ε ) indices m ∈ [ L / k ] such that ε m ≥ ε /2.In particular, since the total number of indices is ( L / k ) we have, E m ∼ [ L / k ] ε m ≤ ε + kL · · k · q k c · ε .This means that there exists an absolute constant c ′ ≥ L ≥ c ′ · k · q k ε ensures E m ∈ [ L / k ] [ ε m ] ≤ ε .To finish our proof, we note that to justify our applications of Corollary 6.8 it suffices toensure λ ( G ℓ , ℓ ) ≤ c · (cid:18) ε k · q k (cid:19) = c · ε · k · q k for all ℓ , ℓ occurring in T as a label. It can be verified that our choice of γ together withTheorem 5.2 (and Remark 6.5) satisfies this, where the constant C ′ ≥ c , c ′ , and the constants hidden within the O -notation in Theorem 5.2. In this Section, we will prove Lemma 6.6.
Proposition 6.9.
Let Y , Z , U , W be random variables where Y and Z ; and U and W are on thesame support. Then, k{ Y }{ U } − { Z }{ W }k ≤ k{ Y } − { Z }k + k{ U } − { W }k . Proof.
Tensoring with the same probability distribution does not change the total variationdistance, i.e. k{ Y } − { Z }k = k{ Y }{ U } − { Z }{ U }k and k{ U } − { W }k = k{ Z }{ U } − { Z }{ W }k .Now, a simple application of the triangle inequality proves the Proposition.A straightforward implication of Proposition 6.9 is the following, which will allowus to bound the correlation along a face a ∈ X ( k ) , using the correlation along sub-faces s , t ⊆ a . 41 orollary 6.10. Let a ∈ X ( ℓ ) and s ∈ X ( ℓ ) , t ∈ X ( ℓ ) be given such that a = s ⊔ t . Then forany k-local PSD ensemble { Y , . . . , Y n } we have k{ Y a } − { Y a } · · · { Y a ℓ }k ≤ k{ Y a } − { Y s }{ Y t }k + (cid:13)(cid:13)(cid:13) { Y s } − { Y s } · · · { Y s ℓ } (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) { Y t } − { Y t } · · · { Y t ℓ } (cid:13)(cid:13)(cid:13) With this, we can go ahead and prove Lemma 6.6
Proof of Lemma 6.6.
Let a ∈ X ( ℓ ) be a fixed face. By Corollary 6.10 and averaging over allthe ( ℓ = ℓ + ℓ ℓ ) ways of splitting a into { s , t } such that s ∈ X ( ℓ ) and t ∈ X ( ℓ ) we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) { Y a } − ℓ = ℓ + ℓ ∏ i = { Y a i } (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ( ℓ + ℓ ℓ ) ∑ { s , t } k{ Y a } − { Y s }{ Y t }k + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) { Y s } − ℓ ∏ i = { Y s i } (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) { Y t } − ℓ ∏ i = { Y t i } (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)! .Now, by taking an average over all the edges a ∈ X ( ℓ ) (with respect to the measure Π ℓ )we obtain, E a ∼ Π ℓ "(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) { Y a } − ℓ ∏ i = { Y a i } (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ( ℓℓ ) · E a ∈ Π ℓ " ∑ { s , t } k{ Y a } − { Y s }{ Y t }k + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) { Y s } − k ∏ i = { Y s i } (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) { Y t } − ℓ ∏ i = { Y t i } (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ! where the indices { s , t } run over the all the ways of splitting a into s and t as before. We cannow see that the RHS can be thought as an average over the (weighted) edges in E ( ℓ , ℓ ) (q.v. Eq. (10)), i.e., E a ∼ Π ℓ "(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) { Y a } − ℓ ∏ i = { Y a i } (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ E { s , t }∼ w ℓ ℓ " k{ Y a } − { Y s }{ Y t }k + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) { Y s } − ℓ ∏ i = { Y s i } (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) { Y t } − ℓ ∏ i = { Y t i } (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Now, note that since Π ℓ , ℓ (q.v. Eq. (9)) is the stationary distribution of the walk definedon G ℓ , ℓ , i.e., 2 Π ℓ , ℓ ( a ) = ∑ b : { a , b }∈ E ( ℓ , ℓ ) w ℓ , ℓ ( a , b ) ,the lemma follows. This is because, we have E a ∈ X ( ℓ ) "(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) { Y a } − ℓ ∏ i = { Y a i } (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ E { s , t }∼ w ℓ ℓ [ k{ Y a } − { Y s }{ Y t }k ] + E { s , t }∼ w ℓ ℓ "(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) { Y s } − ℓ ∏ i = { Y s i } (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) { Y t } − ℓ ∏ i = { Y t i } (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = E { s , t }∼ E ( ℓ , ℓ ) [ k{ Y a } − { Y s }{ Y t }k ] + E s ∼ Π ℓ "(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) { Y s } − ℓ ∏ i = { Y s i } (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + E t ∼ Π ℓ "(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) { Y t } − ℓ ∏ i = { Y t i } (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) High-Dimensional Threshold Rank
In [BRS11], Theorem 6.2 was proven for a more general class of graphs than expandergraphs – namely, the class of low threshold rank graphs.
Definition 7.1 (Threshold Rank of Graphs (from [BRS11])) . Let G = ( V , E , w ) be a weightedgraph on n vertices and A be its normalized random walk matrix. Suppose the eigenvalues of A are = λ ≥ · · · ≥ λ n . Given a parameter τ ∈ (
0, 1 ) , we denote the threshold rank of G by rank ≥ τ ( A ) (or rank ≥ τ ( G ) ) and define it as rank ≥ τ ( A ) : = |{ i | λ i ≥ τ }| .There [BRS11], the authors asked for the correct notion of threshold rank for k -CSPs.In this section, we give a candidate definition of low threshold rank motivated by ourtechniques.To break k -wise correlations it is sufficient to assume that the involved swap graphs inthe foregoing discussion are low threshold rank since this is enough to apply a version ofLemma 6.7, already described in the work of [BRS11].Moreover, we have some flexibility as to which swap graphs to consider as long as theysatisfy some splitting conditions. To define a swap graph it is enough to have a distribu-tions on the hyperedges of a (constraint) hypergraph. Hence, the notion of swap graph isindependent of high-dimensional expansion. HDXs are just an interesting family of objectsfor which the swap graphs are good expanders.To capture the many ways of splitting the statistical distance over hyperedges into thestatistical distance over the edges of swap graphs, we first define the following notion. Wesay that a binary tree T is a k -splitting tree if it has exactly k leaves. Thus, labeling everyvertex with the number of leaves on the subtree rooted at that vertex ensures,- the root of T is labeled with k and all other vertices are labeled with positive integers,- the leaves are labeled with 1, and- each non-leaf vertex satisfy the property that its label is the sum of the labels of itstwo children.Note that, we will think of each non-leaf node with left and right children labeled as a and b as representing the swap graph from X ( a ) to X ( b ) for some simplicial complex X ( ≤ k ) . Let Swap ( T , X ) be the set of all such swap graphs over X finding representationin the splitting tree T . Indeed the tree T used in the proof of Theorem 6.3 is just one specialinstance of a k -splitting tree.Given a threshold parameter τ ≤ A = { A , . . . , A s } , we define the threshold rank of A asrank ≥ τ ( A ) : = max A ∈A rank ≥ τ ( A ) ,where rank ≥ τ ( A ) is denotes usual threshold rank of A as in Definition 7.1.Now, we are ready to define the notion of a k -CSP instance being ( T , τ , r ) -splittable asfollows. 43 efinition 7.2 ( ( T , τ , r ) -splittability) . A k-CSP instance I with the constraint complex X ( ≤ k ) is said to be ( T , τ , r ) -splittable if T is a k-splitting tree and rank ≥ τ ( Swap ( T , X )) ≤ r . If there exists some k-splitting tree T such that I is ( T , τ , r ) -splittable, the instance I will be calleda ( τ , r ) -splittable instance. Now, using this definition we can show that whenever rank τ ( I ) is bounded for the ap-propriate choice of τ , after conditioning on a random partial assignment as in Algorithm 6.1we will have small correlation over the faces a ∈ X ( k ) , i.e., Theorem 7.3.
Suppose a simplicial complex X ( ≤ k ) with X ( ) = [ n ] and an L-local PSD en-semble Y = { Y , . . . , Y n } are given. There exists some universal constants c ≥ and C ′′ ≥ satisfying the following: If L ≥ C ′′ · ( q k · k · r / ε ) , Supp ( Y j ) ≤ q for all j ∈ [ n ] , and I is ( c · ( ε / ( k · q k )) , r ) -splittable. Then, we have E a ∈ X ( k ) h(cid:13)(cid:13) { Y ′ a } − (cid:8) Y ′ a (cid:9) · · · (cid:8) Y ′ a k (cid:9)(cid:13)(cid:13) i ≤ ε , (17) where Y ′ is as defined in Algorithm 6.1 on the input of { Y , . . . , Y n } and Π k . It is important to note that the specific knowledge of the k -splitting tree T that makes I ( T , τ , r ) -splittable is only needed for the proof of Theorem 7.3. The conclusion of Theorem 7.3can be used without the knowledge of the specific k -splitting tree T . The attentive readermight have noticed is that in the proof of Theorem 6.3, the choice of T is not important, asall the splitting tree are guaranteed to have be expanders provided that X is a γ -HDX.The proof of Theorem 7.3, in this light can be thought of an extension of the proof ofTheorem 6.3 to the case where not necessarily every tree is good, and where we can boundthe threshold rank instead of the spectral expansion.This, will readily imply an algorithm Corollary 7.4.
Suppose I is a q-ary k-CSP instance whose constraint complex is X ( ≤ k ) . Thereexists an absolute constant C ′′ ≥ and c ≥ that satisfies the following: If I is ( c · ( ε / ( k · q k )) , r ) -splittable, then there is an algorithm that runs in time n O (cid:18) q k · k · r ε (cid:19) and that is based on ( C ′′ · k · q k · r ε ) -levels of SoS-hierarchy and Algorithm 6.1 that outputs a random assignment η : [ n ] → [ q ] that in expectation ensures SAT I ( η ) = OPT ( I ) − ε . Since the proof of Corollary 7.4 given Theorem 7.3, will be almost identical to the proofof Corollary 6.4, given Theorem 6.3, we will omit the proof of this.
We will need the more general version of Lemma 6.7, already proven in [BRS11].
Lemma 7.5 (Lemma 5.4 from [BRS11]) . Let G = ( V , E , Π ) be a weighted graph, { Y , . . . , Y n } a local PSD ensemble, where we have Supp ( Y i ) ≤ q for every i ∈ V, and q ≥ . If ε ≥ is a lower We give a derivation of this lemma in Appendix A. ound on the expected statistical difference between independent and correlated sampling along theedges,i.e., ε ≤ E { i , j }∼ Π h(cid:13)(cid:13) { Y ij } − { Y i }{ Y j } (cid:13)(cid:13) i . There exists absolute constants c ≥ and c ≥ that satisfy the following: Then, conditioningon a random vertex decreases the variances, E i ∼ Π E j ∼ Π E { Y j } (cid:2) Var (cid:2) Y i | Y j (cid:3)(cid:3) ≤ E i ∼ Π [ Var [ Y i ]] − c · ε q · rank ≥ c ε / q ( G ) .Since we will use this lemma, only with the swap graphs G ℓ , ℓ and ( L / k ) -local PSDensemble { Y a } a ∈ X obtained from the L -local PSD ensemble { Y , . . . , Y n } , for conveniencewe will write the corollary we will use more explicitly Corollary 7.6.
Let ℓ ≥ ℓ ≥ satisfying ℓ + ℓ ≤ k be given parameters, and let G ℓ , ℓ be theswap graph defined for a γ -HDX X ( ≤ k ) . Let { Y a } a ∈ X be a local PSD ensemble; and suppose wehave Supp ( Y a ) ≤ q k for every a ∈ X ( ℓ ) ∪ X ( ℓ ) for some q ≥ . Suppose ε > satisfies, ε k ≤ E { s , t }∈ E ( ℓ , ℓ ) [ k{ Y s ∪ t } − { Y s }{ Y t }k ] . There exists absolute constants c ≥ and c ≥ that satisfy the following:If rank ≥ c · ( ε / ( k · q k )) ( G ℓ , ℓ ) ≤ r, then conditioning on a random face a ∼ Π ℓ , ℓ decreases thevariances, i.e. · E a , b ∼ Π ℓ ℓ (cid:20) E { Y a } [ Var [ Y b | Y a ]] (cid:21) = E a ∈ Π ℓ ℓ " E s ∼ Π ℓ [ Var [ Y s | Y a ]] + E t ∼ Π ℓ [ Var [ Y t | Y a ]] , ≤ E s ∼ Π ℓ [ Var [ Y s ]] + E t ∼ Π ℓ [ Var [ Y t ]] − c · ε · k · q k · r .Here the constant c satisfies c = · c . Proof.
As the proof will mostly follow Theorem 6.3, we will only highlight the relevantdifferences and carry out the relevant computations.Let τ = c · ( ε / ( k · q k )) , and let T be the k -splitting tree certifying that I is ( T , τ , r ) splittable, i.e., the tree T satisfies rank τ ( Swap ( T , X )) ≤ r . This means that all the swapgraphs G ℓ , ℓ finding representation in T satisfy rank τ ( G ℓ , ℓ ) ≤ r .Similarly, as in the proof of we will try to argue that the fraction of indices m ∈ [ L / k ] such that ε m that is large, say ε m ≥ ε /2, is small by arguing about the potential Φ m withboth quantities ε m and Φ m as defined as in the Proof of Theorem 6.3. We assume similarly,that ε m ≥ ε /2 for some m ∈ [ L / k ] .Analogously to Section 7.1 in the proof of Theorem 6.3, from Corollary 7.6 we obtain E S ∼ Π mk E { Y S } " ∑ ℓ ∈ J ( T ) E { t , t }∈ E ( ℓ , ℓ ) [ k{ Y t ⊔ t | Y S } − { Y t | Y S }{ Y t | Y S }k ] ≥ ε τ ( I ) ≤ r and where the set J ( T ) contains all labels ℓ of internal nodes45 ∈ T , and we write ℓ (resp. ℓ ) for the label of the left (resp. right) child of the vertexwith the label ℓ . Similarly, to the proof of Theorem 6.3, there exists some ( ℓ , ℓ ) ∈ J ( T ) that satisfies E S ∼ Π mk E { Y S } E { t , t }∼ w ℓ ℓ k{ Y t ⊔ t | Y S } − { Y t | Y S }{ Y t | Y S }k ≥ ε k .Now, analogously to Eq. (15), using ℓ ≤ k using we have P S ∼ Π mk { Y S } " E a ∈ Π ℓ ℓ " E t ∈ X ( ℓ ) [ Var [ Y t | Y S ] − Var [ Y t | Y S , Y a ]]+ E t ∈ X ( ℓ ) [ Var [ Y t | | Y S ] − Var [ Y t | Y S , Y a ]] ≥ c · ε · k · q k · r ≥ ε k ,(18)Using the same arguments in the proof of Theorem 6.3, we can get that Φ m − Φ m + ≥ k · ε k · c · ε · k · q k · r = c · ε · k · q k · r .Again, this would mean that there can be at most 2048 · k · q k · r / ( ε · c ) indices m suchthat ε m /2 ≥ ε /2. In particular, E m ∈ [ L / k ] [ ε m ] ≤ ε + kL · · k · q k · r ε · c .i.e. there exists a universal constant C ′′ ≥
0, such that L ≥ C ′′ · k · q k · r ε ensures E m ∼ [ L / k ] ε m ≤ ε . Our k -CSP results extend to the quantum setting generalizing the approximation schemefor 2-local Hamiltonians on bounded degree low threshold rank graphs from Brandão andHarrow [BH13] (BH). Before we can make the previous statement more precise we willneed to introduce some notation. A well studied quantum analogue of classical k -CSPs arethe so-called quantum k -local Hamiltonians [AAV13]. Definition 8.1 ( k -local Hamiltonian) . We say that H = E s ∼ Π k H s is an instance of the k-localHamiltonian problem over q-qudits on ground set [ n ] if there is a distribution Π k on subsets ofsize k of [ n ] such that for every s ∈ Supp ( Π k ) there is an Hermitian operator H s on C q n with k H s k op ≤ and acting (possibly) non-trivially on the q-qudits of s and trivially on [ n ] \ s . Given an instance H = E s ∼ Π k H s of the k -local Hamiltonian problem on ground set [ n ] ,the goal is to provide a good (additive) approximation to the ground state energy e ( H ) , i.e.,the smallest eigenvalue of H . Equivalently, the goal is to approximate e ( H ) = min ρ ∈ D ( C qn ) Tr ( H ρ ) ,where D (cid:0) C q n (cid:1) is the set of density operators, PSD operators of trace one, on C q n . Theeigenspace of H associated to e ( H ) is called the ground space of H .46 emark 8.2. The locality k of a k-local Hamiltonian has a similar role as the arity of k-CPSswhereas the qudit dimension q has the role of alphabet size. Observe that for a k-CSP the goal is tomaximize the fraction of satisfied constrains while for a k-local Hamiltonian the goal is to minimizethe energy (constraint violations).
We will need an informationally complete measurement Λ modeled as a channel Λ : D ( C q ) → D (cid:16) C q (cid:17) ,and defined as Λ ( ρ ) : = ∑ y ∈Y Tr ( M y ρ ) · e y e † y ,where { M y } y ∈Y is a POVM and { e y } y ∈Y is an orthonormal basis (see Lemma 8.8 belowfor the properties of Λ ). Recall that an informationally complete measurement is an injec-tive channel, i.e., the probability outcomes p ( y ) = Tr ( M y ρ ) fully determine ρ . By definitiongiven this probability distribution { p ( y ) } y ∈Y we can uniquely determine ρ . We use the no-tation ρ = Λ − (cid:0) { p ( y ) } y ∈Y (cid:1) for the recovered state from probability outcomes { p ( y ) } y ∈Y .BH using the informationally complete measurement Λ reduced the quantum 2-localHamiltonian problem to a classical problem involving PSD ensembles of indicator randomvariables of outcomes Y of Λ . In this reduction, they had to ensure that the local distribu-tions encoded by these indicators random variables are indeed consistent with probabilitydistributions of outcomes arising from actual local density matrices. Note that the channel Λ is only injective, an arbitrary probability distribution on Y may not correspond to a validquantum state. For this reason, they introduced a new SDP hierarchy to find this specialkind of PSD ensemble, which we refer to as quantum PSD ensemble , minimizing the valueof the given input k -local Hamiltonian instance.Using our k -CSP approximation scheme for low threshold rank hypergraphs, we showthat product state approximations close to the ground space of k -local Hamiltonians onbounded degree low threshold rank hypergraphs can be computed efficiently in polyno-mial time by Algorithm 8.3. Our result is a generalization of the k = A POVM is a collection of operators { M y } y ∈Y such that ∑ y ∈Y M y = I and ( ∀ y ∈ Y )( M y (cid:23) ) . lgorithm 8.3 (Quantum Propagation Rounding Algorithm) . Intput
L-local quantum PSD ensemble a { Y , . . . , Y n } and distribution Π on X ( ≤ ℓ ) . Output
A random state ρ = ρ ⊗ . . . ⊗ ρ n where each ρ i ∈ D ( C q ) .1. Choose m ∈ {
1, . . . , L / ℓ } at random.2. Independently sample m ℓ -faces, s j ∼ Π for j =
1, . . . , m.3. Write S = S mj = s j , for the set of the seed vertices.4. Sample assignment η S : S → [ q ] according to the local distribution, { Y S } .5. Set Y ′ = { Y , . . . Y n | Y S = η S } , i.e. the local ensemble Y conditioned on agreeing with η S .6. For all j ∈ [ n ] , set ρ j = Λ − ( { Y ′ j } ) .7. Output ρ = ρ ⊗ . . . ⊗ ρ n . a We define the quantum ensemble as the PSD ensemble produced by the SDP hierarchy of [BH13]
The precise result is given in Theorem 8.4.
Theorem 8.4.
Suppose I = ( H = E s ∼ Π k H s ) is a q-qudit k-local Hamiltonian instance whoseconstraint complex is X ( ≤ k ) and has bounded normalized degree, i.e., Π ≤ δ . Let τ = c · ( ε / ( k q k )) , for ε > . There exists an absolute constant C ′ that satisfies the following:Set L = ( C ′ · k · q k · rank τ ( I ) ε ) . Then there is an algorithm based on L-levels of SoS-hierarchy andAlgorithm 8.3 that outputs a random product state ρ = ρ ⊗ . . . ⊗ ρ n that in expectation ensures Tr ( H ρ ) ≤ e ( H ) + ( q ) k /2 · ε + L · k · δ , where e ( H ) is the ground state energy of H . Remark 8.5.
Similarly to the classical case, Theorem 8.4 serves as a no-go barrier (in its parameterregime) to the quantum local-Hamiltonian version of the quantum PCP Conjecture [AAV13]. Inparticular, k-local Hamiltonians on bounded degree γ -HDXs for γ sufficiently small can be effi-ciently approximated in polynomial time. Now we sketch a proof of Theorem 8.4. We provide a sketch rather than a full proofsince Theorem 8.4 easily follows from the BH analysis once the main result used by them,Theorem 5.6 from [BRS11], is appropriately generalized to “break” k -wise correlations asaccomplished by our Theorem 7.3 (restated below for convenience). Furthermore, a fullproof would require introducing more objects and concepts only needed in this simplederivation (the reader is referred to [BH13] for the quantum terminology and the omitteddetails). Theorem 8.6 (Adaptation of Theorem 7.3) . Suppose a simplicial complex X ( ≤ k ) with X ( ) =[ n ] and an L-local PSD ensemble Y = { Y , . . . , Y n } are given. There exists some universal con-stants c ≥ and C ′′ ≥ satisfying the following: If L ≥ C ′′ · ( q k · k · r / ε ) , Supp ( Y j ) ≤ q forall j ∈ [ n ] , and I is ( c · ( ε / ( k · q k )) , r ) -splittable. Then, we have E a ∈ X ( k ) h(cid:13)(cid:13) { Y ′ a } − (cid:8) Y ′ a (cid:9) · · · (cid:8) Y ′ a k (cid:9)(cid:13)(cid:13) i ≤ ε , (19) We define the constraint complex of a k -local Hamiltonian in the same way we define it for k -CSPs, namely,by taking the downward closure of the support of Π k . here Y ′ is as defined in Algorithm 8.3 on the input of { Y , . . . , Y n } and Π k . Once in possession of the quantum PSD ensemble the problem becomes essentially clas-sical. The key result in the BH approach is Theorem 5.6 from [BRS11] that brings (in ex-pectation under conditioning on a random small seed set of qudits) the local distributions,over the edges of the constraint graph of a 2-local Hamiltonian, close to product distri-butions . Now, using the fact that they have an informationally complete measurement Λ they can “lift” the conditioned marginal distribution on each qudit { Y ′ j } to an actualquantum state as ρ j = Λ − ( { Y ′ j } ) (see Algorithm 8.3). In this lifting process, they pay an average distortion cost of 18 q · ε (for using the marginal over the qudits). For k -local Hamil-tonians, the distortion of k q -qudits is given by Lemma 8.7 (stated next without proof). Lemma 8.7.
Let Z , . . . , Z k be random variables in an L-local quantum PSD ensemble with L ≥ k.Suppose that ε : = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) { Z , . . . , Z k } − k ∏ i = { Z i } (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . Then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:16) Λ ⊗ k (cid:17) − ( { Z , . . . , Z k } ) − (cid:16) Λ ⊗ k (cid:17) − k ∏ i = { Z i } !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ( q ) k /2 · ε .Note that Lemma 8.7 is a direct consequence of Lemma 8.8 from [BH13]. Lemma 8.8 (Informationally complete measurements (Lemma 16 [BH13])) . For every posi-tive integer q there exists a measurement Λ with ≤ q outcomes such that for every positive integerk and every traceless operator ξ , we have k ξ k ≤ ( q ) k /2 (cid:13)(cid:13)(cid:13) Λ ⊗ k ( ξ ) (cid:13)(cid:13)(cid:13) .BH also pay a full cost for each local term in the Hamiltonian that involves a seedqudit since its state was not reconstructed using the full distribution of a qudit given by thequantum ensemble but rather reconstructed from a single outcome y ∈ Y of Λ . Naively,this means that the final state of this qudit may be far from the intended state given by SDPrelaxation. In our case, we assume that the normalized degree satisfies Π ≤ δ . Therefore,the total error from constraints involving seed qudits is at most L · k · δ .Putting the above pieces together we conclude the proof (sketch) of Theorem 8.4. Acknowledgements
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Proceedings of the 5th Conference on Innovations in Theoretical ComputerScience , pages 423–438. ACM, 2014. 4
A From Local to Global Correlation
We include the key result we use from [BRS11], namely, their Lemma 5.4 (below). Whilethey proved the lemma for regular graphs, we include the details in the proof for general52eighted graphs, since even for HDXs regular at the top level, the swap graphs are notnecesarily regular. The extension to general graphs is straighforward (and [BRS11] indi-cated the same) but we include the details for the sake of completeness . Lemma A.1 (Lemma 5.4 from [BRS11] (restatement of Lemma 7.5)) . Let G = ( V , E , Π ) be aweighted graph, { Y , . . . , Y n } a local PSD ensemble, where we have Supp ( Y i ) ≤ q for every i ∈ V,and q ≥ . If ε ≥ is a lower bound on the expected statistical difference between independent andcorrelated sampling along the edges,i.e., ε ≤ E { i , j }∼ Π h(cid:13)(cid:13) { Y ij } − { Y i }{ Y j } (cid:13)(cid:13) i . Then, conditioning on a random vertex decreases the variances, E i , j ∼ Π " E { Y j } (cid:2) Var (cid:2) Y i | Y j (cid:3)(cid:3) ≤ E i ∼ Π [ Var [ Y i ]] − ε q · rank ε / ( q ) ( G ) .The key ingredient in proving Lemma 5.4 is a “local to global” argument generalizingthe expander case to low threshold rank graphs. This new argument is proven in two stepswith Lemma A.2 being the first one. Lemma A.2 (Adapted from Lemma 6.1 of [BRS11]) . Let G be an undirected weighted graph.Suppose v , . . . , v n ∈ R n are such that E i ∼ V ( G ) [ h v i , v i i ] = E ij ∼ E ( G ) (cid:2) h v i , v j i (cid:3) ≥ − ε , but E i , j ∼ V ( G ) (cid:2) h v i , v j i (cid:3) ≤ m . Then for c > , we have λ ( − c ) m ≥ − c · ε . In particular, λ m /4 ≥ − ε .Proof. Let Y be the Gram matrix defined as Y i , j = h v i , v j i . Clearly, Y is positive semi-definite. Without loss of generality suppose that the edge weights { w ( { i , j } ) | ij ∈ E ( G ) } form a probability distribution. Set w ( i ) = ∑ j ∼ i w ( { i , j } ) . Let D to be the diagonal matrixsuch that D ( i , i ) = w ( i ) , i.e., the matrix of generalized degrees. Let A be such that A i , j = w ( { i , j } ) /2 and A G = D − AD − be its normalized adjacency matrix.Suppose A G = ∑ ni = λ i u i u ⊤ i is a spectral decomposition of A G . Set Y ′ = D YD . Forconvenience, define the matrix X as X ( i , j ) = h u i , Y ′ u j i and set p ( i ) = X ( i , i ) . We claim that p is a probability distribution. Since Y ′ is positive semi-definite, we have that p ( i ) ≥ ∑ ni = p ( i ) = = E i ∼ V ( G ) [ h v i , v i i ] = Tr ( Y ′ ) = Tr ( X ) = n ∑ i = X ( i , i ) = n ∑ i = p ( i ) . For expander graphs it is possible to obtain an improved bound of Ω (( ε / q ) ) instead of Ω (( ε / q ) ) givenby Lemma A.1, simply by using the definition of the second smallest eigenvalue of the Laplacian. While BRSanalyzed E i , j [ h v i , v j i ] for low-threshold rank graphs, it is possible to directly analyze the quantity E i , j [ h v i , v j i ] for expanders, leading to the improved bound. m ′ be the largest value in [ n ] satisfying λ m ′ ≥ − c · ε . By Cauchy-Schwarz , q = m ′ ∑ i = p ( i ) ≤ √ m ′ vuut m ′ ∑ i = p ( i ) ≤ √ m ′ s ∑ i , j ( X ( i , j )) ≤ r m ′ m ,where the last inequality follows from our assumption that1 m ≥ E i , j ∼ V ( G ) (cid:2) h v i , v j i (cid:3) = h Y ′ , Y ′ i = h X , X i = ∑ i , j X ( i , j ) .Then 1 − ε ≤ E ij ∼ E ( G ) (cid:2) h v i , v j i (cid:3) = h A , Y i = h A G , X i = n ∑ i = λ i X ( i , i ) ,implies that1 − ε ≤ n ∑ i = λ i · X ( i , i ) ≤ m ′ ∑ i = p ( i ) + ( − c · ε ) n ∑ i = m ′ + p ( i ) = − c · ε ( − q ) .Finally, using the bound on q we obtain (cid:18) − c (cid:19) √ m ≤ √ m ′ ,from which the lemma readily follows.As a corollary it follows that local correlation implies global correlation. Corollary A.3 (Adapted from Lemma 4.1 of [BRS11]) . Let G be an undirected weighted graph.Suppose v , . . . , v n ∈ R n are vectors in the unit ball such that E ij ∼ E ( G ) (cid:2) h v i , v j i (cid:3) ≥ ρ , then E i , j ∼ V ( G ) (cid:2) h v i , v j i (cid:3) ≥ ρ · rank ρ /4 ( G ) . In particular, we have E i , j ∼ V ( G ) (cid:2) |h v i , v j i| (cid:3) ≥ ρ · rank ρ /4 ( G ) . Proof.
If all v , . . . , v n are zero, the result trivially follows so assume that this is not thecase. Then α = E i ∼ V ( G ) [ h v i , v i i ] >
0. Also, α ≤ v ′ i = v i / √ α . By construction E i ∼ V ( G ) (cid:2) h v ′ i , v ′ i i (cid:3) = E ij ∼ E ( G ) h h v ′ i , v ′ j i i ≥ ρα . (20) In [BRS11], there was a minor bug in the application this Cauchy-Schwarz, which led to a bound of ( − c ) instead of ( − c ) in the lemma, leading to a global correlation bound of Ω ( ρ ) instead of Ω ( ρ ) asindicated in Corollary A.3. ρ ′ = ρ / ( α ) , ε = − ρ ′ and c = ( − ρ ′ /2 ) / ( − ρ ′ ) . Then1 − c = ρ ′ /21 − ρ ′ /2 ≤ ρ ′ ,and 1 − c · ε = ρ ′ E i , j ∼ V ( G ) h h v ′ i , v ′ j i i > m ≥ ( ρ ′ ) rank ρ ′ /2 ( G ) ,since rank ρ ′ /2 ( G ) < ( ρ ′ ) m as λ ( ρ ′ ) m < ρ ′ /2. Or equivalently E i , j ∼ V ( G ) " h v i , v j i α = E i , j ∼ V ( G ) h h v ′ i , v ′ j i i ≥ ρ α · rank ρ / ( α ) ( G ) ≥ ρ α · rank ρ /4 ( G ) ,where the last inequality follows form the fact that α ≤ Fact A.4 (Adapted from [BRS11]) . Let { Y , . . . , Y n } be a 2-local PSD ensemble where each Y i can take at most q values. Suppose ε = E { i , j }∼ Π h(cid:13)(cid:13) { Y ij } − { Y i }{ Y j } (cid:13)(cid:13) i . Then there exist vectors v , . . . , v n in the unit ball such that E ij ∼ E ( G ) (cid:2) h v i , v j i (cid:3) ≥ q · E ij ∼ Π h(cid:13)(cid:13) { Y ij } − { Y i }{ Y j } (cid:13)(cid:13) i ≥ ε q , (21) and E i , j ∼ V ( G ) " Var [ Y i ] − E { Y j } (cid:2) Var (cid:2) Y i | Y j (cid:3)(cid:3) ≥ E i , j ∼ V ( G ) (cid:2)(cid:12)(cid:12) h v i , v j i (cid:12)(cid:12)(cid:3) . (22)Now we are ready to prove the key result from [BRS11] used in our proof. Lemma A.5 (Lemma 5.4 from [BRS11] (restatement of Lemma 7.5)) . Let G = ( V , E , Π ) be aweighted graph, { Y , . . . , Y n } a local PSD ensemble, where we have Supp ( Y i ) ≤ q for every i ∈ V,and q ≥ . If ε ≥ is a lower bound on the expected statistical difference between independent andcorrelated sampling along the edges,i.e., ε ≤ E { i , j }∼ Π h(cid:13)(cid:13) { Y ij } − { Y i }{ Y j } (cid:13)(cid:13) i . Then, conditioning on a random vertex decreases the variances, E i , j ∼ Π " E { Y j } (cid:2) Var (cid:2) Y i | Y j (cid:3)(cid:3) ≤ E i ∼ Π [ Var [ Y i ]] − ε q · rank ε / ( q ) ( G ) .55 roof. Using Eq. (21) there exist vectors v , . . . , v n such that Fact A.4 implies E ij ∼ E ( G ) (cid:2) h v i , v j i (cid:3) ≥ ε q .From Corollary A.3 we obtain E i , j ∼ V ( G ) (cid:2)(cid:12)(cid:12) h v i , v j i (cid:12)(cid:12)(cid:3) ≥ ε q · rank ε / ( q ) ( G ) .Finally, using Eq. (22) we get E i , j ∼ V ( G ) " Var [ Y i ] − E { Y j } (cid:2) Var (cid:2) Y i | Y j (cid:3)(cid:3) ≥ E i , j ∼ V ( G ) (cid:2)(cid:12)(cid:12) h v i , v j i (cid:12)(cid:12)(cid:3) ≥ ε q · rank ε / ( q ) ( G ) ,as claimed. B Harmonic Analysis on HDXs
We provide the proofs of known facts used in Section 5.2.
Definition B.1 (From [DDFH18]) . We say that d-sized complex X is proper provided ker ( U i ) istrivial for ≤ i < d. We will need the following decomposition.
Claim B.2.
Let A : V → W where V and W are finite dimensional inner product spaces. ThenV = ker A ⊕ im A † . Proof.
We show that ker A = (cid:0) im A † (cid:1) ⊥ . Recall that v ∈ (cid:0) im A † (cid:1) ⊥ if and only if h A † w , v i = w ∈ W . This is equivalent to h w , A v i = w ∈ W , implying A v = Lemma B.3 (From [DDFH18]) . We haveC k = k ∑ i = C ki . Moreover, if X is proper then C k = k M i = C ki , and dim C ki = | X ( i ) | − | X ( i − ) | .Proof. We induct on k . For k = X ( ) = { ∅ } and C = C . Now suppose k >
0. Since D k and U k − are adjoints, we have C k = ker D k ⊕ im U k − or equivalently C k = ker D k ⊕ U k − C k − . (23)56sing the induction hypothesis C k − = ∑ k − i = C k − i . Note that U k − C k − i = n U k − U k − − i h i | h i ∈ H i o = C ki .Thus C k = C kk + ∑ k − i = C ki . Assuming ker ( U i ) is trivial for 0 ≤ i < k we obtaindim C ki = dim H i = dim C i − dim C i − = | X ( i ) | − | X ( i − ) | ,where the second equality follows from Eq. (23). Hence dim C k = ∑ ki = dim C ki . This im-plies that each C ki ∩ ∑ j = i C kj is trivial and so we have a direct sum as claimed. Corollary B.4 (From [DDFH18]) . Let f ∈ C k . If X is proper, then f can be written uniquely asf = f + · · · + f k , where f i ∈ C ki ..