High Dimensional Expanders: Random Walks, Pseudorandomness, and Unique Games
aa r X i v : . [ c s . CC ] N ov High Dimensional Expanders: Random Walks,Pseudorandomness, and Unique Games
Max Hopkins ∗ Tali Kaufman † Shachar Lovett ‡ November 16, 2020
Abstract
Higher order random walks (HD-walks) on high dimensional expanders have played a crucial rolein a number of recent breakthroughs in theoretical computer science, perhaps most famously in therecent resolution of the Mihail-Vazirani conjecture (Anari et al. STOC 2019), which focuses on HD-walkson one-sided local-spectral expanders. In this work we study the spectral structure of walks on thestronger two-sided variant, which capture wide generalizations of important objects like the Johnson andGrassmann graphs. We prove that the spectra of these walks are tightly concentrated in a small numberof strips, each of which corresponds combinatorially to a level in the underlying complex. Moreover, theeigenvalues corresponding to these strips decay exponentially with a measure we term the depth of thewalk.Using this spectral machinery, we characterize the edge-expansion of small sets based upon the inter-play of their local combinatorial structure and the global decay of the walk’s eigenvalues across strips.Variants of this result for the special cases of the Johnson and Grassmann graphs were recently crucialboth for the resolution of the 2-2 Games Conjecture (Khot et al. FOCS 2018), and for efficient algorithmsfor affine unique games over the Johnson graphs (Bafna et al. Arxiv 2020). For the complete complex,our characterization admits a low-degree Sum of Squares proof. Building on the work of Bafna et al., weprovide the first polynomial time algorithm for affine unique games over the Johnson scheme. The sound-ness and runtime of our algorithm depend upon the number of strips with large eigenvalues, a measurewe call High-Dimensional Threshold Rank that calls back to the seminal work of Barak, Raghavendra,and Steurer (FOCS 2011) on unique games and threshold rank. ∗ Department of Computer Science and Engineering, UCSD, CA 92092. Email: [email protected] . Supported by NSFAward DGE-1650112. † Department of Computer Science, Bar-Ilan University. Email: [email protected] . Supported by ERC and BSF. ‡ Department of Computer Science and Engineering, UCSD, CA 92092. Email: [email protected] . Supported by NSFAward CCF-1909634. Introduction
In recent years, high dimensional expanders have begun to play an increasingly important role in our un-derstanding of a wide range of problems within theoretical computer science, including the resolution of theMihail-Vazirani conjecture [1] and a host of problems across areas such as approximation algorithms [2, 3],sampling [4, 5], agreement testing [6–8], and error correction [9, 10]. These breakthroughs have mostly beendriven by the recent introduction of higher order random walks (HD-walks) [6, 11, 12, 3], random processes onhigh-dimensional objects that generalize the standard vertex-edge-vertex walk on expander graphs—walkingfor instance from a triangle, to a pyramid, and back to a triangle.Most applications of these walks (e.g. to matroids [1], Glauber Dynamics of Markov chains [4, 5]) rely ona spectral notion of high dimensional expansion known as one-sided local-spectral expanders [13, 14]. In thiswork, we study HD-walks on the stronger two-sided local-spectral expanders , a variant introduced by Dinurand Kaufman [11] in the context of agreement testing. Under this stronger setting, HD-walks still captureimportant structure, offering a broad generalization of non-negative matrices in the Johnson scheme. The Johnson scheme, historically studied for its connections to coding theory, and its q -analog theGrassmann scheme have seen a recent resurgence due to their connection with the Unique Games Conjecture(UGC). Indeed, it was a deeper understanding of the spectral and combinatorial structure of the Johnsonand Grassmann graphs (bases of their respective schemes) that finally lead to the resolution of the 2-2Games Conjecture [15], completing a long line of work in this direction [16–20]. Conversely, similar structurerecently allowed Bafna, Barak, Kothari, Schramm, and Steurer (BBKSS) [21] to give the first polynomial-time algorithm for unique games on the Johnson graphs, raising an interesting interplay between hardness,algorithms, and spectral structure. This breakthrough line of work, however, suffers from a lack of generality:its analysis is rooted in tailoring old Fourier analytic machinery to the Johnson and Grassmann graphs.This specificity results in an unfortunate technical barrier towards further progress on both hardness andalgorithms for unique games, where it seems that a broader structural understanding is necessary. In thiswork, we argue that this barrier may be broken (at least in part) by viewing the Johnson, Grassmann,and related graphs as part of a larger overarching process, a random walk performed on some underlyinghigh-dimensional object.We study the interplay of local combinatorial and global spectral structure in HD-walks. We prove thatthe spectra of such walks are tightly concentrated in a small number of strips, each corresponding to a levelin the underlying complex, and moreover that the eigenvalues of these strips decay exponentially with ameasure we call depth . This allows us to give a tight characterization of the edge-expansion of small setsbased upon their local combinatorial structure and the eigenvalue decay across strips, a result which in thespecial cases of the Johnson and Grassmann graphs has been crucially important both for hardness of [16–20]and algorithms for [21] unique games. In greater detail, we show that sets which are locally-pseudorandom at level i of the underlying complex expand near-perfectly as long as the i th eigenstrip has small eigenvalues.Conversely, we show that sets which are locally-structured at level i expand poorly as long as the i th eigenstriphas large eigenvalues.In the special case of the complete complex, our characterization admits a low-degree Sum of Squares proof.Combined with our local-to-global characterization of the expansion of structured sets along with the recentinsights of [21], this allows us to provide the first efficient algorithm for affine unique games whose constraintgraphs are HD-walks (or equivalently, lie in the Johnson scheme). The soundness and running time of ouralgorithm depend on a novel spectral parameter we call High-Dimensional Threshold Rank (HD-ThresholdRank), which measures the number of strips with eigenvalues above a certain size; this generalizes standardthreshold rank, which Barak, Raghavendra, and Steurer [22] showed in a seminal work to be intimately tiedto unique games. To make these results concrete, we study their specification to standard HD-walks fromthe literature, and show in particular that walks which reach deep into the underlying complex have constantHD-Threshold Rank. In fact, not only are affine unique games particularly easy over such walks, but theycarry a stronger characterization of edge-expansion as well, which at a finer-grain level can be seen to dependon the interplay of pseudorandomness and HD-Threshold Rank.Finally, we note that all of our results extend to a broader set of objects called expanding posets, ageneralization of two-sided local-spectral expanders recently introduced by Dikstein, Dinur, Filmus, and The Johnson Scheme consists of matrices indexed by k -sets of [ n ] which depend only on intersection size. HD-walks on thecomplete complex are exactly the non-negative elements of the Johnson Scheme. Before discussing our results in greater detail, we first overview the theory of two-sided local-spectral ex-panders and higher-order random walks.
Two-sided local-spectral expanders are a generalization of spectral expander graphs to weighted, uniformhypergraphs, which we will think of as simplicial complexes.
Definition 1.1 (Weighted, Pure Simplicial Complex) . A d -dimensional, pure simplicial complex X on n vertices is a subset of (cid:0) [ n ] d (cid:1) . We will think of X as the downward closure of these sets, and in particulardefine the level X ( i ) as: X ( i ) = (cid:26) s ∈ (cid:18) [ n ] i (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ∃ t ∈ X, s ⊆ t (cid:27) . We call the elements of X ( i ) i -faces. A simplicial complex is weighted if its top level faces are endowed witha distribution Π . This induces a distribution over each X ( i ) by downward closure: Π i ( x ) = 1 i + 1 X y ∈ X ( i +1): y ⊃ x Π i +1 ( y ) , (1) where Π d = Π . Two-sided local-spectral expanders are based upon a phenomenon called local-to-global structure, whichlooks to propogate information on local neighborhoods of a simplicial complex called links to the entirecomplex.
Definition 1.2 (Link) . Given a weighted, pure simplicial complex ( X, Π) , the link of an i -face s ∈ X ( i ) isthe sub-complex containing s , i.e. X s = { t \ s ∈ X | t ⊇ s } . Π induces a distribution over X s by normalizing over top-level faces. When considering a function on the k -th level of a complex, we also use X s to denote the k -faces which contain s as long as it is clear fromcontext, and refer to X s as an i -link if s ∈ X ( i ) . Two-sided local-spectral expansion simply posits that the graph underlying every link must be a two-sided spectral expander. Definition 1.3 (Local-spectral expansion) . A weighted, pure simplicial complex ( X, Π) is a two-sided γ -local-spectral expander if for every i ≤ d − and every face s ∈ X ( i ) , the underlying graph of X s is atwo-sided γ -spectral expander. We differ here from much of the HDX literature where an i -face is often defined to have i + 1 elements. Since our work ismostly combinatorial rather than topological or geometric, defining an i -face to have i elements ends up being the more naturalchoice. The underlying graph of a simplicial complex X is its -skeleton ( X (0) , X (1)) . A weighted graph G ( V, E ) with edge weights Π E is a two-sided γ -spectral expander if the vertex-edge-vertex random walkwith transition probabilities proportional to Π E has second largest eigenvalue in absolute value at most γ . .1.2 Higher Order Random Walks Weighted simplicial complexes admit a natural generalization of the standard vertex-edge-vertex walk ongraphs known as higher order random walks (HD-walks). The basic idea is simple: starting at some k -set S ⊂ X ( k ) , pick at random a set T ∈ X ( k + 1) such that T ⊃ S , and then return to X ( k ) by selecting some S ′ ⊂ T . Let the space of functions f : X ( k ) → R be denoted by C k . Formally, higher order random walks area composition of two averaging operators: the “Up” operator which lifts a function f ∈ C k to U k f ∈ C k +1 : ∀ y ∈ X ( k + 1) : U k f ( y ) = 1 k + 1 X x ∈ X ( k ): x ⊂ y f ( x ) , and the “Down” operator which lowers a function f ∈ C k +1 to D k +1 f ∈ C k : ∀ x ∈ X ( k ) : D k +1 f ( x ) = 1 k + 1 X y ∈ X ( k +1): y ⊃ x Π k +1 ( y )Π k ( x ) f ( y ) . These operators exist for each level of the complex, and composing them gives a basic set of higher orderrandom walks we call pure (following [2]). We call an affine combination of pure walks which start and endon X ( k ) a k -dimensional HD-walk . Definition 1.4 (HD-walk) . Let ( X, Π) be a pure, weighted simplicial complex. Let Y be a family of purewalks Y : C k → C k on ( X, Π) . We call an affine combination M = X Y ∈Y α Y Y, a k -dimensional HD-walk on ( X, Π) as long as it remains a valid walk (i.e. has non-negative transitionprobabilities). Previous work on HD-walks mainly focuses on two natural classes: canonical walks, and partial-swapwalks.
Definition 1.5 (Canonical Walk) . Given a d -dimensional weighted, pure simplicial complex ( X, Π) , andparameters k + j ≤ d , the canonical walk N jk is: N jk = D k + jk U k + jk , where U kℓ = U k − . . . U ℓ , and D kℓ = D ℓ +1 . . . D k . In other words, the canonical walk N jk takes j steps up and down the complex via the averaging operators.Partial-swap walks are a similar process, but after ascending the complex, we restrict to returning to faceswith a given intersection from the starting point. Definition 1.6 (Partial-Swap walk) . The partial-swap walk S jk is the restriction of N jk to faces with in-tersection k − j . In other words, if | s ∩ s ′ | 6 = k − j, S jk ( s, s ′ ) = 0 , and otherwise S jk ( s, s ′ ) = α s N jk ( s, s ′ ) ,where α s = X s ′ : | s ∩ s ′ | = k − j N k ( s, s ′ ) − is the appropriate normalization factor. It is not hard to see that partial-swap walk S tk on the complete complex J ( n, d ) (all d -subsets of [ n ] endowed with the uniform distribution) is exactly the Johnson graph J ( n, k, k − t ) . While it is not immediatelyobvious that the partial-swap walks are HD-walks, Alev, Jeronimo, and Tulsiani [2] showed this is the caseby expressing them as an alternating hypergeometric sum of canonical walks.4 .1.3 Expansion of HD-Walks In this work, we study the combinatorial edge expansion of HD-walks, a fundamental property of graphs withstrong connections to many areas of theoretical computer science, including both hardness and algorithmsfor unique games. Given a weighted graph G = (( V, E ) , (Π V , Π E )) where Π V is a distribution over vertices,and Π E is a set of non-negative edge weights, the expansion of a subset S ⊂ V is the average edge-weightleaving S . Definition 1.7 (Weighted Edge Expansion) . Given a weighted, directed graph G = (( V, E ) , (Π V , Π E )) , theweighted edge expansion of a subset S ⊂ V is: Φ( G, S ) = E v ∼ Π V | S [ E ( v, V \ S )] , where E ( v, V \ S ) = X ( v,y ) ∈ E : y ∈ V \ S Π E (( v, y )) is the total weight of edges between vertex v and the subset V \ S , and Π V | S is the re-normalized restrictionof Π V to S . In the context of a k -dimensional HD-Walk M on a weighted simplicial complex ( X, Π) , we willalways have V = X ( k ) , Π V = Π k , and E, Π E given by M . Thus when clear from context, we will simplywrite Φ( S ) . Edge expansion in a weighted graph is closely related to the spectral structure of its adjacency matrix.Given a set S ⊂ V of density α = E [ S ] , we may write Φ( G, S ) = 1 − α h S , A G S i Π V , where A G is the adjacency matrix with weights given by Π E , and h f, g i Π V is the expectation of f g over Π V . When considering such an inner product over a weighted simplicial complex ( X, Π) , the associateddistribution will always be Π k , so we will drop it from the corresponding notation. Notice that the right-hand side of this equivalence may be further broken down via a spectral decomposition of S with respect to A G . Thus to understand the edge-expansion of HD-walks, it is crucial to understand the structure of theirspectra. It is well known [23, 2] that HD-walks on two-sided local-spectral expanders admit an approximate eigende-composition —a decomposition in which the i -th subspace consists of near-eigenvectors corresponding to somevalue λ i . We prove a general linear-algebraic theorem about operators which admit such a decomposition:their true spectra lies in strips tightly concentrated around each λ i . Theorem 1.8 (Informal Theorem 2.2: Approximate Eigendecompositions Imply Eigenstripping) . Let M bea self-adjoint operator over an inner product space V , and V = V ⊕ . . . ⊕ V k a decomposition satisfying ∀ ≤ i ≤ k, f i ∈ V i : k M f i − λ i f i k ≤ c i k f i k for some family of constants ( { λ i } ki =1 , { c i } ki =1 ) . Then as long as the c i are sufficiently small, the spectra of M is concentrated around each λ i : Spec ( M ) ⊆ k [ i =1 [ λ i − e, λ i + e ] = I λ i , where e = O k,λ (cid:16)p max i { c i } (cid:17) . In other words, any sufficiently strong approximate eigendecomposition corresponds to a collection ofnon-overlapping eigenstrips , the span of eigenvectors corresponding to each interval I λ i . Such behavior5as previously known [12] only for the most basic HD-walk, N k , on very strong two-sided local-spectralexpanders.The eigenstrips promised by Theorem 1.8 form their own decomposition closely related (though notnecessarily equivalent) to the original V i . Since in our case the approximate eigendecomposition of interestis combinatorial in nature, Theorem 2.2 will help us to view expansion through both a combinatorial andspectral lens. Indeed, eigenstripping and approximate eigendecompositions are quite useful for understandingexpansion. Given such a decomposition on a two-sided local-spectral expander ( X, Π) , we can express theexpansion of a set S ⊂ X ( k ) of density α on an HD-walk M as: Φ( S ) = 1 − α h S , M S i = 1 − α k X i =1 h S , M S,i i , where S = S, + . . . + S,k is the indicator for S , S,i ∈ V i , and the inner product is understood to beover Π k . Since the V i ’s are closely related to the spectrum of M , we can further write Φ( S ) ≈ − α k X i =1 λ i h S , S,i i , (2)where λ i is the center of the i -th eigenstrip and corresponds to the approximate eigenvalue of V i . Theexpansion of S then hinges on the interplay of two spectral quantities: how the weight of S is distributedon our decompositions, and how the eigenvalues of the underlying walk decay. We begin by analyzing the former, which requires understanding the combinatorial structure of HD-walks.HD-walks on sufficiently strong two-sided local-spectral expanders admit a useful combinatorial decomposi-tion due to [23] we call the
HD-Level-Set Decomposition , which breaks functions on X ( k ) down by contribu-tion from each level X ( i ) of the complex for ≤ i ≤ k . Theorem 1.9 (HD-Level-Set Decomposition, Theorem 8.2 [23]) . Let ( X, Π) be a d -dimensional two-sided γ -local-spectral expander, γ < d , ≤ k ≤ d , and let: H = C , H i = Ker ( D i ) , V ik = U ki H i . Then: C k = V k ⊕ . . . ⊕ V kk . In other words, every f ∈ C k has a unique decomposition f = f + . . . + f k such that f i = U ki g i for g i ∈ Ker ( D i ) . Extending the work of [23, 2], we show that this decomposition is in fact an approximate eigendecompo-sition for all HD-walks, and thus by Theorem 1.8 corresponds to a set of disjoint eigenstrips as long as γ is sufficiently small. By leveraging this structure, we prove that for a very broad class of HD-walks (encom-passing pure, canonical, partial-swap walks, and more), the eigenvalues corresponding to the HD-Level-SetDecomposition decrease monotonically (see Proposition 4.10). We will assume this is the case for HD-walkswe consider throughout the remainder of this section. As a result, for any ≤ ℓ ≤ k , we can re-writeEquation (2) as the bound: Φ( S ) & − α − α ℓ X i =1 λ i h S,i , S,i i − λ ℓ +1 , (3)where S,i ∈ V ik , the level- i subspace of the HD-Level-Set Decomposition. Thus we are particularly interestedin how the weight of a subset S ⊆ X ( k ) is distributed on the first ℓ levels of the decomposition. Technically, this may require combining V i with equivalent approximate eigenvalues. i -th (approximate) eigenspace consists offunctions coming from X ( i ) , which manifest as the sum over links of i -faces. One might suspect that theweight of S on V ik is then controlled to some extent by its density across i -links. To formalize this, we borrowa notion of pseudorandomness from [20] which measures the distance between global and local densitieswithin S . Definition 1.10 (Pseudorandom Sets) . Let ( X, Π) be a weighted, pure simplicial complex. We call a set S ⊂ X ( k ) ( ε , . . . , ε ℓ ) -pseudorandom if its local expectations at levels through ℓ do not greatly exceed itsglobal expectation. That is if, for all ≤ i ≤ ℓ , we have: ∀ s ∈ X ( i ) : E X s [ S ] − E [ S ] ≤ ε i , where expectations are taken with respect to Π k . As befits our intuition, we prove that functions which are ( ε , . . . , ε ℓ ) -pseudorandom have bounded weighton levels ≤ i ≤ ℓ (level is the constant part, and thus always has weight E [ S ] ). Theorem 1.11 (Informal Theorem 5.3: Pseudorandomness Controls Eigenweight) . Let ( X, Π) be a two-sided γ -local-spectral expander with γ sufficiently small, and S ⊂ X ( k ) an ( ε , . . . , ε ℓ ) -pseudorandom set ofdensity α . Let S = S, + . . . + S,k be the HD-Level-Set Decomposition of the indicator of S . Then for all < i ≤ ℓ , the weight of S on V ik is at most: h S , S,i i ≤ (1 + O k ( √ γ )) (cid:18) ki (cid:19) ε i α. If the HD-Level-Set is orthogonal, we achieve linear dependence on γ : h S , S,i i ≤ (1 + O k ( γ )) (cid:18) ki (cid:19) ε i α. Using Theorem 1.11, we can re-examine Equation (3), our expression of expansion. If S ⊂ X ( k ) is an ( ε , . . . , ε ℓ ) -pseudorandom set of density α on a sufficiently strong two-sided local-spectral expander, thenwe can bound the expansion by about: Φ( S ) & − α − ℓ X i =1 λ i (cid:18) ki (cid:19) ε i − λ ℓ +1 . (4)Thus we see that as long as α and λ ℓ +1 are small, pseudorandom sets on HD-walks expand near-perfectly.The final piece of the puzzle lies in understanding how the eigenvalues of the HD-Level-Set Decompositiondecay.Before moving to this, however, it is worth pausing to note that Equation (4) provides a tight characteri-zation of expansion for HD-walks on two-sided local-spectral expanders. In particular, for large enough n wecan find a subset of any partial-swap walk on the complete complex (i.e. any Johnson graph) with expansionarbitrarily close to Equation (4) (see Proposition 5.4). Conversely, Equation (4) says nothing about setswhich are not sufficiently pseudorandom, i.e. when some ε i ≫ (cid:0) ki (cid:1) − . In this case we may hope for a boundwith worse (sub-linear) dependence on ε i , but whose coefficient is independent of k . Such a bound is knownfor the special case of the Johnson graphs [20], but remains open for general HD-walks. Equation (4) raises an interesting dynamic between pseudorandomness and the eigenvalues of the HD-Level-Set Decomposition. To formalize this notion, we can rephrase the equation as follows: for any constant δ > ,if M has at most r δ eigenstrips containing eigenvalues greater than δ then: Φ( S ) & − α − δ − r δ X i =1 λ i (cid:18) ki (cid:19) ε i , (5) Here we are considering the constant function to be the th space. S is ( ε , . . . , ε r δ ) -pseudorandom. The interplay between δ and r δ , a notion we formalize through theintroduction of High-Dimensional Threshold Rank (HD-Threshold Rank), is thus crucial for understandingexpansion. Definition 1.12 (High-Dimensional Threshold Rank) . Let M be a linear operator over a vector space V with decomposition V = L i V i denoted D , where each V i is the span of some set of eigenvectors. Given δ ∈ R , the High-Dimensional Threshold Rank with respect to δ and D is: R δ ( M, D ) = (cid:12)(cid:12) { V i ∈ D : ∃ f ∈ V i , M f = λf, λ > δ } (cid:12)(cid:12) . In this work, D will always correspond to the eigenstrips given by applying Theorem 1.8 to the HD-Level-SetDecomposition, so we will write either R δ ( M ) or just R δ when M is clear from context. It is not hard to see that High-Dimensional Threshold Rank is a direct generalization of standard thresholdrank, which measures the total number of eigenvalues (with multiplicity) greater than some parameter δ .This is recovered by letting D be the standard spectral decomposition. For objects like HD-walks which haverelatively few eigenstrips, High-Dimensional Threshold Rank may be substantially smaller than thresholdrank. In particular, k -dimensional HD-walks have at most k + 1 eigenstrips despite having up to n O ( k ) eigenvalues.Viewing Equation (5) in terms of HD-Threshold Rank gives the following theorem regarding expansionof small, pseudorandom sets on HD-walks. Theorem 1.13 (Informal Theorem 5.3: Pseudorandom Sets Expand Near-Perfectly) . Given an HD-walk M on a sufficiently strong two-sided local-spectral expander and small constants α, δ, { ε i } R δ ( M ) i =1 > , we havethat ( ε , . . . , ε R δ ( M ) ) -pseudorandom sets of density α expand near-perfectly. One of the main utilities of this statement lies in its contrapositive: non-expanding sets in HD-walks arecorrelated with low-level links, where “low-level” is determined by the HD-Threshold Rank of the walk.
Corollary 1.14 (Informal Corollary 5.5: Non-expansion Implies Link Correlation) . Let ( X, Π) be a two-sided γ -local-spectral expander with γ sufficiently small and M a k -dimensional HD-walk on ( X, Π) . Then if S ⊂ X ( k ) is a set of density α and expansion: Φ( S ) < − α − O k ( γ ) − δ for some δ > and r = R δ/ ( M ) , S must be non-trivially correlated with some i -link for ≤ i ≤ r : ∃ ≤ i ≤ r, τ ∈ X ( i ) : E X τ [ S ] ≥ α + Ω k ( δ ) It turns out that this viewpoint is crucial both to understanding hardness [15] and algorithms [21] forUnique Games. In the following section we will provide an algorithm for Unique Games on HD-walks over J ( n, d ) whose soundness and runtime depend on a similar characterization of non-expansion, and as a resultdepend on the HD-Threshold Rank of the constraint graph.Since our results depend crucially on R δ , it is natural to ask the following: which HD-walks have smallHigh-Dimensional Threshold Rank? To answer this, we restrict our focus to the main classes of interestin the literature, the canonical and partial-swap walks. In both cases we see that the High-DimensionalThreshold Rank is controlled by the depth of the walk.
Definition 1.15 (Depth) . Depth is a parameter in [0 , measuring how far a canonical or partial-swap walkreaches into its underlying complex. We say the depth of N jk is jj + k since it walks through j out of j+k levelsof the complex. The depth of S jk differs slightly due to its restricted nature, and is defined as jk , the fractionof elements it swaps. We prove that the (stripped) eigenvalues of the canonical and partial-swap walks decay exponentially,with base dependent on depth. 8 heorem 1.16 (Informal Proposition 4.6, Corollary 4.8: Depth Controlled Exponential Eigenvalue Decay) . Let ( X, Π) be a two-sided γ -local-spectral expander with γ sufficiently small. Let M be a canonical or partial-swap walk of depth ≤ β ≤ . Then the eigenvalues corresponding to the eigenstrips of M decay exponentiallyfast: λ max i ≤ e − βi , where λ max i is the maximum eigenvalue of M in the eigenstrip corresponding to V ik promised by Theorem 1.8.Similarly, the HD-Threshold Rank of M is at most: R δ ( M ) ≤ ln (cid:0) δ (cid:1) β . We compute tight bounds on the exact spectra of these walks in Section 4, along with the general formof spectra for any HD-walk on a two-sided local-spectral expander.
We have seen how eigenvalue decay and the structure of the HD-Level-Set Decomposition inform the ex-pansion of locally-pseudorandom sets S ⊂ X ( k ) , but in the case of local-spectral expanders, we are oftenparticularly interested in understanding locally-structured sets as well, i.e. links. Such understanding iscrucial in employing the local-to-global paradigm that underlies almost all work on local-spectral expanders.In our case, we show that the local expansion of links is inversely controlled by the corresponding globaleigenvalues in the HD-Level-Set Decomposition. Theorem 1.17 (Informal Theorem 5.6: Local Expansion vs Global Spectrum) . Let ( X, Π) be a two-sided γ -local-spectral expander with γ ≤ − Ω( k ) , M a k -dimensional HD-walk on ( X, Π) , and λ i ( M ) the approximateeigenvalues corresponding to the HD-Level-Set Decomposition. Then for all ≤ i ≤ k and τ ∈ X ( i ) : Φ( X τ ) ≤ − λ i ( M ) + O k ( γ ) . When γ is sufficiently small, Theorem 1.8 implies that the λ i ( M ) correspond to the true global spectrum of M . Thus in a sense, Theorem 1.17 offers a local-to-global approach for situations where standard expansion-based techinques are obstructed by large global eigenvalues. Since the expansion of corresponding links issmall, this allows us to operate independently at a local level without losing too much from ignoring edgesbetween links. In the next section, we will see a specific application of this technique to unique games, whereTheorem 1.17 allows us to patch together local solutions on links to give a good global solution. Our motivation for studying the spectral structure and non-expansion of HD-walks stems from a simpleclass of 2-CSPs known as unique games, a central object of study in hardness-of-approximation since Khot’sintroduction of the Unique Games Conjecture (UGC) [24] nearly 20 years ago.
Definition 1.18 (Unique Games) . An instance I of unique games over alphabet Σ is a weighted graph G ( V, E ) , and set of permutations Π = { π ( u,v ) ∈ S Σ } ( u,v ) ∈ E . The value of I , val ( I ) , is the maximum fractionof satisfied constraints over all possible assignments Σ V : max x ∈ Σ V E ( u,v ) ∼ E [ x v = π uv ( x u )] , where edges are drawn corresponding to their weight. For an individual assignment x , we refer to thisexpectation as val I ( x ) . Informally, the UGC states that for sufficiently small constants ε, δ , there exists an alphabet size such thatdistinguishing between instances of unique games with value − ε and δ is NP-hard. A positive resolutionto the UGC would resolve the hardness-of-approximation of many important combinatorial optimizationproblems, including CSPs [25], vertex-cover [26], and a host of others [24, 27–31]. In this work, we willconsider affine unique games, a restriction stipulating that Σ is an additive group, and each permutation π ( u,v ) is an additive shift (i.e. π ( u,v ) ( x ) = x − a for some a ∈ Σ ). Since the UGC is equivalent to its restrictionon affine instances [27], this is not a significant loss in generality.9 .6.2 Algorithms for Unique Games on HD-walks Recently, Bafna, Barak, Kothari, Schramm, and Steurer [21] gave the first polynomial time algorithm foraffine unique games over the Johnson graphs. Their method, based upon the Sum of Squares semidefiniteprogramming hierarchy (a method for approximating polynomial optimization problems, see Section 6.1),relies on two core structural properties of the Johnson graphs:1. There exists a low-degree SoS proof that non-expanding sets are concentrated in links.2. There exists a parameter r = r ( ε ) such that:(a) The ( r + 1) -st largest (distinct) eigenvalue is small: λ r ≤ − Ω( ε ) (b) The expansion of any s -link for any s < r is small: ∀ τ ∈ X ( s ) , s < r : Φ( X τ ) ≤ O ( ε ) . This second, somewhat cumbersome parameter is found in [21] by direct computation on the Johnson graphs,and controls both the soundness and runtime of BBKSS’ algorithm. However, viewing the Johnson graph asan HD-walk, it becomes clear that this behavior actually stems from the local-to-global structure exhibitedby Theorem 1.17. As a result, we may replace BBKSS’ second parameter with HD-Threshold Rank andinformally extend their algorithm to affine unique games over HD-walks M satisfying1. There exists a low-degree SoS proof that non-expanding sets in M are concentrated in links.2. M has small HD-Threshold Rank.More formally, we focus on the case of HD-walks over the complete complex (equivalently, non-negativematrices in the Johnson scheme) where a slight variant of Theorem 1.11 proves that the first condition holds.As a result, we give an algorithm for affine unique games over such constraint graphs whose soundness andruntime depend on HD-Threshold Rank, calling back in a sense to Barak, Steurer, and Raghavendra’s [22]seminal work giving an algorithm for unique games in terms of the constraint graph’s standard thresholdrank. Theorem 1.19 (Informal Theorem 6.1: Playing Unique Games on HD-walks) . Let M be a k -dimensionalHD-walk on J ( n, d ) with n ≫ k , ε ∈ [0 , . , and r ( ε ) = R − ε ( M ) . Then if I is an instance of affineunique games over M with alphabet Σ satisfying:1. | Σ | ≥ Ω (cid:18) r (2 ε ) ( kr (2 ε ) ) ε (cid:19) ,2. val ( I ) ≥ − ε ,there exists an algorithm which outputs an Ω (cid:18) ε r ( ε ) ( kr ( ε ) ) (cid:19) -satisfying assignment in time | X ( k ) | poly (( kr ( ε ) ) , ε ) . It is worth noting that the High-Dimensional Threshold Rank of a k -dimensional HD-walk is at most k + 1 . Since k is often considered constant with respect to the number of vertices n , Theorem 1.19 gives apolynomial time algorithm for all constraint graphs in the Johnson scheme. To make concrete the finer-graincomplexity of the algorithm, we again focus on the canonical and partial-swap walks, which we note are bothbases of the Johnson Scheme. As before, the complexity depends on the depth of the walk. Corollary 1.20 (Corollary 6.2: Unique Games on Deep Walks) . Let I be an instance of affine uniquegames on a canonical or partial-swap walk of depth ≤ β ≤ over J ( n, d ) satisfying the conditions ofTheorem 1.19. Then there exists an algorithm that outputs an Ω (cid:18) β ε ( k ⌈ ε/β ⌉ ) (cid:19) -satisfying assignment in time | X ( k ) | poly (( k ⌈ ε/β ⌉ ) , /ε ) . In reality we will focus only on HD-walks over J ( n, d ) . Extending beyond this case is certainly possible, but requires thewalk to be symmetric (i.e. undirected), and to satisfy certain symmetry properties with respect to links which hold for naturalclasses like the canonical and partial-swap walks. β = Θ(1) (corresponding to walking Ω( k ) levels into thecomplex), our algorithm has soundness inverse polynomial in k , and running time exponential in poly ( k ) .On the other hand, for β ≤ O (1 /k ) (walks taking only a constant number of steps into the complex), thesoundness of our algorithm is inverse exponential, and the runtime doubly exponential in k . Finally, we briefly discuss the connection of our results to recent progress on the UGC and argue that ourframework opens an avenue for further progress. The resolution of the 2-2 Games Conjecture [15] hinged ona characterization of non-expanding sets on the Grassmann graph not dissimilar to what we have shown fortwo-sided local-spectral expanders. While we have focused above on HDX which are simplicial complexes,our work extends to a broader set of objects introduced by DDFH [23] called expanding posets. This classof objects includes expanding subsets of the Grassmann poset we call q -eposets. Definition 1.21 ( q -eposet) . A d -dimensional, weighted, pure q -simplicial complex X is a γ - q -eposet if forall ≤ i ≤ d − : (cid:13)(cid:13)(cid:13)(cid:13) D i +1 U i − q − q i +1 − I − q q i − q i +1 − U i − D i (cid:13)(cid:13)(cid:13)(cid:13) ≤ γ The particular choice of parameters for q -eposet stems from analysis of the Grassmann poset—in partic-ular the Grassmann poset is an O d (1 /q n ) - q -eposet (see [23] for details). Theorem 1.13 and Corollary 1.14extend naturally to HD-walks on q -eposets. We state the latter result here since it follows without too muchdifficulty from the arguments in this paper, but the full details (and further generalizations to expandingposets) will appear in our upcoming companion paper. Corollary 1.22 (Non-expansion in q -eposet) . Let ( X, Π) be a two-sided γ - q -eposet with γ sufficiently small, M a k -dimensional HD-walk on ( X, Π) . Then if S ⊂ X ( k ) is a set of density α and expansion: Φ( S ) < − α − O q,k ( γ ) − δ for some δ > and r = R δ/ ( M ) , S must be non-trivially correlated with some i -link for ≤ i ≤ r : ∃ ≤ i ≤ r, τ ∈ X ( i ) : E X τ [ S ] ≥ α + Ω q,k ( δ ) . Since the Grassmann graphs are simply partial-swap walks on the Grassmann poset , Corollary 1.22provides a direct connection to the proof of the 2-2 Games Conjecture [15]. However, due to the dependenceon k , the result is likely too weak to be used in this context as is—improving the bound to be independent of k remains an important open problem. However, we view our method’s generality and simplicity as evidencethat a deeper understanding of higher order random walks and the spectral structure of the HD-Level-SetDecomposition may be key to further progress on the UGC. The spectral structure of higher order random walks has seen significant study in recent years, startingwith the work of Kaufman and Oppenheim [12] who proved bounds on the spectra of N k on one-sided local-spectral expanders. Their result lead not only to the resolution of the Mihail-Vazirani conjecture [1], but to anumber of further breakthroughs in sampling algorithms via a small but consequential improvement on theirbound by Alev and Lau [3]. The spectral structure of N k on the stronger two-sided local-spectral expanderswas further studied by Dikstein, Dinur, Filmus, and Harsha (DDFH) [23] who introduced the HD-Level-SetDecomposition, and Kaufman and Oppenheim [6] who introduced a distinct approximate eigendecomposi-tion with the benefit of orthogonality (though this came at the cost of additional combinatorial complexity). This is the q -analog of a simplicial complex, and can be thought of as the downward closure of a set of d -dimensionalsubspaces of F nq . Seeing that they are HD-walks is non-trivial, and follows from the q -analog of work in [2].
11n recent work, Kaufman and Sharakanski [32] claim that these two decompositions are equivalent on suffi-ciently strong two-sided γ -local-spectral expanders, but their proof relies on [12, Theorem 5.10] which has anon-trivial error. Indeed, it is possible to construct arbitrarily strong two-sided local-spectral expanders forwhich the HD-Level-Set Decomposition is not orthogonal (see Appendix B), so their result cannot hold. Finally, Alev, Jeronimo, and Tulsiani showed that the HD-Level-Set Decomposition is an approximate eigen-decomposition (in a weaker sense than we require) for general HD-walks, a result we strengthen in Section 4,and generalize to expanding posets in the companion paper. For further information on these prior worksand their applications, the interested reader should see [33].
The study of unique games has played a central role in hardness-of-approximation since Khot’s [24] intro-duction of the Unique Games Conjecture. One line of work towards refuting the UGC focuses on buildingefficient algorithms for unique games for certain classes of constraint graphs based off of spectral or spectrally-related properties; these include works employing spectral expansion [34, 35], threshold rank [36, 22, 37, 38],hypercontractivity [39], and small-set-expansion or characterized non-expansion [21]. Our work continuesto expand this direction with polynomial-time algorithms for (affine) unique games over HD-walks and theintroduction of HD-Threshold Rank. On the other hand, recent work towards proving the UGC has focusedon characterizing non-expanding sets in structures such as the Grassmann [15–20] and Shortcode [19, 15]graphs. Our spectral framework based on HD-walks and the HD-Level-Set Decomposition provides a moregeneral method to approach this direction than previous Fourier analytic machinery which we believe maybe key to future progress on the UGC.Finally, it is worth noting a related, recent vein of work connecting high dimensional expansion, Sum ofSquares, and CSP-approximation. In particular, Alev, Jeronimo, and Tulsiani [2] recently showed that for k > , certain natural k -CSP’s on two-sided local-spectral expanders can be efficiently approximated by Sumof Squares. Conversely, Dinur, Filmus, Harsha, and Tulsiani [40] later used cosystolic expanders (a strongervariant) to build explicit instances of 3-XOR that are hard for SoS. While these works are not directly relatedto ours since the CSP’s they study do not encompass unique games, we see a similar pattern where highdimensional expanding structure is useful both for hardness of and algorithms for CSP-approximation. In this section we prove Theorem 1.8: the spectra of any operator with an approximate eigendecompositionis tightly concentrated around the decomposition’s approximate eigenvalues. We begin by formalizing whatwe mean by an approximate eigendecomposition.
Definition 2.1.
Let M be an operator over an inner product space V . We call V = V ⊕ . . . ⊕ V k a ( { λ i } ki =1 , { c i } ki =1 ) -approximate eigendecomposition if for all i and v i ∈ V i , the following holds: k M v i − λ i v i k ≤ c i k v i k . As long as the c i are sufficiently small, we prove each V i (loosely) corresponds to an eigenstrip, the spanof eigenvectors with eigenvalue closely concentrated around λ i . Theorem 2.2 (Eigenstripping) . Let M be a self-adjoint operator over an inner product space V , and V = V ⊕ . . . ⊕ V k a ( { λ i } ki =1 , { c i } ki =1 ) -approximate eigendecomposition. Let c max = max i { c i } , λ dif = min i,j {| λ i − λ j |} , and λ ratio = max i {| λ i |} λ / dif . Then as long as c max is sufficiently small: c max ≤ λ dif k , the spectra of M is concentrated around each λ i :Spec ( M ) ⊆ k [ i =1 [ λ i − e, λ i + e ] = I λ i , It is worth noting that the main results of [12, 32] are unaffected by this error, as an approximate version of [12, Theorem5.10] sufficient for these results continues to hold. here e = O (cid:16) k · λ ratio · c / max (cid:17) . It is worth noting that a version of Theorem 2.2 holds with no assumption on c max , but the assumptionsubstantially simplifies the bounds and is sufficient for our purposes. Before proving Theorem 2.2, we notea useful property of approximate eigendecompositions of self-adjoint operators: they are approximatelyorthogonal. Lemma 2.3.
Let M be a self-adjoint operator over an inner product space V . Further, let V = V ⊕ . . . ⊕ V k be a ( { λ i } ki =1 , { c i } ki =1 ) -approximate eigen-decomposition. Then for i = j , V i and V j are nearly orthogonal.That is, for any v i ∈ V i and v j ∈ V j : |h v i , v j i| ≤ c i + c j | λ i − λ j | k v i k k v j k . Proof.
This follows from the fact that M is self-adjoint, and V i and V j are approximate eigenspaces. Inparticular, notice that for any v i ∈ V i and v j ∈ V j we can bound the interval in which h M v i , v j i = h v i , M v j i lies by Cauchy-Schwarz: h M v i , v j i ∈ λ i h v i , v j i ± c i k v i k k v j k and h v i , M v j i ∈ λ j h v i , v j i ± c j k v i k k v j k . Since these terms are equal, the right-hand intervals must overlap. As a result we get: | ( λ i − λ j ) h v i , v j i| ≤ ( c i + c j ) k v i k k v j k , as desired.Using Lemma 2.3, we can modify [12, Theorem 5.9] to prove Theorem 2.2. Given an eigenvalue µ of M ,the idea is to find a probability distribution over [ k ] for which the expectation of | µ − λ i | is small, where i ∈ [ k ] is sampled from the aforementioned distribution. Proof.
The proof follows mostly along the lines of [12, Theorem 5.9], modifying where necessary due to lackof orthogonality. Let φ be an eigenvector of M with eigenvalue µ . Our goal is to prove the existence ofsome λ i such that | µ − λ i | is small. To do this, we appeal to an averaging argument. In particular, denotingthe component of φ in V i by φ i , we bound the expectation of | µ − λ i | over a distribution P φ given by the(normalized) squared norms k φ i k : E i ∼ P φ (cid:2) | µ − λ i | (cid:3) = 1 k P j =1 k φ j k k X i =1 | µ − λ i | k φ i k . (6)If we can upper bound this expectation by some value c , then by averaging there must exist λ i such that | µ − λ i | ≤ √ c , and thus the spectra of M must lie in strips λ i ± √ c . To upper bound Equation (6), considerthe result of pushing the outer summation inside the norm: k X i =1 | µ − λ i | k φ i k = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k X i =1 ( µ − λ i ) φ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − X ≤ i = j ≤ k ( µ − λ i )( µ − λ j ) h φ i , φ j i . (7)We will separately bound the two resulting terms, the former by the fact that the φ i are approximateeigenvectors, and the latter by their approximate orthogonality. We start with the former, which follows by13 simple application of Cauchy-Schwarz: (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k X i =1 ( µ − λ i ) φ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) µφ − k X i =1 λ i φ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M φ − k X i =1 λ i φ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k X i =1 ( M φ i − λ i φ i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ k k X i =1 k ( M φ i − λ i φ i ) k ≤ kc max k X i =1 k φ i k . The latter takes a bit more effort. Let λ max be max i {| λ i |} , then by Lemma 2.3 we have: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ≤ i = j ≤ k ( µ − λ i )( µ − λ j ) h φ i , φ j i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X ≤ i = j ≤ k | µ − λ i || µ − λ j | c i + c j | λ i − λ j | k φ i k k φ j k≤ c max λ − dif ( λ max + k M k ) k X i =1 k φ i k ! ≤ kc max λ − dif ( λ max + k M k ) k X i =1 k φ i k Since we’d like our bound to depend only on λ i and c i , we must further bound k M k which will follow similarlyfrom approximate orthogonality. Let v be a unit eigenvector with eigenvalue k M k and v i be v ’s componenton V i , then we have: k M k = k M v k = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k X i =1 M v i − λ i v i + λ i v i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ k X i =1 ( λ i + c i ) k v i k≤ ( λ max + c max ) k X i =1 k v i k≤ ( λ max + c max ) vuut k k X i =1 k v i k ≤ ( λ max + c max ) √ k. c max : k X i =1 k v i k = k v k + X ≤ i = j ≤ k h v i , v j i≤ c max λ dif X ≤ i = j ≤ k k v i k k v j k≤ c max λ dif k X i =1 k v i k ! ≤ kc max λ dif k X i =1 k v i k ≤ k X i =1 k v i k . Together, these bounds imply the existence of some λ i ′ such that: | µ − λ i ′ | ≤ s kc max (cid:18) c max + 2 λ − dif (cid:16) λ max + ( λ max + c max ) √ k (cid:17) (cid:19) , which implies the desired result when accounting for our assumption on c max .Notice that if c max is sufficiently small, the intervals I λ i are disjoint. As a result, each V i corresponds toan eigenstrip W i : W i = Span { φ : M φ = µφ, µ ∈ I λ i } . The approximate eigenspaces V i are closely related to the resulting eigenstrips. Indeed, it is possible to showthat most of the weight of a function in V i must lie on W i , though we will not need this result in whatfollows. Previous works [12, 32] make stronger claims for the specific case of the HD-Level-Set Decomposition,most notably that V i and W i are in fact equivalent on sufficiently strong two-sided local-spectral expanders.Unfortunately, these results are based off of [12, Theorem 5.10], whose proof has a non-trivial error wediscuss further in Appendix B. Indeed, were their proof correct, it would imply (due to the generality oftheir argument) that V i = W i for any approximate eigendecomposition. However, it is easy to see thiscannot be the case by considering a diagonal × matrix with an approximate eigendecomposition givenby a slight rotation of the standard basis vectors in R . Now that we have seen how approximate eigendecompositions relate to an operator’s spectrum, we takea closer look at the combinatorial structure of the HD-Level-Set Decomposition itself, characterizing howfunctions project onto each space. In particular, we prove in this section a generalization of Theorem 1.11 toarbitrary functions in C k . To do this, we will first need to extend our definition of pseudorandomness fromsets (boolean functions) to arbitrary functions. Definition 3.1 (Pseudorandom) . A function f ∈ C k is ( ε , . . . , ε ℓ ) -pseudorandom if its local expectation isclose to its global expectation. That is if, for all ≤ i ≤ ℓ , we have: ∀ s ∈ X ( i ) : (cid:12)(cid:12)(cid:12)(cid:12) E X s [ f ] − E [ f ] (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε i . Our analysis of the projection of such a function onto the HD-Level-Set Decomposition is based uponGarland’s method [41], a way of decomposing global information to local information across links. As aresult, our (initial) analysis will assume f satisfies a useful local-consistency property.15 efinition 3.2. Let ( X, Π) be a weighted, pure simplicial complex. We say a function f ∈ C k has ℓ -localconstant sign if:1. E [ f ] = 0 ,2. ∀ s ∈ X ( ℓ ) s.t. E X s [ f ] = 0 : sign (cid:18) E X s [ f ] (cid:19) = sign ( E [ f ]) . Note that any function may be shifted by some constant to have locally constant sign—this will allowus to generalize our result to all functions. We now state the generalized version of Theorem 1.11, whichshows that pseudorandom functions have small projection onto low levels of the HD-Level-Set Decomposition(which, moreover, correspond to the worst eigenvalues for most walks).
Theorem 3.3.
Let ( X, Π) be a γ -local-spectral expander with γ ≤ − Ω( k ) and let f ∈ C k have HD-Level-SetDecomposition f = f + . . . + f k . If f is ( ε , . . . , ε ℓ ) -pseudorandom, then for any a ∈ R + and all ≤ i ≤ ℓ : h f, f i i ≤ (1 + c ( k ) γ ) a + 1 a (cid:18) ki (cid:19) ε i + ac ( k ) γ k f k , where c ( k ) ≤ O ( k ) , and if f has locally constant sign then: h f, f i i ≤ (1 + c ( k ) γ ) a + 1 a (cid:18) ki (cid:19) ε i | E [ f ] | + ac ( k ) γ k f k . Finally, if the HD-Level-Set Decomposition is orthogonal, we can be rid of a . For arbitrary functions wehave: h f, f i i ≤ (1 + c ( k, i ) γ ) (cid:18) ki (cid:19) ε i , where c ( k, i ) = O (cid:16) k (cid:0) ki (cid:1)(cid:17) , and if f has locally constant sign then: h f, f i i ≤ (1 + c ( k, i ) γ ) (cid:18) ki (cid:19) ε i | E [ f ] | . It’s worth noting that due the the approximate orthogonality of the HD-Level-Set Decomposition (seeLemma 3.6), this bound holds in absolute value as well since h f, f i i cannot be too negative. It is alsoworth noting that in the case that the decomposition is orthogonal, we can improve the dependence on γ by pushing exponential dependence on k to the second order γ term via more careful analysis of errorpropagation. However since the analysis is complicated and only gives a substantial improvement for a smallrange of relevant γ , we relegate such discussion to Appendix A.In dealing with expansion, we will mainly be interested in boolean-valued functions, which always havelocally-constant sign and satisfy h f, f i = E [ f ] . Thus in the boolean case, setting a = 1 / √ γ implies Theo-rem 1.11: h f, f i i ≤ (cid:16) O ( k ) √ γ (cid:17) (cid:18) ki (cid:19) ε i E [ f ] . This is particularly useful when considering expansion (which we recall may be written as − E [ f ] h f, M f i )to cancel the normalization by E [ f ] –indeed, in Section 5 we will show this gives a tight characterization.Finally, it should be noted that in the case f is non-negative, the absolute value can be removed from thedefinition of pseudorandomness in this argument, which will later allow us to show that non-expanding setsare locally denser than expected.We prove Theorem 3.3 through three main steps. First and foremost, we show how weight on V ℓk (the ℓ -thHD-Level-Set Decomposition space) implies an imbalance between local and global expectation at level X ( ℓ ) for any function f with locally constant sign. Second, we analyze how this generalizes to general functionsvia a simple constant shift. Finally, we simplify the bound by analyzing the relation between h f, f ℓ i and k g ℓ k based on the machinery of DDFH [23]. We start with the core of our result: any non-trivial projection onto V ℓk implies a disparity between the local and global behaviour of f .16 roposition 3.4. Let ( X, Π) be a two-sided γ -local-spectral expander with γ < /k , and let f ∈ C k be afunction on k -faces with HD-Level-Set Decomposition f = f + . . . + f k . Then for all < ℓ ≤ k , if f has ℓ -local constant sign, there exists a face s ∈ X ( ℓ ) such that the difference between local and global expectationis lower bounded by the weight of f ’s projection onto V ℓk : (cid:12)(cid:12)(cid:12)(cid:12) E X s [ f ] − E [ f ] (cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h f, f ℓ i k g ℓ k E [ f ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where we recall g ℓ ∈ H ℓ satisfies f ℓ = U kℓ g ℓ . If f is non-negative, the inequality holds without absolute valuesigns.Proof. We begin by examining the squared norm of E ℓf , the vector of link expectations over X ( ℓ ) : ∀ s ∈ X ( ℓ ) : E ℓf ( s ) = E X s [ f ] . Let Π k ( X s ) be shorthand for P t ∈ X s Π k ( t ) (i.e. the normalization factor for the above restricted expectation).Since f has locally constant sign, we may rewrite the squared norm of E ℓf as an expectation over a relateddistribution P ℓ : E [ f ] h E ℓf , E ℓf i = X s ∈ X ( ℓ ) Π ℓ ( s ) E [ f ] X t ∈ X s Π k ( t ) f ( t )Π k ( X s ) ! E ℓf ( s ) (8) = X s ∈ X ( ℓ ) E [ f ] X t ∈ X s Π k ( t ) f ( t ) (cid:0) kℓ (cid:1) ! E ℓf ( s ) (9) = E P ℓ [ E ℓf ] , (10)where we have used the fact that Π k ( X s ) = (cid:0) kℓ (cid:1) Π ℓ ( s ) by Equation (1). To understand P ℓ ( s ) more intuitively,consider the special case when f is non-negative. Here, Π and f induce a distribution P k over X ( k ) , where P k ( t ) = Π k ( t ) f ( t ) E [ f ] .P k then induces the distribution P ℓ on X ( ℓ ) via the following process: draw a face t ∈ X ( k ) from P k , andthen choose a ℓ -face s ⊂ t uniformly at random. Replacing the non-negativity of f with the conditions inthe theorem statement still leaves P ℓ ( s ) a valid distribution, albeit one with a less intuitive description.If we can lower bound h E ℓf , E ℓf i , Equation (10) then implies a lower bound on max s ( | E ℓf ( s ) | ) by averaging.In particular, it is enough to show that: h E ℓf , E ℓf i ≥ h f, f ℓ i k g ℓ k + E [ f ] . (11)To see why this is sufficient, assume Equation (11) and that E [ f ] > (the negative case follows similarly).Then we have: E P ℓ [ E ℓf ] ≥ h f, f ℓ i k g ℓ k E [ f ] + E [ f ] , and further that by averaging there exists an ℓ -face s ∈ X ( ℓ ) satisfying the desired property: E X s [ f ] − E [ f ] ≥ h f, f ℓ i k g ℓ k E [ f ] .
17o show Equation (11), consider splitting the norm into two parts: (cid:10) E ℓf , E ℓf (cid:11) = D g ℓ , E ℓf E h g ℓ , g ℓ i + E E ℓf − h g ℓ , E ℓf ih g ℓ , g ℓ i g ℓ ! (12) ≥ D g ℓ , E ℓf E h g ℓ , g ℓ i + E " E ℓf − h g ℓ , E ℓf ih g ℓ , g ℓ i g ℓ (13) = D g ℓ , E ℓf E h g ℓ , g ℓ i + E [ f ] , (14)where the last step follows from recalling that g ℓ is in the kernel of the down operator (and hence E [ g ℓ ] = 0 ),and E [ E ℓf ] = E [ f ] . To relate this to the projected weight h f, f ℓ i , we use Garland’s method [41] to show thatthe numerator of the left-hand term is exactly h f, f ℓ i . For any ≤ i ≤ k and u ∈ X ( i ) , let y u ∈ C k be theindicator for the link X u : y u ( t ) = ( t ⊃ u ) . Then: (cid:10) g ℓ , E ℓf (cid:11) = X s ∈ X ( ℓ ) Π ℓ ( s ) g ℓ ( s ) E X s [ f ]= X s ∈ X ( ℓ ) g ℓ ( s ) X t ∈ X s Π k ( t ) (cid:0) kℓ (cid:1) f ( t )= X s ∈ X ( ℓ ) g ℓ ( s ) X t ∈ X ( k ) Π k ( t ) (cid:0) kℓ (cid:1) f ( t ) y s ( t )= * X t ∈ X ( k ) f ( t ) y t , X s ∈ X ( ℓ ) g ℓ ( s ) U kℓ y s + = h f, U kℓ g ℓ i = h f, f ℓ i Plugging this into Equation (14) completes the proof.As an immediate corollary, we can upper bound the projection of any function onto the HD-Level-SetDecomposition in terms of its pseudorandomness.
Corollary 3.5.
Let ( X, Π) be a γ -local-spectral expander, and let f ∈ C k be a function on k -faces withHD-Level-Set Decomposition f = f + . . . + f k . If f is ( ε , . . . , ε ℓ ) -pseudorandom, then for all ≤ i ≤ ℓ : h f, f i i ≤ ε i k g i k . Further, if f has i -local constant sign, then: h f, f i i ≤ ε i | E [ f ] | k g i k . Proof.
The latter bound follows immediately from Proposition 3.4. For the former, assume for simplicity that E [ f ] ≥ (the negative case follows from a similar argument). Notice that as long as ε i = 0 , f + ( ε i − E [ f ]) has positive expectation over all i links and non-zero expectation, allowing us to apply Proposition 3.4. Notefurther that in the HD-Level-Set Decomposition, f corresponds to the constant part of f , and we may thussimilarly decompose f + ( ε i − E [ f ]) as: f + ( ε i − E [ f ]) = f ′ + f + . . . + f k , where f ′ = f + ( ε i − E [ f ]) . Applying Proposition 3.4 then gives: h f + ( ε i − E [ f ]) , f i i ≤ ε i E [ f + ( ε i − E [ f ]) ] k g i k . Noting that, for i > , f i is orthogonal to then gives the desired result. We are left to deal with the casethat ε i = 0 , which follows from a limiting argument applying the above to any ε > .18 priori, it is not clear that Corollary 3.5 is particularly useful due to its dependence on k g i k . In fact, k g i k is closely related to both k f i k and h f, f i i on two-sided local-spectral expanders, a fact proved byDikstein, Dinur, Filmus, and Harsha [23]. Lemma 3.6 (Lemmas 8.10, 8.13, Theorem 4.6 [23]) . Let ( X, Π) be a d -dimensional γ -local-spectral expanderwith γ < /d , f ∈ C k a function with HD-Level-Set Decomposition f + . . . + f k . Then for all ≤ ℓ ≤ k ≤ d : k f ℓ k = 1 (cid:0) kℓ (cid:1) (1 ± c ( k, ℓ ) γ ) k g ℓ k , where c ( k, ℓ ) = O ( k (cid:0) kℓ (cid:1) ) . Further for all ≤ i = j ≤ k : h f i , f j i ≤ O ( k ) k f i k k f j k , and if γ ≤ − Ω( k ) , then: k f ℓ k ≤ k f k − O ( k ) γ . The version of this result appearing in [23] does not have explicit coefficients, but they follow from directcomputation. With these in hand, the proof of Theorem 3.3 amounts to a few lines of computation.
Proof of Theorem 3.3.
For both the orthogonal and non-orthogonal cases we prove only the result for arbi-trary functions. The result for i -local constant sign follows from exactly the same arguments. We start withthe case where the HD-Level-Set Decomposition is orthogonal. Here the result is almost immediate fromcombining Corollary 3.5 and Lemma 3.6: h f, f i i ≤ ε i k g i k h f, f i i = ε i k g i k k f i k ≤ ε i (cid:0) ki (cid:1) (1 − c ( k, i ) γ ) where c ( k, i ) = O ( k (cid:0) ki (cid:1) ) . Since we assume γ ≤ − Ω( k ) , Taylor expanding − c ( k,i ) γ ) gives the desiredresult.When the HD-Level-Set Decomposition is not orthogonal, we cannot cancel terms on the right andleft-hand side, but still have the relation: h f, f i i ≤ ε i (cid:0) ki (cid:1) (1 − c ( k, i ) γ ) k f i k = ε i (cid:0) ki (cid:1) (1 − c ( k, i ) γ ) h f, f i i − ε i (cid:0) ki (cid:1) (1 − c ( k, i ) γ ) X j = i h f i , f j i≤ (1 + c ( k, i ) γ ) ε i (cid:18) ki (cid:19) (cid:16) h f, f i i + c ( k, i ) γ k f k (cid:17) , where c ( k, i ) , c ( k, i ) ≤ O ( k ) , and the last step follows from the approximate orthogonality given byLemma 3.6 and a Taylor expansion. To get the final result, notice that we are done if h f, f i i ≤ a O ( k ) γ k f k .Otherwise, we can write: h f, f i i ≤ (1 + c ( k, i ) γ ) ε i (cid:18) ki (cid:19) (cid:18) h f, f i i + 1 a h f, f i i (cid:19) , which implies the desired result. We now show that the HD-Level-Set Decomposition is an approximate eigendecomposition for any HD-Walk,and thus by Theorem 2.2 corresponds to a decomposition of the walk’s spectrum into tightly concentrated19igenstrips. As a result, we give explicit bounds on the spectra of HD-walks, paying special attention to thecanonical and partial-swap walks. Finally, we show that the approximate eigenvalues (and thus the values intheir corresponding eigenstrips) of the HD-Level-Set Decomposition decrease monotonically for a broad classof HD-Walks we call complete walks which, to our knowledge, encompass all walks used in the literature. Aswe will see in the following section, such decay is crucial for understanding edge expansion.To start, we recall the definition of pure and HD-walks along with introducing some useful notation.
Definition 4.1 ( k -Dimensional Pure Walk) . Given a weighted, simplicial complex ( X, Π) , a k -dimensionalpure walk Y : C k → C k on ( X, Π) is a composition: Y = Z h ( Y ) ◦ · · · ◦ Z , where each Z i is a copy of D or U , and h ( Y ) is the height of the walk, measuring the total number of down(or up) operators. Definition 4.2 ( k -Dimensional HD-Walk) . Given a weighted, simplicial complex ( X, Π) , a k -dimensionalHD-walk on ( X, Π) is an affine combination of pure walks M = X Y ∈Y α Y Y which gives a valid walk on ( X, Π) (i.e. has non-negative transition probabilities). We say the height of M , h ( M ) , is the maximal height of any Y with a non-zero coefficient, and say the weight of M , w ( M ) , is theone norm of the α Y (namely, w ( M ) = P | α Y | ). Our proofs in this section rely mainly on a useful observation of [23], who show that the up and downoperators on two-sided γ -local-spectral expanders satisfy the following relation: (cid:13)(cid:13)(cid:13)(cid:13) D i +1 U i − i + 1 I − ii + 1 U i − D i (cid:13)(cid:13)(cid:13)(cid:13) ≤ γ. (15)This fact leads to a particularly useful structural lemma showing the effect of flipping D through multiple U operators. Lemma 4.3 (Claim 8.8 [23]) . Let ( X, Π) be a d -dimensional γ -local-spectral expander. Then for all j < k Let ( X, Π) be a two-sided γ -local-spectral expander with γ ≤ − Ω( k ) and Y : C k → C k apure walk: Y = Z h ( Y ) ◦ · · · ◦ Z . Let i ≤ . . . ≤ i h ( Y ) denote the h ( Y ) indices at which Z i is a down operator. Then for all ≤ ℓ ≤ k, f ∈ V ℓk : (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Y f − h ( Y ) Y s =1 (cid:18) − ℓ max { ℓ, i s − s + k + 1 } (cid:19) f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ O (cid:18) γh ( Y )( k + h ( Y )) (cid:18) kℓ (cid:19) k f k (cid:19) . roof. We prove a slightly stronger statement to simplify the induction. For b > , let Y bj : C ℓ → C ℓ + b denote an unbalanced walk with j down operators, and j + b up operators. If Y bj has down operators inpositions i ≤ . . . ≤ i j and g ℓ ∈ H ℓ , we claim: (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Y bj g ℓ − j Y s =1 (cid:18) − ℓ max { ℓ, i s − s + ℓ + 1 } (cid:19) U b + ℓℓ g ℓ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ γj ( b + j ) k g ℓ k . (16)Notice that since f ∈ V ℓk may be written as U kℓ g ℓ for g ℓ ∈ H ℓ , then we may write Y f as Y k − ℓh ( Y ) g ℓ where Y k − ℓh ( Y ) has down operators in positions i + k − ℓ ≤ . . . ≤ i j + k − ℓ . Combining Equation (16) with Lemma 3.6then implies the result.We prove Equation (16) by induction. The base case j = 0 is trivial. Assume the inductive hypothesisholds for all Y bi , i < j . Notice first that if i = 1 , we are done since g ℓ ∈ H ℓ , and j Y s =1 (cid:18) − ℓ max { ℓ, i s − s + ℓ + 1 } (cid:19) Y b g ℓ = 0 , as i s − s + ℓ + 1 = ℓ for s = 1 . Otherwise, it must be the case that one or more copies of the up operatorappear before the first down operator, and we may therefore apply Lemma 4.3 to get: Y bj g ℓ = (cid:18) i − i + ℓ − (cid:19) Y bj − g ℓ + Γ g ℓ , where we can (loosely) bound the spectral norm of Γ by k Γ k ≤ ( b + j ) γ since at worst the first down operator D passes through b + j up operators. By the form of Lemma 4.3, Y bj − has down operators at indices i − ≤ . . . ≤ i j − . Then by the fact that i + ℓ − > ℓ and the inductivehypothesis: Y bj g ℓ = i − { ℓ, i + ℓ − } j − Y s =1 i s +1 − s − { ℓ, i s +1 − s + ℓ − } ! Y b g ℓ + i − i + ℓ − h + Γ g ℓ = i − { ℓ, i + ℓ − } j Y s =2 i s − s + 1max { ℓ, i s − s + ℓ + 1 } ! Y b g ℓ + i − i + ℓ − h + Γ g ℓ = j Y s =1 i s − s + 1max { ℓ, i s − s + ℓ + 1 } ! Y b g ℓ + i − i + ℓ − h + Γ g ℓ , where k h k ≤ γ ( j − b + j − k g ℓ k and we have used the (vacuous) fact that max { ℓ, i + ℓ − } = i + ℓ − .Finally, we can bound the norm of the right-hand error term by: (cid:13)(cid:13)(cid:13)(cid:13) i − i + ℓ − h + Γ g ℓ (cid:13)(cid:13)(cid:13)(cid:13) ≤ k h k + k Γ k k g ℓ k≤ ( j − b + j − k g ℓ k + ( b + j ) k g ℓ k≤ j ( b + j ) k g ℓ k as desired.Since HD-walks are simply affine combinations of pure walks, the triangle inequality immediately impliesthe result carries over to this more general setting. Corollary 4.5. Let ( X, Π) be a two-sided γ -local-spectral expander with γ ≤ − Ω( k ) and M = P i α i Y i a k -dimensional HD-walk on ( X, Π) . Then for all ≤ ℓ ≤ k, f ∈ V ℓk : k M f − λ ℓ ( M ) f k ≤ O (cid:18) γw ( M ) h ( M )( k + h ( M )) (cid:18) kℓ (cid:19) k f k (cid:19) , here λ ℓ ( M ) = X α i λ ℓ ( Y i ) and λ ℓ ( Y i ) is the approximate eigenvalue of Y i given in Proposition 4.4. It is worth noting that the resulting approximate eigenvalues in Corollary 4.5 are exactly the eigenvaluesof M when considered on a sequentially differential poset with ~δ i = i/ ( i + 1) . We discuss this generalizationin more depth and give tighter bounds on the approximate spectra in our upcoming companion paper. Itshould be noted that this result is similar to one appearing in [2], where a weaker notion of approximateeigenspaces based on the quadratic form h f, M f i is analyzed. Plugging Corollary 4.5 into Theorem 2.2, weimmediately get that for small enough γ the true spectra of HD-walks lie in strips around each λ i ( M ) , andthus that that the approximate eigenvalues of the HD-Level-Set Decomposition and the spectra of HD-walksare essentially interchangeable.For concreteness, we now turn our attention to computing the approximate eigenvalues (and thereby thetrue spectra) of the canonical and swap walks. Proposition 4.6 (Spectrum of Canonical Walks) . Let ( X, Π) be a d -dimensional γ -local-spectral expanderwith γ satisfying γ ≤ − Ω( k + j ) , k + j ≤ d , and f ℓ ∈ V ℓk . Then: (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N jk f ℓ − (cid:0) kℓ (cid:1)(cid:0) k + jℓ (cid:1) f ℓ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ c ( k, ℓ, j ) k f ℓ k , where c ( k, ℓ, j ) = O (cid:16) γj ( j + k ) (cid:0) kℓ (cid:1)(cid:17) . Moreover:Spec ( N jk ) = { } ∪ k [ j =1 " (cid:0) kℓ (cid:1)(cid:0) k + jℓ (cid:1) ± O ( j + k ) √ γ . Proof. By Proposition 4.4, N jk is an (cid:0) { λ ℓ } kℓ =0 , { c ( k, ℓ, j ) } kℓ =0 (cid:1) -approximate eigendecomposition for λ ℓ = j Y s =1 (cid:18) − ℓ max { k − s + i s + 1 , ℓ } (cid:19) , where i ≤ . . . ≤ i s denote the indices of down operators. By the definition of N jk we have i s = j + s , andtherefore λ ℓ = j Y s =1 (cid:18) − ℓk − s + j + 1 (cid:19) = (cid:0) kℓ (cid:1)(cid:0) k + jℓ (cid:1) as desired. The bounds on Spec ( N jk ) follow immediately from plugging the above into Theorem 2.2.A priori, it is not obvious how to bound the spectra of the partial-swap walks, or indeed even that theyare HD-walks. However, Alev, Jeronimo, and Tulsiani [2] proved that partial-swap walks may be written asa alternating hypergeometric sum of canonical walks. Proposition 4.7 (Corollary 4.13 [2]) . Let ( X, Π) be a two-sided γ -local-spectral expander with γ < /k .Then for ≤ j ≤ k : S jk = 1 (cid:0) kk − j (cid:1) j X i =0 ( − j − i (cid:18) ji (cid:19)(cid:18) k + ii (cid:19) N ik . As a result, we can use Proposition 4.6 to bound their approximate eigenvalues and true spectrum. Corollary 4.8. Let X be d -dimensional two-sided γ -local-spectral expander, γ < − Ω( k ) , k + j ≤ d , and f ℓ ∈ V ℓk . Then: (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) S jk f ℓ − (cid:0) k − jℓ (cid:1)(cid:0) kℓ (cid:1) f ℓ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ c ( k ) k f ℓ k , here c ( k ) = γ O ( k ) . Moreover, Spec ( S jk ) = { } ∪ k [ j =1 " (cid:0) k − jℓ (cid:1)(cid:0) kℓ (cid:1) ± O ( k ) √ γ . Proof. By Corollary 4.5, L kℓ =0 V ℓk is a (cid:0) { λ ℓ } kℓ =0 , { c ′ ( k, ℓ, j ) } kℓ =0 (cid:1) -approximate eigendecomposition for S jk with λ ℓ = 1 (cid:0) kk − j (cid:1) j X i =0 ( − j − i (cid:18) ji (cid:19)(cid:18) k + ii (cid:19) λ ℓ ( N ik )= 1 (cid:0) kk − j (cid:1) j X i =0 ( − j − i (cid:18) ji (cid:19)(cid:18) k + ii (cid:19) (cid:0) kℓ (cid:1)(cid:0) k + iℓ (cid:1) = 1 (cid:0) kk − j (cid:1) j X i =0 ( − j − i (cid:18) ji (cid:19)(cid:18) k − ℓ + ii (cid:19) = (cid:0) k − ℓj (cid:1)(cid:0) kk − j (cid:1) = (cid:0) k − jℓ (cid:1)(cid:0) kℓ (cid:1) , and c ′ ( k, ℓ, j ) = γ O ( k ) . This latter fact follows from noting that k ~α k = j X i =0 (cid:18) ji (cid:19)(cid:18) k + ii (cid:19) ≤ j + k where ~α consists of the hypergeometric coefficients of Proposition 4.7. The bounds on Spec( S jk ) then followfrom Theorem 2.2.Together, Proposition 4.6 and Corollary 4.8 prove Theorem 1.16 (assuming γ is sufficiently small).In the introduction, we mentioned that for a broad class of HD-walks, the approximate eigenvalues of theHD-Level-Set Decomposition exhibit a further property key for understanding expansion: monotonic decay.We now formally define this class and prove the desired property. Definition 4.9 (Complete HD-Walk) . Let ( X, Π) be a weighted, pure simplicial complex and M = P Y ∈Y α Y Y an HD-walk on ( X, Π) . We call M complete if for all n ∈ N there exist n > n and d such that P Y ∈Y α Y Y isalso an HD-walk when taken to be over J ( n , d ) . To our knowledge, all walks considered in the literature (pure, canonical, partial-swap) are complete.We can prove that the eigenstrips of complete HD-walks corresponding to the HD-Level-Set Decompositionexhibit eigenvalue decay by noting that the approximate eigenvalues of Corollary 4.5 are independent of theunderlying complex. Proposition 4.10. Let ( X, Π) be a two-sided γ -local-spectral expander, M = P Y ∈Y α Y Y a complete HD-walkover ( X, Π) , and γ small enough to apply the conditions of Theorem 2.2. Then for all ≤ i < j ≤ k , λ i ( M ) ≥ λ j ( M ) roof. The proof follows from two observations. First, recall from Corollary 4.5 that λ i ( M ) is independentof the underlying complex. Second, any HD-walk on the complete complex can be written as a non-negativesum of partial-swap walks, which satisfy the monotonic decrease property. Let n ∈ N be any parameter suchthat applying P Y ∈Y α Y Y to J ( n, d ) results in a valid walk (i.e. a non-negative matrix). By the symmetryof J ( n, d ) , the transition probabilities of this walk depends only on size of intersection, and it may thus bewritten as some convex combination of partial-swap walks: M = X Y ∈Y α Y Y = k X i =0 β i S ik . Since these walks are equivalent over J ( n, d ) , their spectra must match. Then by Theorem 2.2, it must bethe case that for every ≤ ℓ ≤ k and n sufficiently large, the intersection of P Y ∈Y α Y λ ℓ ( Y ) ± O (1 /n ) and P Y ∈Y β i λ ℓ ( S ik ) ± O (1 /n ) is non-empty. Since we may take n arbitrarily large, this implies the two quantitiesare in fact equivalent. Finally, by Corollary 4.8 λ ℓ ( S ik ) decreases monotonically in ℓ for all i , which impliesthat the λ i ( M ) = P α Y λ i ( Y ) = P βλ i ( S jk ) decrease monotonically as desired. In this section we prove Theorem 1.13, Corollary 1.14, and Theorem 1.17, capturing the tradeoffs between lo-cal structure, HD-Threshold Rank, expansion, and non-expansion. We first recall the definitions of expansionand HD-Threshold Rank from the introduction specified to HD-walks for simplicity. Definition 5.1 (Weighted Edge Expansion) . Given a weighted simplicial complex ( X, Π) , a k -dimensionalHD-Walk M over ( X, Π) , and a subset S ⊂ X ( k ) , the weighted edge expansion of S is Φ( S ) = E v ∼ Π k | S [ M ( v, X ( k ) \ S )] , where M ( v, X ( k ) \ S ) = X y ∈ X ( k ) \ S M ( v, y ) and M ( v, y ) is the transition probability from v to y . We will control the expansion of sets in HD-walks partially through the walk’s HD-Threshold Rank, ameasure of how many eigenstrips contain large eigenvalues. Definition 5.2 (High-Dimensional Threshold Rank) . Let ( X, Π) be a two-sided γ -local-spectral expander and M a k -dimensional HD-walk with γ small enough that the HD-Level-Set Decomposition has a correspondingdecomposition of disjoint eigenstrips C k = L W ik . The HD-Threshold-Rank of M with respect to δ is thenumber of strips containing an eigenvector with eigenvalue at least δ : R δ ( M ) = |{ W ik : ∃ f ∈ V i , M f = λf, λ > δ }| . We often write just R δ when M is clear from context. It should be noted that when the HD-Level-Set Decomposition has spaces with the same approximateeigenvalue, their corresponding eigenstrips technically must be merged. However, since this detail has noeffect on our arguments, we ignore it in what follows. We now show how to express the expansion of a set S ⊂ X ( k ) with respect to an HD-walk M in terms of the pseudorandomness of S and the HD-ThresholdRank of M . Theorem 5.3. Let ( X, Π) be a two-sided γ -local-spectral expander, M a k -dimensional, complete HD-walk,and let γ be small enough that the eigenstrip intervals of Theorem 2.2 are disjoint. For any δ > , let r = R δ ( M ) − . Then the expansion of a set S ⊂ X ( k ) of density α is at least: Φ( S ) ≥ − α − δ − c √ γ − r X i =1 λ i ( M ) (cid:18) ki (cid:19) ε i , here λ i ( M ) is the approximate eigenvalue given by Corollary 4.5, S is ( ε , . . . , ε r ) -pseudorandom, and c ≤ w ( M ) h ( M ) O ( k ) . Further, if the HD-Level-Set Decomposition is orthogonal, then: Φ( S ) ≥ − α − δ − cγ − r X i =1 λ i ( M ) (cid:18) ki (cid:19) ε i . Proof. Recall that the expansion of S may be written as: Φ( S ) = 1 − E [ S ] h S , M S i . Decomposing S = S, + . . . + S,k by the HD-Level-Set Decomposition, we have: Φ( S ) = 1 − E [ S ] − E [ S ] k X i =1 h S , M S,i i = 1 − E [ S ] − E [ S ] k X i =1 λ i ( M ) h S , S,i i + 1 E [ S ] k X i =1 h S , Γ i i where by Corollary 4.5 k Γ i k ≤ O (cid:16) w ( M ) h ( M )( h ( M ) + k ) (cid:0) ki (cid:1) γ k S,i k (cid:17) . Using Cauchy-Schwarz and the factthat k S,i k ≤ (1 + 2 O ( k ) γ ) k S k we can simplify this to Φ( S ) ≥ − E [ S ] − E [ S ] k X i =1 λ i ( M ) h S , S,i i − eγ, where e ≤ w ( M ) h ( M ) O ( k ) . Since M is a complete walk, we know the λ i ( M ) decrease monotonically andas long as γ is sufficiently small, correspond to the eigenvalues in strip W i as well. Thus we may write: Φ( S ) ≥ − eγ − E [ S ] − E [ S ] r X i =1 λ i ( M ) h S , S,i i − δ E [ S ] k X i = R δ h S , S,i i = 1 − e γ − E [ S ] − E [ S ] r X i =1 λ i ( M ) h S , S,i i − δ, where e ≤ w ( M ) h ( M ) O ( k ) and error from the rightmost term has been absorbed into e . Finally, applyingthe corresponding version of Theorem 3.3 (with a = O (1 / √ γ ) ) depending on whether the HD-Level-SetDecomposition is orthogonal and combining e γ with the resulting error term gives the desired results.Theorem 5.3 gives a tight characterization of expansion in the regime of linear dependence on ε i . Inparticular, we can find a copy of J ( m, k, t ) ⊂ J ( n, k, t ) whose expansion is arbitrarily close to the bound ofTheorem 5.3 by taking n ≫ m . Proposition 5.4. Let X = J ( n, d ) be the Johnson complex, k − t ≤ d , m | n , and B m be the the set of all k -faces (cid:0) [ n/m ] k (cid:1) . Then for any t , Theorem 5.3 is tight for S k − tk as n, m → ∞ .Proof. It is not hard to see by direct computation that the expansion of B m with respect to S k − tk is: Φ( B m ) = 1 − (cid:0) nm − kk − t (cid:1)(cid:0) n − kk − t (cid:1) = 1 − m t m k + O k,m (1 /n ) On the other hand, we can directly compute that B m is ( ε , . . . , ε k ) -pseudorandom, where: ε i ≤ (cid:0) nm − ik − i (cid:1)(cid:0) n − ik − i (cid:1) O (1 /n ) -local-spectral expander whose HD-Level-Set Decomposi-tion is orthgonal [23], for large enough n Theorem 5.3 gives the bound: Φ( B m ) ≥ − t X i =0 (cid:18) ti (cid:19) (cid:0) nm − ik − i (cid:1)(cid:0) n − ik − i (cid:1) − O k,m (1 /n )= 1 − (cid:0) n/mk (cid:1)(cid:0) nk (cid:1) t X i =0 (cid:18) ti (cid:19) (cid:0) ni (cid:1)(cid:0) n/mi (cid:1) − O k,m (1 /n ) ≥ − m − k t X i =0 (cid:18) ti (cid:19) m i − O k,m (1 /n )= 1 − ( m + 1) t m k − O k,m (1 /n ) Thus we see that for large n , the bound is tight up to the leading term in m .Note that this tightness does not preclude a version of Theorem 5.3 with sub-linear dependence on ε i and a corresponding coefficient better than (cid:0) ki (cid:1) , or even independent of k . Such a bound is known for theJohnson graphs [20], but is difficult to extend to two-sided local-spectral expanders due to its reliance onsymmetry. The difference between the two regimes is perhaps most stark when examining the contrapositiveof Theorem 5.3, which states that non-expanding sets must be concentrated inside links. Corollary 5.5. Let ( X, Π) be a two-sided γ -local-spectral expander, M a k -dimensional, complete HD-walk,and let γ be small enough to satisfy the requirements of Theorem 2.2. Then for any δ > , if S ⊂ X ( k ) is aset of density α and expansion: Φ( S ) < − α − cγ − δ for c ≤ w ( M ) h ( M ) O ( k ) , then S is non-trivially correlated with an i -link for ≤ i ≤ R δ/ : ∃ ≤ i ≤ R δ/ , τ ∈ X ( i ) : E X τ [ S ] ≥ α + δc R δ/ (cid:0) ki (cid:1) λ i ( M ) , where c > is some small absolute constant. Notice that the excess correlation implied by Corollary 5.5 shrinks with k (except in the case of very deepwalks like S k − O (1) k ); this is one of the main obstructions to using results like Corollary 5.5 for hardness ofunique games. Moreover, since we proved in Proposition 5.4 that this is unavoidable in the regime of lineardependence on ε i , further progress likely requires new techniques beyond analyzing our strategy of analyzing L -mass (such as the hypercontractive analysis of [20]). On the other hand, the regime we consider doesturn out to useful for considering algorithms for unique games. Before presenting such an algorithm, however, we first discuss what is, in a sense, a converse to the above:a characterization of expansion for structured sets (links). Understanding this structure will allow us tosolve global problems like unique games by operating at a local scale. In particular, we show that the localexpansion of links in HD-walks is inversely related to the walk’s global spectra. Theorem 5.6 (Local Expansion vs Global Spectra) . Let ( X, Π) be a two-sided γ -local-spectral expander with γ ≤ − Ω( k ) , and M a k -dimensional, complete HD-walk. Then for all ≤ i ≤ k and τ ∈ X ( i ) : Φ( X τ ) ≤ − λ i ( M ) + cγ, where c ≤ w ( M ) h ( M ) O ( k ) . roof. Recall that the expansion of X τ may be written as: Φ( X τ ) = 1 − α h X τ , M X τ i = 1 − α k X s =0 h X τ , M X τ ,s i , where α is the density of X τ and X τ ,s ∈ V sk . Notice that because X τ is a link, it comes from level at most i : X τ = (cid:18) ki (cid:19) U ki τ ∈ V k ⊕ . . . ⊕ V ik Then similar to Theorem 5.3, we may write: Φ( X τ ) = 1 − α i X s =0 h X τ , M X τ ,s i = 1 − α i X s =0 λ j ( M ) h X τ , X τ ,s i + 1 α i X s =0 h X τ , Γ s i , where by Corollary 4.5 k Γ s k ≤ O (cid:16) w ( M ) h ( M )( h ( M ) + k ) (cid:0) ks (cid:1) γ k X τ ,s k (cid:17) . Since the approximate eigenvaluesof M decrease monotonically, we can further write: Φ( X τ ) ≤ − λ i ( M ) α i X s =0 h X τ , X τ ,s i + 1 α i X s =0 h X τ , Γ s i , = 1 − λ i ( M ) + 1 α i X s =0 h X τ , Γ s i + cγ, where c ≤ w ( M ) h ( M ) O ( k ) and we have dealt with the error term as in the proof of Theorem 5.3.Theorem 5.6 will play a crucial role in our algorithm for unique games, allowing us to patch together localsolutions over links corresponding to bad eigenvalues of the constraint graph. We believe this paradigm ofexploiting local expansion corresponding to large eigenvalues is of independent interest, and may be usefulfor related problems like agreement testing and PCPs. Recently, BBKSS [21] proposed a polynomial-time algorithm based on the Sum of Squares (SoS) semidefiniteprogramming hierarchy for affine unique games on the Johnson graphs. Their strategy relies on two corestructural properties of the underlying constraint graphs:1. There exists a low-degree SoS proof that non-expanding sets are concentrated in links.2. There exists a parameter r = r ( ε ) such that:(a) The ( r + 1) -st largest (distinct) eigenvalue is small: λ r ≤ − Ω( ε ) (b) The expansion of any s -link with s < r is small: ∀ τ ∈ X ( s ) , s < r : Φ( X τ ) ≤ O ( ε ) . Theorem 6.1. Let M be a k -dimensional HD-walk on X = J ( n, d ) , n ≥ Ω( k ) , ε ∈ [0 , . , and r ( ε ) = R − ε ( M ) . Then given an instance of affine unique games over M with alphabet Σ such that:1. | Σ | ≥ Ω (cid:18) r (2 ε ) ( kr (2 ε ) ) ε (cid:19) ,2. val ( I ) ≥ − ε ,there exists an algorithm outputting an Ω (cid:18) ε r (2 ε ) ( kr (2 ε ) ) (cid:19) -satisfying assignment in time | X ( k ) | poly (( kr ( ε ) ) , ε ) . For concreteness, we examine the specification of Theorem 6.1 to standard HD-walks, the canonical andpartial-swap walks. While for fixed k Theorem 6.1 provides a polynomial time algorithm for affine uniquegames for all HD-walks, we see its fine-grain performance depends on the depth of the walk. Corollary 6.2. Let X = J ( n, d ) , n > Ω( k ) , ε ∈ [0 , . . Then there exists a universal constant c > suchthat if I is an instance of affine unique games on a k -dimensional canonical or partial-swap walk of depth ≤ β ≤ with alphabet size at least | Σ | ≥ Ω ( kc εβ ) β ! and value at least − ε , there exists an algorithmoutputting an Ω β ε ( kc εβ ) ! -satisfying assignment in time | X ( k ) | poly (cid:18) ( kεβ ) , ε (cid:19) . Theorem 6.1 relies on the Sum of Squares semidefinite programming hierarchy and its relation to UniqueGames, which we now overview before giving the proof. Proving Theorem 6.1 from the ground up requires substantial and non-trivial background in the SoS frame-work. However, since we mostly rely on a number of higher level results from [21] for the SoS side of ourwork, we cover here only background necessary to understand our methods, and refer the reader to Sections1, 2, and A of [21] for additional information. The Sum of Squares framework is a method for approximating polynomial optimization problemsthrough semi-definite programming relaxations. In particular, given the problem:Maximize p ∈ R [ x , . . . , x n ] constraint to { q i = 0 } mi =1 , for q i ∈ R [ x , . . . , x n ] , the Degree- D Sum of Squares semidefinite programming relaxation outputs in time n O ( D ) a pseudo-expectation operator ˜ E : X ≤ D → R over monomials in R [ x , . . . , x n ] of degree at most D satisfying:1. Scaling: ˜ E [1] = 1 2. Linearity: ˜ E [ af ( x ) + bg ( x )] = a ˜ E [ f ( x )] + b ˜ E [ g ( x )] 3. Non-negativity (for squares): ˜ E [ f ( x ) ] ≥ 4. Program constraints: ˜ E [ f ( x ) q i ( x )] = 0 5. Optimality: ˜ E [ p ( x )] ≥ max x { p ( x ) : { q i = 0 } mi =1 } { q i = 0 } mi =1 ),whereas the fifth is promised by the SoS relaxation. A Degree- D Sum of Squares proof of a polynomial inequality f ( x ) ≤ g ( x ) (where f, g are polyno-mials of degree at most D ) is a method for ensuring the inequality continues to hold over a degree- D pseudo-expectation. In particular, given constraints { q i = 0 } mi =1 , a degree- D sum of squares proof of f ≤ g ,denoted by: { q i = 0 } mi =1 ⊢ D f ≤ g, is a certificate of the form g ( x ) = f ( x )+ P s ( x ) + P i t ( x ) q i ( x ) where all terms are at most degree- D . Noticethat properties 2, 3, and 4 then immediately imply ˜ E [ f ( x )] ≤ ˜ E [ g ( x )] . Unique games can be written as a polynomial optimization problem. In particular, given an instance I of unique games with alphabet Σ and constraints Π over G ( V, E ) , consider the following quadratic opti-mization problem over variables { X v,s } V × Σ that computes val ( I ) :Maximize: E ( u,v ) ∼ E "X s ∈ Σ X u,s X v,π uv ( s ) Constraint to: X v,s = X v,s ∀ v ∈ V, s ∈ Σ X v,a X v,b = 0 ∀ v ∈ V, a = b ∈ Σ X s ∈ Σ X v,s = 1 ∀ v ∈ V Following BBKSS, we will call the constraints of this program A I . We will work with the Degree- D Sumof Squares relaxation of this program, which outputs a degree- D pseudo-expectation satisfying A I in time | V | O ( D ) such that ˜ E [ val I ( x )] ≥ val ( I ) . Throughout our proof, we will modify this pseudo-expectation in twoways. Conditioning is a standard algorithmic technique in the SoS paradigm used to improve the value of inde-pendently sampling a solution from the output of an SoS semidefinite relaxation (see e.g. [22, 21]). Givena pseudo-expectation ˜ E and a sum of squares polynomial s ( x ) , we can define a new pseudo-expectation by conditioning on s ( x ) as follows: ˜ E [ f ( x ) | s ( x )] = ˜ E [ f ( x ) s ( x )]˜ E [ s ( x )] . In this work, we will only be interested in the case where s ( x ) is the indicator of some binary variable. Inother words, in our context this process can be thought of as conditioning on the value of some X u,a . Symmetrization is an operation on pseudo-expectations introduced in [21] to take advantage of the sym-metric structure of affine unique games. The idea is to symmetrize over shifts s ∈ Σ . Formally, given apseudo-expectation ˜ E , define the s -shifted pseudo-expectation ˜ E s to be: ˜ E s [ X u ,a · · · X u t ,a t ] = ˜ E [ X u ,a − s · · · X u t ,a t − s ] . BBKSS then define the symmetrization operator which maps ˜ E to: ˜ E sym = 1 | Σ | X s ∈ Σ ˜ E s , and call a pseudo-expectation shift-symmetric if it is invariant under this operation. Let ˜ E be the pseudodis-tribution output by the degree- D SoS relaxation of unique games. BBKSS note three important properties:1. ˜ E sym is a degree- D pseudo-expectation satisfying A I .29. Symmetrization can be performed in time subquadratic in the descrption of ˜ E .3. The objective value is invariant under symmetrization: ˜ E [ val ( I )] = ˜ E sym [ val ( I )] .As a result, we may freely assume throughout that we are working with a shift-symmetric pseudo-expectation. We have now covered sufficient background to present the algorithm behind Theorem 6.1, Iterated Conditionand Round. The algorithm follows the strategy presented in [21, Algorithm 6.1], differing mainly in that theparameter r ( ε ) satisfying their second condition has been replaced with HD-Threshold Rank. Condition and Round: We start with a basic sub-routine common to the SoS literature, which takesa pseudo-expectation ˜ E for an affine unique games instance ( G ( V, E ) , Π) on alphabet Σ and outputs anassignment x ∈ Σ V via the following process:1. Sample a vertex v ∈ V uniformly at random, and condition on event X v, = 1 to get the conditionalpseudo-expectation ˜ E [ ·| X v, = 1] ,2. Sample x ∈ Σ V by independently sampling each coordinate x u from the categorical distribution Pr[ x u = s ] = ˜ E [ X u,s | X v, = 1] .Following [21], we call the expected value of the solution output by Condition and Round the CR-Value of the instance I , denoted CR-Val ( I ) . It is worth noting that Condition and Round, and thus the entirealgorithm, can be de-randomized by standard techniques like the method of conditional expectations [21]. Iterated Condition and Round: The full algorithm builds a solution by iteratively applying Conditionand Round to links. Let M be a k -dimensional HD-walk on X = J ( n, d ) , I = ( M, Π) an instance of affineunique games over alphabet Σ with val ( I ) ≥ − ε , and r = R − ε ( M ) . Further, given a subset H ⊂ X ( k ) ,let I H denote the restriction of the instance I to the subgraph vertex-induced by H . The following processreturns an Ω ε,r,k (1) satisfying assignment.1. Let δ ( ε ) = εr ( kr ) . Solve the Degree- D = ˜ O (1 /δ ( ε )) SoS SDP relaxation of unique games, and symmetrizethe resulting pseudo-expectation to get ˜ E . Set j = 1 .2. Let Dif ( j ) = ˜ E [ val I ( x )] − ˜ E j − [ val I ( x )] . While Dif ( j ) ≤ ε :(a) Find an r ′ -link X τ for r ′ ≤ r such that the CR-Value of I | X τ is at least δ ( ε + Dif ( j )) .(b) Let S j be the subgraph of X τ induced by the vertices which have not yet been assigned a valuein any partial assignment f i , i ≤ j , and perform Condition and Round on S j to get partialassignment f j .(c) Create a new pseudo-expectation ˜ E j by making the marginal distribution over assigned verticesuniform and independent of others, i.e. for all degree ≤ D monomials let ˜ E j be: ˜ E j [ X h ,a . . . X h t ,a t X u ,b . . . X u m ,b m ] = 1 | Σ | t ˜ E j − [ X u ,b . . . X u m ,b m ] , where h i ∈ S j and u i ∈ X ( k ) \ S j . Increment j ← j + 1 . The remainder of this section is devoted to proving Theorem 6.1. Our main technical contribution lies inshowing how key structural properties of the Johnson graphs exploited by [21] are in fact inherent local-to-global properties of HD-walks. On the algorithmic side, we appeal directly to [21], employing the followinglemma which shows how to reduce Iterated Condition and Round to Condition and Round on subgraphswith poor expansion. 30 emma 6.3 (Lemma 6.12 [21]) . Let ε ∈ (0 , be a universal constant, ε < ε / , δ : [0 , → [0 , afunction, and δ min = min δ ( η ) ∈ [ ε, ε ] . Let G be a random walk on any graph and I be any affine uniquegames instance on G with alphabet size | Σ | ≥ Ω (cid:16) δ min (cid:17) and value at least − ε .Suppose we have a subroutine A which given as input ˜ E , a shift-symmetric degree- D pseudo-expectationsatisfying A I with ˜ E [ val I ( x )] ≥ − η ≥ − ε returns a vertex-induced subgraph H such that:1. The CR-Value of I H is at least δ ( η ) .2. The expansion of H is O ( η ) .Then if A runs in time T ( A ) , Iterated Condition and Round outputs a solution for I satisfying an Ω( δ min ε ) -fraction of edges of G in time | V ( G ) | ( T ( A ) + | V ( G ) | O ( D ) ) . Our task is thus reduced to efficiently finding a subgraph with poor expansion on which Condition andRound has high expected value. BBKSS prove this fact for the Johnson graph in part by exploiting the twoproperties we discussed at the beginning of the section, that is:1. There exists a low-degree SoS proof that non-expanding sets are concentrated in links.2. There exists a parameter r = r ( ε ) such that:(a) The ( r + 1) -st largest (distinct) eigenvalue is small: λ r ≤ − Ω( ε ) (b) The expansion of any s -link with s < r is small: ∀ τ ∈ X ( s ) , s ≤ r : Φ( X τ ) ≤ O ( ε ) . In essence, we have already shown these properties hold for HD-walks in Section 5. All that remains isto show our methods fit into the SoS framework, and to strengthen the results for the special case of thecomplete complex. While it may be possible to extend our arguments beyond the complete complex (atleast for some subset of HD-walks), it is not clear whether or not this more general case fits into the SoSframework. We discuss this issue in more detail after proving property 1 for the complete complex. Proposition 6.4. Let M be a k -dimensional HD-walk on X = J ( n, d ) , n ≥ Ω( k ) , and { λ i } ki =0 be eigenval-ues corresponding to the HD-Level-Set Decomposition. Finally, let r ≤ ⌈ k/ ⌉ and λ b be parameters such that ∀ i ≥ r + 1 , λ i ≤ λ b . Then the expansion with respect to M of any function f ∈ C k which is not correlatedwith any s -link for s ≤ r is large: ⊢ h f, ( I − M ) f i ≥ (1 − λ b ) E [ f ] − (cid:0) kr (cid:1)(cid:0) n − rk − r (cid:1)(cid:0) n − rk − r (cid:1) r X i =0 h E if , E if i X ( i ) ! + B ( f ) ! , where B ( f ) = E [ f − f ] measures the booleanity of f , and we recall for τ ∈ X ( i ) , E if ( τ ) = E X τ [ f ] .Proof. Let f = f + . . . + f k give the HD-Level-Set Decomposition of f . Recall from the proof of Theorem 1.11that: h E if , E if i = h f i , f i i h g i , g i i + E "(cid:18) E if − h f i , f i ih g i , g i i g i (cid:19) , (17) BBKSS only state the result for regular graphs, but their proof only requires that the weight is spread evenly across vertices,a fact which holds for any random walk. Technically the algorithm is slightly more general, replacing links with any subgraph along with a number of parameters.See [21, Algorithm 6.13]. Here we use a slightly different notion of expansion which will be useful later in the proof. E if is the vector of densities of f in i -links, that is for τ ∈ X ( i ) , E if ( τ ) = E X τ [ f ] . Further, it is wellknown [23] that for J ( n, d ) the ratio of h f i , f i i to h g i , g i i is in fact a constant: h f i , f i ih g i , g i i = (cid:0) n − ik − i (cid:1)(cid:0) ki (cid:1)(cid:0) n − ik − i (cid:1) . Since E if and g i are both linear in the coefficients of f , Equation (17) then provides the following degree- SoS proof: ⊢ h f i , f i i ≤ (cid:0) ki (cid:1)(cid:0) n − ik − i (cid:1)(cid:0) n − ik − i (cid:1) h E if , E if i . In fact, it’s worth noting that we can even replace h E if , E if i on the RHS with Var ( E if ) , tightening the bound.However, since the term does not provide much additional advantage in what follows, we drop it for simplicity.Expanding h f, ( I − M ) f i , we get the desired bound: h f, ( I − M ) f i = h f, f i − k X i =0 λ i h f i , f i i = h f, f i − r X i =0 λ i h f i , f i i − k X i = r +1 λ i h f i , f i i≤ h f, f i − r X i =0 h f i , f i i − λ b k X i = r +1 h f i , f i i = (1 − λ b ) E [ f ] + B ( f ) − r X i =0 h f i , f i i ! ≤ (1 − λ b ) E [ f ] + B ( f ) − (cid:0) kr (cid:1)(cid:0) n − rk − r (cid:1)(cid:0) n − rk − r (cid:1) r X i =0 h E if , E if i ! where both inequalities are degree-2 SoS (in the coefficients of f ), and the last relies further on our assumptionthat n ≥ Ω( k ) .It is worth giving a brief qualitative comparison of this result to a similar version given in [21], who employthe old Fourier analytic techniques used originally by [20]. Proposition 6.4 not only gives a tighter bound(by a factor of exp( r )), but perhaps more importantly shows how viewing the problem from the frameworkof high dimensional expansion drastically simplifies the proof (which takes up a 10-page appendix in [21]).It is also worth highlighting here that the key barrier to extending Theorem 6.1 to HD-walks beyond thecomplete complex lies in our use of the fact that h f i , f i i / h g i , g i i is a constant. While this term is tightlybounded for two-sided local-spectral expanders, to our knowledge it is not necessarily a constant, and theinequality no longer fits into the SoS framework as a result. On the other hand, h f i , f i i / h g i , g i i is alwaysconstant for objects called sequentially differential posets (objects introduced by Stanley [42] and discussedin the context of high dimensional expansion by [23]). We will discuss the extension of our results to suchobjects in our upcoming companion paper.We now turn our attention to the second structural property necessary for BBKSS’ analysis: the existenceof a parameter r ( ε ) such that λ r ≤ − Ω( ε ) , while the expansion of s -links for s < r is at most O ( ε ) . Forstrong enough two-sided local-spectral expanders, it is not hard to see that applying Theorem 5.6 implies notonly that r ( ε ) exists, but further that it is exactly the HD-Threshold Rank of the constraint graph. However,depending on the constraint graph, it is possible that the complete complex is not a sufficiently strong two-sided local-spectral expander for this to be the case, so we must modify our approach. In fact, it is possibleto prove a stronger, exact version of Theorem 5.6 by recalling that the HD-Level-Set Decomposition is anexact eigendecomposition for J ( n, d ) [23], which will give us a sufficiently strong result for any HD-walk.32 roposition 6.5. Let M be a k -dimensional HD-walk over X = J ( n, d ) , { λ i } ki =0 be eigenvalues correspond-ing to the HD-Level-Set Decomposition , and r a parameter such that: λ ≥ . . . ≥ λ r . Then for all ≤ i ≤ r , the expansion of i -links is inversely related to λ i : ∀ τ ∈ X ( i ) , Φ( X τ ) ≤ − λ i . Proof. Recall that the expansion of X τ may be written as: Φ( X τ ) = 1 − α h X τ , M X τ i = 1 − α k X j =0 h X τ , M X τ ,j i = 1 − α k X j =0 λ i h X τ , X τ ,j i where α is the density of X τ , and X τ ,j ∈ V j . Notice that because X τ is a link, it comes from level at most i : X τ = (cid:18) ki (cid:19) U ki τ ∈ V k ⊕ . . . ⊕ V ik . Thus we may write: Φ( X τ ) = 1 − α i X j =0 λ j h X τ , X τ ,j i≤ − λ i α i X j =0 h X τ , X τ ,j i = 1 − λ i as desired.Then as long as the eigenvalues of the HD-Level-Set decrease, we see that both of our desired local-to-global properties are satisfied. Unfortunately, there is a slight technical issue: the eigenvalues of theHD-Level-Set Decomposition don’t necessarily decrease monotonically. For instance, the eigenvalues ofthe partial-swap walks, a basis for HD-walks over the complete complex, instead decrease until they become O k (1 /n ) , at which point they begin to alternate in sign. This, however, is not much more than a technicalannoyance since it is true that sufficiently large eigenvalues of the HD-Level-Set Decomposition decreasemonotonically as long as the walk is not too lazy. Lemma 6.6. Let M be an HD-walk on X = J ( n, d ) , n > Ω( k ) , where the probability of staying at avertex is at most / . Let { λ i } ki =0 be eigenvalues corresponding to the HD-Level-Set Decomposition, and let r = R . ( M ) . Then the following three properties hold:1. r ≤ ⌈ k/ ⌉ λ ≥ . . . ≥ λ r ∀ i ≥ r : λ r − ≥ λ i , and λ i < . . For J ( n, d ) , it is well known that each space in the HD-Level-Set Decomposition consists of eigenvectors with a singleeigenvalue λ i , see e.g. [23]. While this may seem to contradict Proposition 4.10, the issue is actually caused by spaces with equivalent approximateeigenvalues. roof. Any HD-walk on J ( n, d ) may be expressed as a convex combination of partial-swap walks: M = k X i =0 α i S ik , where we are further guaranteed that the lazy term α ≤ / . Let λ ij denote the eigenvalue of V ik withrespect to S ik . Since the partial-swap walks share eigenspaces, we have: λ j = k X i =0 α i λ ij . Since J ( n, k ) is a two-sided O k (1 /n ) -local-spectral expander, Corollary 4.8 implies that λ ij = (cid:0) k − ij (cid:1)(cid:0) kj (cid:1) ± O ( k ) n . Using this fact, we start by proving the second and third properties (technically the third follows immediatelyfrom the second, but we prove them jointly). Note that since these properties imply that eigenvalues decreaseuntil they are bounded from above by .68, we can then directly check the first property (that r is at most ⌈ k/ ⌉ ) by computing the worst-case value of λ ⌈ k/ ⌉ . To prove the latter properties, notice that it is sufficientto show the following for all j :1. If λ j ≥ . , ∀ i > j , λ i < λ j ∀ i > j, λ i − λ j < O ( k ) n < . If these conditions hold, the eigenvalues of M must decrease until they reach one of value less than . , atwhich point the remaining values are upper bounded by . .The second of these facts is trivial, following directly from the form of the λ j and letting n be sufficientlylarge. To prove the first, notice that if λ j ≥ . , M must have non-trivial weight α i for some ≤ i ≤ k − j .In particular, for large enough n we have that: ∃ ≤ i ≤ k − j : α i > Ω(1 /k ) . Further, for such an i , the j -th eigenvalue of S ik is substantially larger than all subsequent eigenvalues: ∀ t > j : λ ij − λ it > − O ( k ) . We can now bound the difference of λ j and subsequent eigenvalues. For all t > j we have: λ j − λ t = k X m =0 α m (cid:0) λ mj − λ mt (cid:1) ≥ − O ( k ) + k X m =0 ,m = i α m (cid:0) λ mj − λ mt (cid:1) ≥ − O ( k ) − k O ( k ) n ≥ , as desired. It is left to show that for any M satisfying α ≤ / , λ ⌈ k/ ⌉ < . . Recall that λ ⌈ k/ ⌉ = P α i λ i ⌈ k/ ⌉ . For large enough n , this is maximized by letting α = 1 / , and α = 49 / , so it is sufficientto check that / 50 + 49 / λ ⌈ k/ ⌉ < . . This can be checked directly by noting that λ ⌈ k/ ⌉ ≤ k −⌈ k/ ⌉ k + O k (1 /n ) . 34ince affine unique games are trivial over walks with a constant lazy component, we can assume withoutloss of generality that the lazy component of all walks we consider is at most / . We are finally ready toshow how the discussed properties can be leveraged to find a link with high CR-Value. The argument, whichfollows BBKSS [21, Lemma 6.9], centers around applying Proposition 6.4 to a potential function whose valuelower bounds the CR-value of the instance. The exact form of the potential is not particularly important,but we need it to satisfy a few properties in order to successfully apply Proposition 6.4. Theorem 6.7 (BBKSS [21] Sections 4,6) . Let ε ∈ (0 , / and β, ν ∈ (0 , be parameters such that β ≥ ε ,and ν < ε . Let I = ( G ( V, E ) , Π) be an instance of affine unique games, and ˜ E a Degree- D = ˜ O (1 /ν ) shift-symmetric pseudo-expectation satisfying the corresponding constraints A I with value at least − ε . Finally,let H be a vertex-induced subgraph of G such that for all v ∈ H , the weight of edges leaving v is equivalentand at most ε . Then there exists a potential function Φ I,Hβ,ν in the variables of ˜ E such that:1. If ˜ E h Φ I,Hβ,ν i ≥ δ , then the CR-value of I H is at least ( δ − ν )( β − ε − ν ) 2. There exists a family of degree ≤ D polynomials F β,ν independent of H such that: Φ I,Hβ,ν = X f ∈F β,ν E H [ f ] and the following properties hold:(a) The pseudo-expectation of P E [ f ] is large: ˜ E X f ∈F β,ν E [ f ] ≥ − ε − β − ν − ν (b) The pseudo-expectation of the expansion of P f is small: ˜ E X f ∈F β,ν h f, ( I − A G ) f i ≤ ε + 2 ε − β − ν + 2 ν (c) The pseudo-expectation of P B ( f ) is small: ˜ E X f ∈F β,ν E [ f − f ] ≤ ε − β − ν + ν As written, Theorem 6.7 does not appear in [21], but follows directly from a number of claims andpropositions throughout Sections 4 and 6 of their work. The key to finding a link with high CR-valueis then to notice the connection between the form of the potential in Theorem 6.7 and Proposition 6.4’scharacterization of expansion. Proposition 6.8. Let M be a k -dimensional HD-walk on X = J ( n, d ) , n ≥ Ω( k ) , ε ∈ [0 , . , r = r ( ε ) = R − ε ( M ) − , and I be an affine unique games instance over M with value at least − ε . Then given adegree- ˜ O (cid:16) ε r (cid:0) kr (cid:1)(cid:17) pseudo-expectation satisfying the axioms A I , we can find in time n O ( r ) an s -link X τ for ≤ s ≤ r with CR-Value Ω (cid:18) εr ( kr ) (cid:19) . Self-edge constraints are either always satisfied or unsatisfiable, so for a lazy enough walk with high value any assignmentto the nodes will satisfy sufficient constraints. Technically this pseudo-expectation is actually split over two “independent samples” of the variables corresponding to theinstance I . Since this does not particularly matter for our arguments, we ignore this detail. See [21, Definition 2.2] for moreinformation. roof. As long as the conditions of Theorem 6.7 are satisfied, it is sufficient to find a link X τ with highpotential. To find such a link, recall that the potential promised by Theorem 6.7 may be written as: Φ I,Hβ,ν = X f ∈F E H [ f ] for some family of functions F . This is closely related to the characterization of expansion in Proposition 6.4,where the term h E if , E if i may be re-written as: h E if , E if i = E τ ∈ X ( i ) (cid:20) E X τ [ f ] (cid:21) . Using this connection, we can apply Proposition 6.4 across all functions in the family to lower bound theexpected potential across small links in our complex. Using properties (a), (b), and (c) of Theorem 6.7 tolower bound this expectation, we complete the proof by an averaging argument.More formally, let F β,ν be the family of functions corresponding to the potentials promised by Theo-rem 6.7. Notice that by our restriction on ε and Lemma 6.6, r = R − ε ( M ) − and λ b = 1 − ε satisfythe conditions of Proposition 6.4. Applying this to each f ∈ F β,ν yields: ⊢ ε X f ∈F β,ν h f, ( I − M ) f i ≥ X f ∈F β,ν E [ f ] − (cid:0) kr (cid:1)(cid:0) n − rk − r (cid:1)(cid:0) n − rk − r (cid:1) r X i =0 E τ ∈ X ( i ) (cid:20) E X τ [ f ] (cid:21)! + B ( f ) ! = X f ∈F β,ν E [ f ] − (cid:0) kr (cid:1)(cid:0) n − rk − r (cid:1)(cid:0) n − rk − r (cid:1) r X i =0 E τ ∈ X ( i ) X f ∈F β,ν E X τ [ f ] + X f ∈F β,ν B ( f )= X f ∈F β,ν E [ f ] − (cid:0) kr (cid:1)(cid:0) n − rk − r (cid:1)(cid:0) n − rk − r (cid:1) r X i =0 E τ ∈ X ( i ) h Φ I,X τ β,ν i! + X f ∈F β,ν B ( f ) , which in turn gives the aforementioned lower-bound on the potential of low-level links: ⊢ r X i =0 E τ ∈ X ( i ) h Φ I,X τ β,ν i ≥ (cid:0) n − rk − r (cid:1)(cid:0) kr (cid:1)(cid:0) n − rk − r (cid:1) X f ∈F β,ν E [ f ] + X f ∈F β,ν B ( f ) − X f ∈F β,ν h f, ( I − M ) f i ε . (18)Since r is the first index such that λ r +1 < − ε , it follows from Lemma 6.6 and our restrictions on ε that λ r ≥ − ε , and further by Proposition 6.5 that for any s -link X τ , ≤ s ≤ r , Φ( X τ ) ≤ ε . Sincethis additionally holds vertex-by-vertex by symmetry, all links in Equation (18) satisfy the requirements ofTheorem 6.7, and taking the pseudo-expectation implies: r X i =0 E τ ∈ X ( i ) h ˜ E h Φ I,X τ β,ν ii ≥ (cid:0) n − rk − r (cid:1)(cid:0) kr (cid:1)(cid:0) n − rk − r (cid:1) (cid:18) − ε − β − ν − ν − ε (cid:18) ε + 2 ε − β − ν + 2 ν (cid:19)(cid:19) . by setting β = 17 ε and ν = ε r ( kr ) and recalling our assumptions on n and ε , this may be further simplifiedto: r X i =0 E τ ∈ X ( i ) h ˜ E h Φ I,X τ β,ν ii ≥ (cid:0) kr (cid:1) . As a result, we see by averaging that there must exist some s -link X τ for s ≤ r with high potential: ∃ X τ , | τ | ≤ r : ˜ E h Φ I,X τ β,ν i ≥ r + 1) (cid:0) kr (cid:1) . (19)By Theorem 6.7, the CR-Value of I | X τ is then at leastCR-Val ( I | X τ ) ≥ r + 1) (cid:0) kr (cid:1) − ε r (cid:0) kr (cid:1) ! ε − ε r (cid:0) kr (cid:1) ! ≥ Ω εr (cid:0) kr (cid:1) ! as desired. 36he proof of Theorem 6.1 follows almost immediately from Lemma 6.3, Proposition 6.5, and Proposi-tion 6.8. Proof. We begin by setting the parameters for Lemma 6.3. Let the function δ be given by: δ ( η ) = 150 r ( η ) (cid:0) kr ( η ) (cid:1) . Then by Proposition 6.8 and Proposition 6.5, there exists a sub-routine which finds an s -link X τ , ≤ s ≤ r ( ε ) ,in time n O ( r ( ε )) such that:1. The CR-Value of I X τ is at least δ ( ε ) .2. The expansion of X τ is small: Φ( X τ ) < ε .Thus we are in position to apply Lemma 6.3, which completes the proof. This work stemmed in part from collaboration at the Simons Institute of Theory of Computing, Berkeleyduring the 2019 summer cluster: “Error-Correcting Codes and High-Dimensional Expansion”. 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Explicit sos lower bounds fromhigh-dimensional expanders. arXiv preprint arXiv:2009.05218 , 2020.[41] Howard Garland. p-adic curvature and the cohomology of discrete subgroups of p-adic groups. Annalsof Mathematics , pages 375–423, 1973.[42] Richard P Stanley. Differential posets. Journal of the American Mathematical Society , 1(4):919–961,1988. 39 Proof of Lemma 4.3 In this section, we prove a strengthening of the main technical lemma of DDFH Section 8 [23, Claim 8.8],which allows for better control of error propagation. Lemma A.1 (Strengthened Claim 8.8 [23]) . Let ( X, Π) be a d -dimensional two-sided γ -local-spectral ex-pander. Then for all j < k < d : D k +1 U k +1 k − j − j + 1 k + 1 U kk − j − k − jk + 1 U kk − j − D k − j = j − X i = − k − ik + 1 U kk − − i Γ i U k − − ik − j where k Γ i k ≤ γ .Proof. The proof follows by a simple induction. The base cases, j = 0 and k < d , follow immediately fromEquation (15). For the inductive step, consider: D k +1 U k +1 k − ( j +1) = (cid:18) D k +1 U k +1 k − j − j + 1 k + 1 U kk − j − k − jk + 1 U kk − j D k − j (cid:19) U k − j − + j + 1 k + 1 U kk − j − + k − jk + 1 U kk − j − D k − j U k − j − By the inductive hypothesis, the first term on the RHS may be written as: (cid:18) D k +1 U k +1 k − j − j + 1 k + 1 U kk − j − k − jk + 1 U kk − j D k − j (cid:19) U k − j − = j − X i = − k − ik + 1 U kk − i − Γ i U k − − ik − j − , where k Γ i k ≤ γ . For the latter term, consider flipping DU and U D . By Equation (15) we have: k − jk + 1 U kk − j − D k − j U k − j − = U kk − j − (cid:18) k + 1 I + k − j − k + 1 U k − j − D k − j − + k − jk + 1 Γ j (cid:19) , for some Γ j satisfying k Γ j k ≤ γ . Combining these observations yields the desired result: D k +1 U k +1 k − ( j +1) − ( j + 1) + 1 k + 1 U kk − j +1 + k − ( j + 1) k + 1 U kk − ( j +1) − D k − j +1 = D k +1 U k +1 k − ( j +1) − j + 1 k + 1 U kk − j − − U kk − j − (cid:18) k + 1 I − k − j − k + 1 U k − j − D k − j − (cid:19) = j − X i = − k − ik + 1 U kk − i − Γ i U k − − ik − j − ! + k − jk + 1 U kk − ( j +1) Γ j = j X i = − k − ik + 1 U kk − − i Γ i U k − − ik − ( j +1) . We now show how to use this strengthened result to prove tighter bounds on the quadratic form h f, N jk f i which implies a stronger version of Lemma 3.6 as an immediate corollary. This improvement mainly mattersin the regime where γ ≤ − ck for c a small constant. Proposition A.2. Let ( X, Π) be a d -dimensional γ -local-spectral expander with γ satisfying γ ≤ − Ω( k + j ) , k + j ≤ d , and f ℓ ∈ V ℓk . Then: h f ℓ , N jk f ℓ i = (cid:0) kℓ (cid:1)(cid:0) k + jℓ (cid:1) (cid:18) ± j ( j + 2 k + 2 ℓ + 3)4 γ ± c ( k, j, ℓ ) γ (cid:19) h f ℓ , f ℓ i where c ( k, j, ℓ ) = O (( k + j ) (cid:0) k + jℓ (cid:1) ) . roof. We proceed by induction on j . We will prove a slightly stronger statement for the base-case j = 1 : h f ℓ , D k +1 U k f ℓ i = (cid:18) k + 1 − ℓk + 1 ± ( k − ℓ + 1)( k + ℓ + 2)2( k + 1) γ ± c ( k, ℓ ) γ (cid:19) h f ℓ , f ℓ i , where c ( k, ℓ ) = O ( k (cid:0) kℓ (cid:1) ) . Recall that f ℓ may be expressed as U kℓ g ℓ , for g ℓ ∈ H ℓ . For notational convenience,we write f iℓ = U iℓ g ℓ . Then we may expand the inner product based on Lemma 4.3, and simplify based onapplying the naive bounds on N ik given by Corollary 4.5: h f ℓ , D k +1 U k f ℓ i = h f ℓ , D k +1 U k +1 ℓ g ℓ i = k − ℓ + 1 k + 1 h f ℓ , f ℓ i + k − ℓ − X i = − h f ℓ , k − ik + 1 U kk − − i Γ i U k − − iℓ g ℓ i = k − ℓ + 1 k + 1 h f ℓ , f ℓ i + k − ℓ − X i = − k − ik + 1 h N i +1 k − i − f k − − iℓ , Γ i f k − − iℓ i = k − ℓ + 1 k + 1 h f ℓ , f ℓ i + k − ℓ − X i = − k − ik + 1 (cid:0) k − i − ℓ (cid:1)(cid:0) kℓ (cid:1) h f k − − iℓ , Γ i f k − − iℓ i + k − ℓ − X i = − k − ik + 1 h h i , Γ i f k − − iℓ i where k h i k ≤ γ ( k − ℓ )( i + 1) k g ℓ k . We now apply Cauchy-Schwarz, and Lemma 3.6 to collect terms in h f ℓ , f ℓ i : h f ℓ , D k +1 U k f ℓ i = k − ℓ + 1 k + 1 h f ℓ , f ℓ i ± γ k − ℓ − X i = − k − ik + 1 (cid:0) k − i − ℓ (cid:1)(cid:0) kℓ (cid:1) h f k − − iℓ , f k − − iℓ i ± a ( k, ℓ ) γ h g ℓ , g ℓ i = k − ℓ + 1 k + 1 h f ℓ , f ℓ i ± γ k − j − X i = − k − ik + 1 1 (cid:0) kj (cid:1) h g ℓ , g ℓ i ± a ( k, ℓ ) γ h g ℓ , g ℓ i = k − ℓ + 1 k + 1 h f ℓ , f ℓ i ± γ k − j − X i = − k − ik + 1 h f ℓ , f ℓ i (1 − c ( k, ℓ ) γ ) ± a ( k, ℓ ) γ h f ℓ , f ℓ i (1 − c ( k, ℓ ) γ )= k − ℓ + 1 k + 1 (cid:18) ± ( k + ℓ + 2)2 γ ± a ( k, ℓ ) γ (cid:19) h f ℓ , f ℓ i where the final step comes from a Taylor expansion assuming γ sufficiently small, and a ( k, ℓ ) = O ( k (cid:0) kℓ (cid:1) ) .The inductive step follows from noting that the canonical walk essentially acts like a product of upperwalks from lower levels in the following sense: h f ℓ , N jk f ℓ i = h U k + j − k f ℓ , D k + j U k + jk f ℓ i = h U k + j − k f ℓ , N k + j − ( U k + j − k f ℓ ) i . Thus by the base-case and inductive hypothesis we get: h f ℓ , N jk f ℓ i = h U k + j − k f ℓ , N k + j − ( U k + j − k f ℓ ) i = (cid:18) k + j − ℓk + j ± ( k + j − ℓ )( k + j + ℓ + 1)2( k + j ) γ ± c ( k + j − , ℓ ) γ (cid:19) h f ℓ , N j − k f ℓ i = (cid:0) kℓ (cid:1)(cid:0) k + jℓ (cid:1) (cid:18) ± ( k + j + ℓ + 1)2 γ ± k + jk + j − ℓ c ( k + j − , ℓ ) γ (cid:19) · (cid:18) ± ( j − j + 2 k + 2 ℓ + 2)4 γ ± c ( k, j − , ℓ ) γ (cid:19) h f ℓ , f ℓ i = (cid:0) kℓ (cid:1)(cid:0) k + jℓ (cid:1) (cid:18) ± j ( j + 2 k + 2 ℓ + 3)4 γ ± c ( k, j, ℓ ) γ (cid:19) h f ℓ , f ℓ i , h U kℓ g ℓ , U kℓ g ℓ i = h N k − ℓℓ g ℓ , g ℓ i .Finally, we conjecture that a stronger result is true, and the error dependence on γ should in fact beexp ( − poly ( k ) γ ) . Proving this would require a more careful and involved analysis of how the error termpropogates. B Orthogonality and the HD-Level-Set Decomposition In this section we discuss in a bit more depth the error in [12, Theorem 5.10], and further show by directcounter-example that its implication [32] that the HD-Level-Set is orthogonal does not hold. In [12], Kaufmanand Oppenheim analyze an approximate eidgendecomposition of the upper walk N k for two-sided local-spectral expanders. They prove a specialized version of Theorem 2.2 for this case, and in particular that forsufficiently strong two-sided local-spectral expanders, the spectra of N k is divided into strips concentratedaround the approximate eigenvalues of their decomposition. They call the span of each strip W i , and notethat the W i form an orthogonal decomposition of the space. Let V i be the space in the original approximateeigendecomposition corresonding to strip W i . Kaufman and Oppenheim claim in [12, Theorem 5.10] thatthe W i are closely related to the original approximate decomposition in the following sense: ∀ φ ∈ C k : k P W i φ k ≤ c k P V i φ k for some constant c > , where P W i and P V i are projection operators. Unfortunately, this relation cannothold, as it implies [32] that the HD-Level-Set Decomposition is orthogonal for sufficiently strong two-sidedlocal-spectral expanders, which we will show below is false by direct example. In slightly greater detail, theissue in the argument is the following. The authors show that for any j = i : k P W j P V i k ≤ c ′ , for some small constant c ′ , and then claim that this fact implies for any φ ∈ C k : k P W j P V i φ k ≤ c ′ k P V j φ k . Unfortunately, this is not true—the righthand side should read P V i rather than P V j for the relation to hold,but this makes it impossible to compare P W i φ solely to P V i φ .We now move to showing that for any γ > , there exists a two-sided γ -local-spectral expander suchthat the HD-Level-Set Decomposition is not orthogonal, which implies [12, Theorem 5.10] cannot hold byarguments of [32]. Proposition B.1. For any γ > , there exists a two-sided γ -local-spectral expander such that the HD-Level-Set Decomposition is not orthogonal.Proof. Our construction is based off of a slight modification of the complete complex J ( n, . In particular,we consider the uniform distribution Π over triangles X = (cid:0) [ n ]3 (cid:1) \ (123) . It is not hard to see through directcomputation that ( X, Π) is a two-sided O (1 /n ) -local-spectral expander. Recall that the link of , U , liesin V ⊕ V . Our goal is to prove the existence of a function f = U g ∈ V such that the inner product: h U , f i ∝ X (1 xy ) ∈ X g (1 x ) + g (1 y ) + g ( xy ) (20)is non-zero. To do this, we first simplify the above expression assuming g ∈ Ker ( D ) , which we recall impliesthe following relations: ∀ y ∈ [ n ] : X ( xy ) ∈ X (2) Π ( xy ) g ( xy ) = 0 . In particular, summing over all y ∈ [ n ] gives X ( xy ) ∈ X (2) Π ( xy ) g ( xy ) = 0 . Π , we have Π (12) = Π (13) = Π (23) = n − ( n ) − , and otherwise Π ( xy ) = n − ( n ) − . We then may write: X (1 x ) ∈ X (2): x/ ∈ [3] g (1 x ) = − n − n − g (12) + g (13)) , X ( xy ) ∈ X (2): ( xy ) / ∈ [3] × [3] g ( xy ) = − n − n − g (12) + g (13) + g (23)) . Plugging this into Equation (20), the inner product drastically simplifies to depend only on g (23) . To seethis, we separate the inner product into two terms and deal with each separately: X (1 xy ) ∈ X g (1 x ) + g (1 y ) + g ( xy ) = X (1 xy ) ∈ X g (1 x ) + g (1 y ) + X (1 xy ) ∈ X g ( xy ) We start with the former. Notice that each face (1 z ) in this term is counted exactly the number of times itappears in a triangle in X , and further that this is exactly how Π is defined. Thus we have: X (1 xy ) ∈ X g (1 x ) + g (1 y ) ∝ X (1 x ) ∈ X (2) Π (1 x ) g (1 x ) = 0 . It is left to analyze the latter term. Since (123) is not in our complex, we may write: X (1 xy ) ∈ X g ( xy ) = X ( xy ) ∈ X (2): ( xy ) / ∈ [3] × [3] g ( xy ) − X (1 x ) ∈ X (2): x/ ∈ [3] g (1 x ) = − n − n − g (12) + g (13) + g (23)) + n − n − g (12) + g (13))= − n − n − g (23) . Thus it remains to show that there exists g ∈ Ker ( D ) such that g (23) = 0 . Note that the kernel of D isexactly the space of solutions to the underdetermined linear system of equations given by D g ( i ) = 0 for all ≤ i ≤ n . Thus we can check if a solution exists with g (23) = c for c = 0 by ensuring that this constraint islinearly independent of the D g ( i ))