The Complexity of Linear Tensor Product Problems in (Anti-) Symmetric Hilbert Spaces
aa r X i v : . [ m a t h . NA ] A ug The Complexity of linear Tensor Product Problems in(Anti-) Symmetric Hilbert Spaces ∗ Markus Weimar † October 24, 2018
Abstract
We study linear problems S d defined on tensor products of Hilbert spaces withan additional (anti-) symmetry property. We construct a linear algorithm that usesfinitely many continuous linear functionals and show an explicit formula for its worstcase error in terms of the eigenvalues λ of the operator W = S † S of the univariateproblem. Moreover, we show that this algorithm is optimal with respect to a wideclass of algorithms and investigate its complexity. We clarify the influence of different(anti-) symmetry conditions on the complexity, compared to the classical unrestrictedproblem. In particular, for symmetric problems with λ ≤ λ and theamount of the assumed symmetry. Finally, we apply our results to the approximationproblem of solutions of the electronic Schr¨odinger equation. Keywords:
Antisymmetry, Hilbert spaces, Tensor Products, Complexity.
In the theory of linear operators S d : H d → G d defined between Hilbert spaces it is well-knownthat we often observe the the so-called curse of dimensionality if we deal with d -fold tensorproduct problems. That is, the complexity of approximating the operator S d by algorithmsusing finitely many pieces of information increases exponentially fast with the dimension d . ∗ This is an extended version of a same-named paper by the author which was published in the Journal ofApproximation Theory [8]. Here all the proofs, as well as some additional assertions, are explicitly included. † Mathematisches Institut, Universit¨at Jena, Ernst-Abbe-Platz 2, 07743 Jena, Germany. Email:[email protected]. Web: http://users.minet.uni-jena.de/˜weimar.
1n the last years there have been various approaches to break this exponential dependenceon the dimension, e.g., we can relax the error definitions. Another way to overcome the curseis to introduce weights in order to shrink the space of problem elements H d . In the case offunction spaces this approach is motivated by the assumption that we have some additionala priori knowledge about the importance of several (groups of) variables.In the present paper we describe an essentially new kind of a priori knowledge. Weassume the problem elements f ∈ H d to be (anti-) symmetric . This allows us to vanquishthe curse and obtain different types of tractability.The problem of approximating wave functions , e.g., solutions of the electronic Schr¨odingerequation , serves as an important example from computational chemistry and physics. Inquantum physics wave functions Ψ describe quantum states of certain d -particle systems.Formally, these functions depend on d blocks of variables y j , which represent the spacialcoordinates and certain additional intrinsic parameters, e.g., the spin , of each particle withinthe system. Due to the Pauli principle , the only wave functions Ψ which are physicallyadmissible are antisymmetric in the sense that Ψ( y ) = ( − | π | Ψ( π ( y )) for all y and allpermutations π on a subset I ⊂ { , . . . , d } of particles with the same spin. Here ( − | π | denotes the sign of π . The above relation means that Ψ only changes its sign if we replaceparticles by each other which possess the same spin. For further details on this topic werefer to Section 5 of this paper and the references given there. Inspired by this applicationwe illustrate our results with some simple toy examples at the end of this section.To this end, let H and G be infinite dimensional separable Hilbert spaces of univariatefunctions f : D ⊂ R → R and consider a compact linear operator S : H → G with singularvalues σ = ( σ j ) j ∈ N . Further, let λ = ( λ j ) j ∈ N = ( σ j ) j ∈ N denote the sequence of the squaresof the singular values of S . Finally, assume S d : H d → G d to be the d -fold tensor productproblem. We want to approximate S d by linear algorithms using a finite number of continuouslinear functionals.By n ent ( ε, d ) we denote the minimal number of information operations needed to achievean approximation with worst case error at most ε > H d . The integer n ent ( ε, d ) is called information complexity of the entire tensor product problem. Further,consider the subspace of all f ∈ H d that are fully symmetric , i.e., f ( x ) = f ( π ( x )) for all x ∈ D d and all permutations π of { , . . . , d } . The minimal number of linear functionals needed to achieve an ε -approximation for thissubspace is denoted by n sym ( ε, d ). Finally, define the subspace of all functions f ∈ H d thatare fully antisymmetric by the condition f ( x ) = ( − | π | f ( π ( x )) for all x ∈ D d and all π n asy ( ε, d ).Since H d is a Hilbert space, the optimal algorithm for the entire tensor product prob-lem is well-known. Moreover, it is known that its worst case error, and therefore also theinformation complexity, can be expressed in terms of λ , i.e. in terms of the squared singularvalues of the univariate problem operator S , see, e.g., Sections 4.2.3 and 5.2 in Novak andWo´zniakowski [4]. It turns out that this algorithm, applied to the (anti-) symmetric problem,calculates redundant pieces of information. Hence, it can not be optimal in this setting.In preparation for our algorithms, Section 2 is devoted to (anti-) symmetric subspacesin a more general fashion than in this introduction. Moreover, there we study some basicproperties. In Section 3 we conclude formulae of algorithms for linear tensor product prob-lems defined on these subspaces. We show their optimality in a wide class of algorithmsand deduce an exact expression for the n -th minimal error in terms of the squared singularvalues of S . Theorem 1 summarizes the main results. Finally, we use this error formula toobtain tractability results in Section 4 and apply them to wave functions in Section 5.Our results yield that in any case (if we deal with the absolute error criterion) n asy ( ε, d ) ≤ n sym ( ε, d ) ≤ n ent ( ε, d ) for every ε > d ∈ N , where for d = 1 the terms coincide, since then we do not claim any (anti-) symmetry.To see that additional (anti-) symmetry conditions may reduce the information complexitydramatically consider the simple case of a linear operator S with singular values σ such that λ = λ = 1 and λ j = 0 for j ≥
3. Then the information complexity of the entire tensorproduct problem can be shown to be n ent ( ε, d ) = 2 d for all d ∈ N and ε < . Hence, the problem suffers from the curse of dimensionality and is therefore intractable . Onthe other hand, our results show that in the fully symmetric setting we have polynomialtractability , because n sym ( ε, d ) = d + 1 for all d ∈ N and ε < . It can be proved that in this case the complexity of the fully antisymmetric problem decreaseswith increasing dimension d and, finally, the problem even gets trivial. In detail, we have n asy ( ε, d ) = max { − d, } for all d ∈ N and ε < , which yields strong polynomial tractability .Next, let us consider a more challenging problem where λ = λ = . . . = λ m = 1 and λ j = 0 for every j > m ≥
2. For m = 2 this obviously coincides with the example studied3bove, but letting m increase may tell us more about the structure of (anti-) symmetrictensor product problems. In this situation it is easy to check that n ent ( ε, d ) = m d and n asy ( ε, d ) = ((cid:0) md (cid:1) , d ≤ m , d > m, for every d ∈ N and all ε < (cid:0) md (cid:1) ≥ d − for d ≤ ⌊ m/ ⌋ , this means that for large m the complexity in the antisym-metric case increases exponentially fast with d up to a certain maximum. Beyond this pointit falls back to zero. The information complexity in the symmetric setting is much harderto calculate for this case. However, it can be seen that we have polynomial tractability, but n sym ( ε, d ) needs to grow at least linearly with d such that the symmetric problem can not bestrongly polynomially tractable, whereas this holds in the antisymmetric setting. The entireproblem again suffers from the curse of dimensionality.The reason why antisymmetric problems are that much easier than their symmetriccounterparts is that from the antisymmetry condition it follows that f ( x ) = 0 if there existcoordinates j and l such that x j = x l . Another explanation for the good tractability behaviorof antisymmetric tensor product problems might be the initial error ε init d . For every choiceof λ it tends to zero as d grows, what is not necessarily the case for the corresponding entireand the symmetric problem, respectively. In fact, we have ε init d, ent = ε init d, sym = λ d/ , whereas ε init d, asy = d Y j =1 λ / j . For a last illustrative example consider the case λ = 1 and λ j +1 = j − β for some β ≥ j ∈ N . That means that we have the two largest singular values σ = σ of S equal to one. The remaining series decays like the inverse of some polynomial. If β = 0 theoperator S is not compact, since ( λ m ) m ∈ N does not tend to zero. Hence, all the informationcomplexities are infinite in this case. For β >
0, any δ >
C > n ent ( ε, d ) ≥ d , n sym ( ε, d ) ≥ d + 1 and n asy ( ε, d ) ≤ Cε − (2 /β + δ ) , for all ε < , d ∈ N . Thus, again for the entire problem we have the curse, whereas the antisymmetric problemis strongly polynomially tractable. Once more, the symmetric problem can shown to bepolynomially tractable. Note that in this example the antisymmetric case is not trivial,because all λ j are strictly positive. If we replace j − β by log − ( j + 1) in this example weobtain (polynomial) intractability even in the antisymmetric setting.Altogether these examples show that exploiting an a priori knowledge about (anti-) sym-metries of the given tensor product problem can help to obtain tractability, but it does notmake the problem trivial in general. We conclude the introduction with a partial summaryof our main complexity results. 4 heorem. Let λ = ( λ m ) m ∈ N denote the non-increasing sequence of the squared singularvalues of S : H → G and assume λ >
0. Then for the information complexity of (anti-)symmetric linear tensor product problems S d we obtain the following characterizations: • The fully symmetric problem is strongly polynomially tractable w.r.t. the normalizederror criterion iff λ ∈ ℓ τ for some τ > λ > λ . Furthermore, in the case λ ≤ λ ∈ ℓ τ and λ < • The fully antisymmetric problem is strongly polynomially tractable w.r.t. the absoluteerror criterion iff λ ∈ ℓ τ for some τ > • the entire tensor product problem is never (strongly) polynomially tractable w.r.t. tonormalized error criterion. Moreover, the problem is strongly polynomially tractablew.r.t. the absolute error criterion iff λ ∈ ℓ τ for some τ > λ < Motivated by the example of wave functions in Section 1, we mainly deal with functionspaces in this section. To this end, we start by defining (anti-) symmetry properties forfunctions which will lead us to orthogonal projections, mapping the function space onto itssubspace of (anti-) symmetric functions. It will turn out that these projections applied toa given basis in the tensor product Hilbert function space lead us to handsome formulaefor orthonormal bases of the subspaces. In a final remark we generalize our approach anddefine (anti-) symmetry conditions for arbitrary tensor product Hilbert spaces based on thededuced results for function spaces.We use a general approach to (anti-) symmetric functions, as it can be found in Section 2.5of Hamaekers [1]. Therefore, for a moment, consider an abstract separable Hilbert space F of real-valued functions defined on a domain Ω ⊂ R d . In this part of the paper let d ≥ F is denoted by h· , ·i F . Moreover, let I = I ( d ) ⊂ { , . . . , d } bean arbitrary given non-empty subset of coordinates. Then we define the set S I = { π : { , . . . , d } → { , . . . , d } | π bijective and π (cid:12)(cid:12) { ,...,d }\ I = id } of all permutations on { , . . . , d } that leave the complement of I fixed. Obviously, thecardinality of this set is given by S I = ( I )!, where π ∈ S I we define the mapping π ′ : Ω → R d , x = ( x , . . . , x d ) π ′ ( x ) = ( x π (1) , . . . , x π ( d ) ) . To abbreviate the notation we identify π and π ′ with each other.For an appropriate definition of partial (anti-) symmetry of functions f ∈ F we need thefollowing simple assumptions. For every π ∈ S I we assume(A1) x ∈ Ω implies π ( x ) ∈ Ω,(A2) f ∈ F implies f ( π ( · )) ∈ F and(A3) there exists c π ≥ f ) such that k f ( π ( · )) | F k ≤ c π k f | F k .Note that these assumptions always hold if F is a d -fold tensor product Hilbert space H d = H ⊗ . . . ⊗ H equipped with a cross norm, as described in the examples of the previoussection.Now we call a function f ∈ F partially symmetric with respect to I (or I -symmetric forshort) if a permutation π ∈ S I applied to the argument x does not affect the value of f .Hence, f ( x ) = f ( π ( x )) for all x ∈ Ω and every π ∈ S I . (1)Moreover, we call a function f ∈ F partially antisymmetric with respect to I (or I -antisymmetric , respectively) if f changes its sign by exchanging the variables x i and x j witheach other, where i, j ∈ I . That is, we have f ( x ) = ( − | π | f ( π ( x )) for all x ∈ Ω and every π ∈ S I , (2)where | π | denotes the inversion number of the permutation π . The term ( − | π | thereforecoincides with the sign , or parity of π and is equal to the determinant of the associatedpermutation matrix. In the case I = 1 we do not claim any (anti-) symmetry, since theset S I = { id } is trivial. For I = { , . . . , d } functions f which satisfy (1) or (2), respectively,are called fully (anti-) symmetric .Note that, in particular, formula (2) yields that the value f ( x ) of (partially) antisym-metric functions f equals zero if x i = x j with i = j and i, j ∈ I . For (partially) symmetricfunctions such an implication does not hold. Therefore, the (partial) antisymmetry propertyis a somewhat more restrictive condition than the (partial) symmetry property with respectto the same subset I . As we will see in the next sections this will also affect our complexityestimates. 6ext, we define the so-called symmetrizer S FI and antisymmetrizer A FI on F with respectto the subset I by S FI : F → F, f S FI ( f ) = 1 S I X π ∈S I f ( π ( · ))and A FI : F → F, f A FI ( f ) = 1 S I X π ∈S I ( − | π | f ( π ( · )) . If there is no danger of confusion we use the notation S I and A I instead of S FI and A FI ,respectively. The following lemma collects together some basic properties which can beproved easily. For details see the appendix of this paper. Lemma 1.
Both the mappings P I ∈ { S I , A I } define bounded linear operators on F with P I = P I . Thus, S I and A I provide orthogonal projections of F onto the closed linearsubspaces S I ( F ) = { f ∈ F | f satisfies (1) } and A I ( F ) = { f ∈ F | f satisfies (2) } (3)of all partially (anti-) symmetric functions (w.r.t. I ) in F , respectively. Hence, F = S I ( F ) ⊕ ( S I ( F )) ⊥ = A I ( F ) ⊕ ( A I ( F )) ⊥ . (4)Note that the notion of partially (anti-) symmetric functions can be extended to morethan one subset I . Therefore, consider two non-empty subsets of coordinates I, J ⊂ { , . . . , d } with I ∩ J = ∅ . Then we call a function f ∈ F multiple partially (anti-) symmetric withrespect to I and J if f satisfies (1), or (2), respectively, for I and J . Since I and J aredisjoint we observe that π ◦ σ = σ ◦ π for all π ∈ S I and σ ∈ S J . Thus, the linear projections P I ∈ { S I , A I } and P J ∈ { S J , A J } commute on F , i.e. P I ◦ P J = P J ◦ P I .Further extensions to more than two disjoint subsets of coordinates are possible. Wewill restrict ourselves to the case of at most two (anti-) symmetry conditions, because inparticular wave functions can be modeled as functions which are antisymmetric with respectto I and and J = I C , where I C denotes the complement of I in { , . . . , d } ; see, e.g., Section 5of this paper.Up to this point the function space F was an arbitrary separable Hilbert space of d -variatereal-valued functions. Indeed, for the definition of (anti-) symmetry we did not claim anyproduct structure. On the other hand, it is also motivated by applications to consider tensor7roduct function spaces; see, e.g., Section 3.6 in Yserentant [10]. In detail, it is well-knownthat so-called spaces of dominated mixed smoothness, e.g. W (1 ,..., ( R d ), can be representedas certain tensor products; see Section 1.4.2 in Hansen [2].Nevertheless, if we take into account such a structure, i.e., assume F = H d = H ⊗ . . . ⊗ H ( d times), where H is a suitable Hilbert space of functions f : D → R , it is known that wecan construct an orthonormal basis (ONB) of F out of a given ONB of H . In fact, if { η i | i ∈ N } is an ONB of the underlying Hilbert function space H then the set of all d -foldtensor products { η d,j = N dl =1 η j l | j = ( j , . . . , j d ) ∈ N d } , η d,j ( x ) = d Y l =1 η j l ( x l ) , x = ( x , . . . , x d ) ∈ D d , is mutually orthonormal in H d and forms a basis. To exploit this representation we startwith a simple observation.Let j ∈ N d and x ∈ D d , as well as a non-empty subset I of { , . . . , d } be arbitrarily fixed.If we define σ = π − ∈ S I then( A I η d,j )( x ) = 1 S I X π ∈S I ( − | π | η d,j ( π ( x )) = 1 S I X π ∈S I ( − | π | d Y m =1 η j m ( x π ( m ) )= 1 S I X π ∈S I ( − | π | d Y m =1 η j σ ( m ) ( x m ) = 1 S I X π ∈S I ( − | σ − | η d,σ ( j ) ( x ) (5)= 1 S I X σ ∈S I ( − | σ | η d,σ ( j ) ( x ) . For simplicity, once again we identified π ( j ) = π ( j , . . . , j d ) with ( j π (1) , . . . , j π ( d ) ) for multi-indices j ∈ N d . Obviously, the same calculation can be made for S I without the factor ( − x ∈ D d was arbitrary we obtain S I η d,j = 1 S I X σ ∈S I η d,σ ( j ) and A I η d,j = 1 S I X σ ∈S I ( − | σ | η d,σ ( j ) for all j ∈ N d . (6)Note that in general, i.e. for arbitrary j ∈ N d and σ ∈ S I , the tensor products η d,σ ( j ) and η d,j do not coincide, because taking the tensor product is not commutative in general. Therefore, S I is not simply the identity on { η d,j | j ∈ N d } . On the other hand, we see that for different j ∈ N d many of the functions S I η d,j coincide. Of course the same holds true for A I η d,j , atleast up to a factor of ( − P I ∈ { S I , A I } a linearly independent subset of allprojections { P I η d,j | j ∈ N d } equipped with suitable normalizing constants can be used as anONB of the linear subspace P I ( H d ) of I -(anti-)symmetric functions in H d . To this end, weneed a further definition. For fixed d ≥ I ⊂ { , . . . , d } , let us introduce a function M I = M I,d : N d → { , . . . , I } I which counts how often different integers occur in a given multi-index j ∈ N d among thesubset I of coordinates, ordered with respect to their rate. To give an example let d = 7and I = { , . . . , } . Then M I, applied to j = (12 , , , , , , ∈ N gives the I = 6dimensional vector M I, ( j ) = (3 , , , , , j contains the number “4” three timesamong the coordinates j , . . . , j , “12” two times and so on. Since in this example there areonly three different numbers involved, the fourth to sixth coordinates of M I, ( j ) equal zero.Obviously, M I is invariant under all permutations π ∈ S I of the argument. Thus, M I ( j ) = M I ( π ( j )) for all j ∈ N d and π ∈ S I . In addition, since M I ( j ) is again a multi-index, we see that | M I ( j ) | = I and M I ( j )! arewell-defined for every j ∈ N d . With this tool we are ready to state the following assertionwhich can be shown using elementary arguments as well as Lemma 1; see the appendix. Lemma 2.
Assume { η d,j | j ∈ N d } to be a given orthonormal tensor product basis of thefunction space H d and let ∅ 6 = I = { i , . . . , i I } ⊂ { , . . . , d } . Moreover, for P I ∈ { S I , A I } define functions ξ j : D d → R , ξ j = s S I M I ( j )! · P I ( η d,j ) for j ∈ N d . Then the set { ξ k | k ∈ ∇ d } builds an orthonormal basis of the partially (anti-) symmetricsubspace P I ( H d ), where ∇ d is given by ∇ d = ( { k ∈ N d | k i ≤ k i ≤ . . . ≤ k i I } , if P I = S I , { k ∈ N d | k i < k i < . . . < k i I } , if P I = A I . (7)Observe that in the antisymmetric case the definition of ξ j for j ∈ ∇ d simplifies, sincethen M I ( j )! = 1 for all j ∈ ∇ d . Moreover, note that in the special case I = { , . . . , I } wehave P I ( H d ) = P I O j ∈ I H ! ⊗ O j / ∈ I H . I -(anti-)symmetric functions f ∈ H d as the tensorproduct of the set of all fully (anti-) symmetric I -variate functions with the ( d − I )-foldtensor product of H . Modifications in connection with multiple partially (anti-) symmetricfunctions are obvious.Finally, note that Lemma 2 also holds if the index set N of the univariate basis { η i | i ∈ N } is replaced by a more general countable set equipped with a total order. But let us shortlyfocus on another generalization of the previous results. Remark 1 (Arbitrary tensor product Hilbert spaces) . Up to now we exclusively dealt withHilbert function spaces. However, the proofs of Lemma 1 and Lemma 2 yield that there areonly a few key arguments in connection with (anti-) symmetry such that we do not need thisrestriction.Starting from the very beginning we need to adapt the definition of I -(anti-)symmetrydue to (1) and (2). Of course it is sufficient to define this property at first only for basiselements. Therefore, if E d = { η d,k | k ∈ N d } denotes a tensor product ONB of H d and ∅ 6 = I ⊂ { , . . . , d } is given then we call an element η d,k = N dl =1 η k l partially symmetric withrespect to I (or I -symmetric for short), if η d,k = η d,π ( k ) for all π ∈ S I , where S I and π ( k ) = ( k π (1) , . . . , k π ( d ) ) are defined as above. Analogously we define I -antisymmetry with an additional factor ( − | π | . Moreover, an arbitrary element in H d iscalled I -(anti-)symmetric if in its basis expansion every element with non-vanishing coeffi-cient possesses this property.Next, the antisymmetrizer A I is defined as the uniquely defined continuous extension ofthe linear mapping A I : E d → H d , A I ( η d,k ) = 1 S I X π ∈S I ( − | π | η d,π ( k ) from E d to H d . Again the symmetrizer S I is given in a similar way. Hence, in the generalsetting we define the mappings using formula (6), which we derived for the special case. Notethat the triangle inequality yields k P I k ≤
1, for P I ∈ { S I , A I } .Once more we denote the sets of all I -(anti-)symmetric elements of H d by P I ( H d ), where P I ∈ { S I , A I } . Observe, that this can be justified since the operators P I again provideorthogonal projections onto closed linear subspaces as described in Lemma 1.Finally, also the proof of Lemma 2 can be adapted to the general Hilbert space case.10 Optimal algorithms
In the present section we conclude optimal algorithms for linear problems defined on (anti-)symmetric subsets of tensor product Hilbert spaces as described in the previous paragraph.Moreover, we deduce formulae for the n -th minimal errors of these (anti-) symmetric prob-lems and recover the known assertions for the entire tensor product problem. Throughout the whole section we use the following notation. Let H be a (infinite dimen-sional) separable Hilbert space with inner product h· , ·i H and let G be some arbitraryHilbert space. Furthermore, assume S : H → G to be a compact linear operator betweenthese spaces and consider its singular value decomposition. That is, define the compactself-adjoint operator W = S † S : H → H and denote its eigenpairs with respect to anon-increasing ordering of the eigenvalues by { ( e i , λ i ) | i ∈ N } , i.e. W ( e i ) = λ i e i , and h e i , e j i H = δ i,j with λ ≥ λ ≥ . . . ≥ . (8)Then λ = ( λ i ) i ∈ N coincides with the sequence of the squared singular values σ = ( σ i ) i ∈ N of S and the set { e i | i ∈ N } forms an ONB of H ; see, e.g., Section 4.2.3 in Novak andWo´zniakowski [4]. In the following we will refer to S as the univariate problem or univariatecase .For d ≥
2, let H d = H ⊗ . . . ⊗ H be the d -fold tensor product space of H . This meansthat H d is the closure of the set of all linear combinations of formal objects f = N dl =1 f l with f l ∈ H , called simple tensors or pure tensors . Here the closure is taken with respect to theinner product in H d which is defined such that * d O l =1 f l , d O l =1 g l + H d = d Y l =1 h f l , g l i H for f l , g l ∈ H . With these definitions H d is also an infinite dimensional Hilbert space and it is easy to checkthat E d = ( η d,j = d O l =1 η j l ∈ H d | j = ( j , . . . , d ) ∈ N d ) (9)forms an orthonormal basis in H d if { η i ∈ H | i ∈ N } is an arbitrary ONB in the underlyingspace H . Similarly, let G d = G ⊗ . . . ⊗ G , d times, and define S d as the tensor productoperator S d = S ⊗ . . . ⊗ S : H d → G d .
11n detail, we define the bounded linear operator e S d : E d → G d such that for all j ∈ N d wehave e S d ( η d,j ) = e S d ( N dl =1 η j l ) = N dl =1 S ( η j l ) ∈ G d . Then S d is assumed to be the uniquelydefined linear, continuous extension of e S d from E d to H d .We refer to the problem of approximating S d : H d → G d as the entire d -variate problem .In contrast, we are interested in the restriction S d (cid:12)(cid:12) P I ( H d ) : P I ( H d ) → G d of S d to some (anti-)symmetric subspace P I ( H d ) with P I ∈ { S I , A I } as described in the previous section. Toabbreviate the notation we denote this restriction again by S d and refer to it as the I -(anti-)symmetric problem .For the singular value decomposition of the entire problem operator S d we consider theself-adjoint, compact operator W d = S d † S d : H d → H d . Its eigenpairs { ( e d,j , λ d,j ) | j = ( j , . . . , j d ) ∈ N d } are given by the set of all d -fold (tensor)products of the univariate eigenpairs (8) of W , i.e., e d,j = d O l =1 e j l and λ d,j = d Y l =1 λ j l for j = ( j , . . . , j d ) ∈ N d . (10)It is well-known how these eigenpairs can be used to construct a linear algorithm A ′ n,d whichis optimal for the entire d -variate tensor product problem. In detail, A ′ n,d minimizes the worst case error e wor ( A n,d ; H d ) = sup f ∈B ( H d ) k A n,d ( f ) − S d ( f ) | G d k among all adaptive linear algorithms A n,d using n continuous linear functionals. Here B ( H d )denotes the unit ball of H d . In other words, A ′ n,d achieves the n -th minimal error e ( n, d ; H d ) = inf A n,d e wor ( A n,d ; H d ) . With this notation our main result reads as follows.
Theorem 1.
Let { ( e m , λ m ) | m ∈ N } denote the eigenpairs of W given by (8). Moreover, for d > ∅ 6 = I = { i , . . . , i I } ⊂ { , . . . , d } and assume S d to be the linear tensor productproblem restricted to the I -(anti-)symmetric subspace P I ( H d ), where P I ∈ { S I , A I } , of the d -fold tensor product space H d . Finally, let ∇ d be given by (7) and define { ( ξ ψ ( v ) , λ d,ψ ( v ) ) | v ∈ N } = { ( ξ k , λ d,k ) | k ∈ ∇ d } (11)12y ξ k = p S I /M I ( k )! · P I ( e k ⊗ . . . ⊗ e k d ) and λ d,k = Q dl =1 λ k l , for k ∈ ∇ d , where ψ : N → ∇ d provides a non-increasing rearrangement of { λ d,k | k ∈ ∇ d } .Then for every d > W d (cid:12)(cid:12) P I ( H d ) = S d † S d . Thus, forevery n ∈ N , the linear algorithm A ∗ n,d : P I ( H d ) → P I ( G d ), A ∗ n,d f = n X v =1 (cid:10) f, ξ ψ ( v ) (cid:11) H d · S d ξ ψ ( v ) , (12)which uses n linear functionals, is n -th optimal for S d on P I ( H d ) with respect to the worstcase setting. Furthermore, it is e ( n, d ; P I ( H d )) = e wor ( A ∗ n,d ; P I ( H d )) = q λ d,ψ ( n +1) . (13)Let us add some remarks on this theorem. First of all, the sum over an empty index set isto be interpreted as zero such that A ∗ ,d f ≡
0. Further, note that the worst case error can beattained with the function ξ ψ ( n +1) . It can be improved neither by non-linear algorithms usingcontinuous information, nor by linear algorithms using adaptive information. Moreover,observe that the classical entire tensor product problem is included as the case I = 1,where we do not claim any (anti-) symmetry. Then ∇ d = N d and the ξ k ’s simply equal thetensor products e d,k = ⊗ dl =1 e k l . Hence, A ∗ n,d = A ′ n,d .The remainder of this section is devoted to the proof of the main result Theorem 1. We start with an auxiliary result which shows that any optimal algorithm A ∗ for S d needsto preserve the (anti-) symmetry properties of its domain of definition, i.e. A ∗ f ∈ P I ( G d ) forall f ∈ P I ( H d ). The following proposition generalizes Lemma 10.2 in Zeiser [11] where thisassertion was shown for the approximation problem, i.e. for S d = id. A comprehensive proofcan be found in the appendix of this paper. Proposition 1.
Let d > ∅ 6 = I ⊂ { , . . . , d } . Furthermore, for X ∈ { H, G } ,let P XI denote the (anti-) symmetrizer P I ∈ { S I , A I } on X d with respect to I and suppose A : P HI ( H d ) → G d to be an arbitrary algorithm for S d . Then, for g ∈ H d ,( S d ◦ P HI )( g ) = ( P GI ◦ S d )( g ) , (14)and for all f ∈ P HI ( H d ) it holds k S d f − Af | G d k = (cid:13)(cid:13) S d f − P GI ( Af ) | G d (cid:13)(cid:13) + (cid:13)(cid:13) Af − P GI ( Af ) | G d (cid:13)(cid:13) . (15)13ence, an optimal algorithm A ∗ for S d preserves (anti-) symmetry, i.e. A ∗ f ∈ P GI ( G d ) for all f ∈ P HI ( H d ) . Beside this qualitative assertion we are interested in explicit error bounds. Therefore,the next proposition shows an upper bound on the worst case error of the algorithm A ∗ n,d given by (12). Proposition 2 (Upper bound) . Under the assumptions of Theorem 1 the worst case errorof A ∗ n,d given by (12) is bounded from above by e wor ( A ∗ n,d ; P I ( H d )) ≤ q λ d,ψ ( n +1) . Proof.
By Lemma 2 we have for all f ∈ P I ( H d ) the unique representation f = X k ∈∇ d h f, ξ k i · ξ k . Therefore, the boundedness of S d together with (11) implies that S d f = X k ∈∇ d h f, ξ k i H d · S d ξ k = X v ∈ N (cid:10) f, ξ ψ ( v ) (cid:11) H d · S d ξ ψ ( v ) for every f ∈ P I ( H d ) . (16)Furthermore, in the case P I = A I it is easy to see that we have h S d ξ j , S d ξ k i G d = S I p M I ( j )! · M I ( k )! · h S d A I e d,j , S d A I e d,k i G d = 1 S I p M I ( j )! · M I ( k )! X π,σ ∈S I ( − | π | + | σ | (cid:10) S d e d,π ( j ) , S d e d,σ ( k ) (cid:11) G d for i, j ∈ ∇ d , because of the commutativity of S d and A I due to (14) in Proposition 1.Obviously, the same calculation can be done in the symmetric case, where P I = S I . Since e d,π ( j ) and e d,σ ( k ) are orthonormal eigenelements of W d = S d † S d : H d → H d , see (10), itis (cid:10) S d e d,π ( j ) , S d e d,σ ( k ) (cid:11) G d = λ d,π ( j ) (cid:10) e d,π ( j ) , e d,σ ( k ) (cid:11) H d = λ d,π ( j ) δ π ( j ) ,σ ( k ) . Hence, similar to theproof of the mutual orthonormality of { ξ k | k ∈ ∇ d } for Lemma 2 we obtain h S d ξ j , S d ξ k i G d = λ d,j δ j,k for all j, k ∈ ∇ d . (17)Therefore, we calculate for n ∈ N and f ∈ P I ( H d ) (cid:13)(cid:13) S d f − A ∗ d,n f | G d (cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X v>n (cid:10) f, ξ ψ ( v ) (cid:11) H d · S d ξ ψ ( v ) | G d (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = X v>n (cid:10) f, ξ ψ ( v ) (cid:11) H d · λ d,ψ ( v ) .
14n the other hand, for f ∈ B ( P I ( H d )), we have by Parseval’s identity1 ≥ k f | P I ( H d ) k = k f | H d k = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X v ∈ N (cid:10) f, ξ ψ ( v ) (cid:11) H d · ξ ψ ( v ) | H d (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = X v ∈ N (cid:10) f, ξ ψ ( v ) (cid:11) H d . Thus, because of the non-increasing ordering of ( λ d,ψ ( v ) ) v ∈ N due to the choice of the rear-rangement ψ , we can estimate the worst case error e wor ( A ∗ n,d ; P I ( H d )) = sup f ∈B ( P I ( H d )) (cid:13)(cid:13) S d f − A ∗ n,d f | G d (cid:13)(cid:13) ≤ λ d,ψ ( n +1) , as claimed. (cid:4) Note that formula (17) in the proof of Proposition 2 together with Lemma 2 yields thatthe set (11) describes the eigenpairs of the self-adjoint operator W d (cid:12)(cid:12) P I ( H d ) = S d † S d : P I ( H d ) → P I ( H d )as stated in Theorem 1. Therefore, the upper bound given in Proposition 2 is sharp and A ∗ n,d in (12) is n -th optimal, due to the general theory; see, e.g., Corollary 4.12 in Novakand Wo´zniakowski [4]. From the general theory it also follows that adaption does not helpto improve this n -th minimal error, see [4, Theorem 4.5], and that linear algorithms are bestpossible; see [4, Theorem 4.8]. Hence, the proof of Theorem 1 is complete.Since it seems to be a little bit unsatisfying to refer to these deep results for the proofof such an easy theorem we refer the reader to the appendix where a nearly self-containedproof of the remaining facts can be found. Moreover, there we describe what we mean byadaption in this context. In this part of the paper we investigate tractability properties of approximating the lineartensor product operator S d on certain (anti-) symmetric subsets P I ( H d ) = P I d ( H d ), where P ∈ { S , A } and ∅ 6 = I d ⊂ { , . . . , d } . Therefore, as usual, we express the n -th minimal errorderived in formula (13) in terms of the information complexity , i.e. the minimal number ofinformation operations needed to achieve an error smaller than a given ε > n ( ε, d ; P I ( H d )) = min { n ∈ N | e ( n, d ; P I ( H d )) ≤ ε } . To abbreviate the notation we write n ent ( ε, d ) if we deal with the entire tensor productproblem. Furthermore, as in the introduction, we denote the information complexity of thefully (anti-) symmetric problem by n asy ( ε, d ) and n sym ( ε, d ), respectively.15 .1 Preliminaries From Theorem 1 we obtain for any ε > d ∈ N n ( ε, d ; P I ( H d )) = min (cid:8) n ∈ N | λ d,ψ ( n +1) ≤ ε (cid:9) = ( k ∈ ∇ d | d Y l =1 λ k l > ε ) by solving (13) for ψ . Using this expression we can easily conclude the results for the firsttwo problems in the introduction. There we dealt with the case λ = . . . = λ m = 1 and λ j = 0 for j > m ≥ n ( ε, d ) increases exponentially in the dimension d we say the problem suffersfrom the curse of dimensionality . That is, there exist constants c > C > ε > n ( ε, d ) ≥ c · C d for infinitely many d ∈ N . More generally, if the information complexity depends exponen-tially on d or ε − we call the problem intractable . Since there are many ways to measure thelack of exponential dependence we distinguish between different types of tractability. Themost important type is polynomial tractability . We say that the problem is polynomiallytractable if there exist constants C, p >
0, as well as q ≥
0, such that n ( ε, d ) ≤ C · ε − p · d q for all d ∈ N , ε ∈ (0 , . If this inequality holds with q = 0, the problem is called strongly polynomially tractable . Ifpolynomial tractability does not hold we say the problem is polynomially intractable . Formore specific definitions and relations between these and other classes of tractability see,e.g., the monographs of Novak and Wo´zniakowski [4, 5, 6].In the following we distinguish two cases. First we consider the absolute error criterion ,where we investigate the dependence of n ( ε, d ; P I ( H d )) on 1 /ε and on the dimension d forevery ε ∈ (0 ,
1] and d ∈ N . Note that without loss of generality we can restrict ourselves to ε ≤ min (cid:8) , ε init d (cid:9) since obviously n ( ε, d ; P I ( H d )) = 0 for all ε ≥ ε init d . Here ε init d = e (0 , d ; P I ( H d )) = q λ d,ψ (1) = (p λ d , if P = S , q λ b d · λ · . . . · λ a d , if P = A describes the initial error of the d -variate problem on the subspace P I ( H d ) where ψ : N →∇ d again is a non-increasing rearrangement of the set of eigenvalues { λ d,k | k ∈ ∇ d } of16 d (cid:12)(cid:12) P I ( H d ) = S d † S d and b d = d − a d denotes the number of coordinates without (anti-)symmetry conditions in dimension d , i.e. a d = I d and b d = d − I d , respectively.Afterwards, we deal with the normalized error criterion , where we especially investigatethe dependence of n ( ε ′ · ε init d , d ; P I ( H d )) on 1 /ε ′ for ε ′ ∈ (0 , ε ′ less than one.To avoid triviality we will assume ε init d >
0, for every d ∈ N , in both cases, becauseotherwise we have strong polynomial tractability by default. From this assumption it followsthat λ >
0, which simply means that S d is not the zero operator. Moreover, note that inthe case of antisymmetric problems, if the number of antisymmetric coordinates, i.e. the set I = I ( d ), grows with the dimension, the condition ε init d > d ∈ N ) even impliesthat λ ≥ λ ≥ . . . > . Finally, we always assume λ >
0, because otherwise S d is equivalent to a continuous lin-ear functional which can be solved exactly with one information operation; see Novak andWo´zniakowski [4, p.176].For the study of tractability for the absolute error criterion we use a slightly modifiedversion of Theorem 5.1, [4]. It deals with the more general situation of arbitrary compactlinear operators between Hilbert spaces. In contrast to Novak and Wo´zniakowski we drop the(hidden) condition ε init d = 1 for the initial error in dimension d . For the sake of completeness aproof can be found in the appendix. If we denote Riemann’s zeta function by ζ the assertionreads as follows. Proposition 3.
Consider a family of compact linear operators { T d : F d → G d | d ∈ N } be-tween Hilbert spaces and the absolute error criterion in the worst case setting. Furthermore,for d ∈ N let ( λ d,i ) i ∈ N denote the non-negative sequence of eigenvalues of T d † T d w.r.t. anon-increasing ordering. • If { T d } is polynomially tractable with the constants C, p > q ≥ τ > p/ C τ = sup d ∈ N d r ∞ X i = f ( d ) λ τd,i /τ < ∞ , (18)where r = 2 q/p and f : N → N with f ( d ) = ⌈ (1 + C ) d q ⌉ .In this case C τ ≤ C /p ζ (2 τ /p ) /τ . 17 If (18) is satisfied for some parameters r ≥ τ > f : N → N suchthat f ( d ) = l C · (cid:0) min (cid:8) ε init d , (cid:9)(cid:1) − p · d q m , where C > p, q ≥
0, then the problemis polynomially tractable and n ( ε, d ) ≤ ( C + C ττ ) ε − max { p, τ } d max { q,rτ } for every d ∈ N and any ε ∈ (0 , ≤ (cid:0) min (cid:8) ε init d , (cid:9)(cid:1) − p for all p ≥ ε init d . Hence, if ε init d is sufficiently small then we can conclude polynomial tractability while ignoring a largerset of eigenvalues in the summation (18).Observe that the first statement does not cover any assertion about the initial error, since f ( d ) ≥
2. Hence, it might happen that we have (strong) polynomial tractability though thelargest eigenvalue λ d, = ( ε init d ) tends faster to infinity than any polynomial. To this end,for d ∈ N , consider the sequences ( λ d,m ) m ∈ N given by λ d, = e d and λ d,m = 1 m , for m ≥ . Here, obviously, the initial error grows exponentially fast to infinity, but nevertheless thesecond point of Proposition 3 shows that { S d } is strongly polynomially tractable, since (18)holds with r = p = q = 0, and C = τ = 2.Let us now return to our I -(anti-)symmetric tensor product problems S d as defined inSection 3. Therefore, let ∅ 6 = I d = { i , . . . , i I d } ⊂ { , . . . , d } and P I d ∈ { S I d , A I d } for every d >
1. We start by using Proposition 3 to conclude a simple necessary condition for (strong)polynomial tractability of { S d } in the worst case setting w.r.t. the absolute error criterion.Recall that ψ : N → ∇ d defines a rearrangement of the parameter set ∇ d given in (7). Thatis, { λ d,ψ ( v ) | v ∈ N } = ( λ d,k = d Y l =1 λ k l | k ∈ ∇ d ) (19)denotes the set of eigenvalues of S d † S d with respect to a non-increasing ordering, see Theorem 1.18 emma 3 (General necessary conditions) . The fact that { S d } is polynomially tractable withthe constants C, p > q ≥ λ = ( λ m ) m ∈ N ∈ ℓ τ for all τ > p/
2. Moreover,for any such τ and all d ∈ N the following estimate holds:1 λ τd,ψ (1) X k ∈∇ d λ τd,k ≤ (1 + C ) d q + C τ/p ζ (cid:18) τp (cid:19) (cid:18) d q/p λ d,ψ (1) (cid:19) τ . Proof.
From Proposition 3 we know that for τ > p/ r = 2 q/p it issup d ∈ N d r ∞ X v = f ( d ) λ τd,ψ ( v ) /τ < ∞ , (20)where the function f : N → N is given by f ( d ) = ⌈ (1 + C ) d q ⌉ .Note that for the proof of the first assertion we only need to consider the case where all λ m are strictly positive. Then the condition (20), in particular, implies that the sum in thebrackets converges for every fixed d ∈ N . If we denote the subset of indices j ∈ ∇ d of the f ( d ) − λ d,ψ ( v ) by L d then there exists a natural number s = s ( d ) ≥ d such that L d is completely contained in the cube Q d,s = { , . . . , s } d . (21)Hence, we can crudely estimate the sum from below by X j ∈∇ d \ Q d,s λ τd,j ≤ X j ∈∇ d \ L d λ τd,j = ∞ X v = f ( d ) λ τd,ψ ( v ) < ∞ . Since R d,s = { j = (1 , , . . . , d − , m ) ∈ N d | m > s } is a subset of ∇ d \ Q d,s , independentlyof the concrete (anti-) symmetrizer P I d , where P ∈ { S , A } , we obtain( λ · λ · . . . · λ d − ) τ ∞ X m = s +1 λ τm = X j ∈ R d,s λ τd,j ≤ X j ∈∇ d \ Q d,s λ τd,j . Thus, for each fixed d ∈ N the tail series P ∞ m = s ( d )+1 λ τm is finite, which is only possible if k λ | ℓ τ k < ∞ . Hence, λ ∈ ℓ τ is necessary for (strong) polynomial tractability.Let us turn to the second assertion. Obviously, (20) implies the existence of some constant C > ∞ X v = f ( d ) λ τd,ψ ( v ) ≤ C d rτ for all d ∈ N . C = C τ/p ζ (2 τ /p ). The rest of the sum canalso be bounded easily for any d ∈ N , f ( d ) − X v =1 λ τd,ψ ( v ) ≤ λ τd,ψ (1) ( f ( d ) − , due to the ordering provided by ψ . Since P k ∈∇ d λ τd,k = P ∞ v =1 λ τd,ψ ( v ) , it remains to show that f ( d ) − ≤ (1 + C ) d q for every d ∈ N with λ d,ψ (1) >
0, which is also obvious due to thedefinition of f . (cid:4) Since we know that antisymmetric problems are easier than symmetric problems we haveto distinguish these cases in order to conclude sharp conditions for tractability.
Beside the general assertion λ ∈ ℓ τ , we start with necessary conditions for (strong) poly-nomial tractability in the symmetric setting. By b d we denote the amount of coordinateswithout symmetry conditions in dimension d . Proposition 4 (Necessary conditions, symmetric case) . Let { S d } be the problem consideredin Lemma 3 and assume P = S . • If { S d } is polynomially tractable and λ ≥ b d ∈ O (ln d ). • If { S d } is strongly polynomially tractable and λ ≥ b d ∈ O (1) and λ < /λ . Proof.
Assume λ ≥ τ be given by Lemma 3. Then, independent of the amount ofsymmetry conditions, we have λ d,ψ (1) = λ d ≥ r ≥ C > λ τd X k ∈∇ d λ τd,k ≤ C d r , d ∈ N , (22)due to Lemma 3. In the case of strong polynomial tractability we even have r = 0. For d ≥ λ d,k , k ∈ ∇ d , and split the sum on the left w.r.t. thecoordinates with and without symmetry conditions. Hence, we conclude X k =( h,j ) ∈∇ d λ τd,k = X j ∈ N bd λ τb d ,j X h ∈ N ad ,h ≤ ... ≤ h ad λ τa d ,h = ∞ X m =1 λ τm ! b d X h ∈ N ad ,h ≤ ... ≤ h ad λ τa d ,h , d = a d + b d ≥ , (23)20hich leads to ∞ X m =1 (cid:18) λ m λ (cid:19) τ ! b d X h ∈ N ad ,h ≤ ... ≤ h ad a d Y l =1 (cid:18) λ h l λ (cid:19) τ ≤ C d r . In any case the second sum in the above inequality is bounded from below by 1. Thus,we conclude that (1 + λ τ /λ τ ) b d ≤ ( P ∞ m =1 λ τm /λ τ ) b d needs to be polynomially bounded fromabove. Since we always assume λ > b d in the case of(strong) polynomial tractability.It remains to show that λ < /λ is necessary for strong polynomial tractability. Tothis end, assume for a moment λ ≥ /λ . Then it is easy to see that (independent of thenumber of symmetry conditions) there are at least 1 + ⌊ d/ ⌋ different k ∈ ∇ d such that λ d,k ≥
1. Namely, for l = 0 , . . . , ⌊ d/ ⌋ we can take the first d − l coordinates of k ∈ ∇ d equalto one. To the remaining coordinates we assign the value two.In other words, we have λ d,ψ (1+ ⌊ d/ ⌋ ) ≥
1. On the other hand, strong polynomial tractabil-ity implies P ∞ v = ⌈ C ⌉ λ τd,ψ ( v ) ≤ C for some absolute constants τ, C, C > d ∈ N ;see (20). Hence, for every d ≥ ⌈ C ⌉ , C ≥ ∞ X v = ⌈ C ⌉ λ τd,ψ ( v ) ≥ ⌊ d/ ⌋ X v = ⌈ C ⌉ λ τd,ψ ( v ) ≥ λ τd,ψ (1+ ⌊ d/ ⌋ ) (2 + ⌊ d/ ⌋ − ⌈ C ⌉ ) ≥ ⌊ d/ ⌋ + 1 − ⌈ C ⌉ , because of the ordering provided by ψ . Obviously, this is a contradiction. Thus, we have λ < /λ and the proof is complete. (cid:4) Note in passing that the previous argument can also be used to show that (independentof the number of symmetry conditions) the information complexity n ( ε, d ) needs to grow atleast linearly in d if we assume λ ≥ /λ . In particular, we cannot have strong polynomialtractability if λ = λ = 1.We continue the analysis of I -symmetric problems with respect to the absolute errorcriterion by proving that the stated necessary conditions are also sufficient for (strong)polynomial tractability. To this end, we need a rather technical preliminary lemma that canbe proven by elementary induction arguments. For the convenience of the reader we includedalso this proof in the appendix. 21 emma 4. Let ( µ m ) m ∈ N be a non-increasing sequence of non-negative real numbers with µ >
0. Then, for all V ∈ N and every d ∈ N , it holds X k ∈ N d , ≤ k ≤ ... ≤ k d µ d,k ≤ µ d d V V + d X L =1 µ − L X j ( L ) ∈ N L ,V +2 ≤ j ( L )1 ≤ ... ≤ j ( L ) L µ L,j ( L ) . (24)Now the sufficient conditions read as follows. Once again, we denote the number ofcoordinates without symmetry conditions in dimension d by b d . Proposition 5 (Sufficient conditions, symmetric case) . Let { S d } be the problem consideredin Lemma 3, assume P = S and let λ = ( λ m ) m ∈ N ∈ ℓ τ for some τ ∈ (0 , ∞ ). • If λ < { S d } is strongly polynomially tractable. • If λ = 1 > λ and b d ∈ O (1) then { S d } is strongly polynomially tractable. • If λ = 1 and b d ∈ O (ln d ) then { S d } is polynomially tractable. Proof. Step 1 . We start the proof by exploiting the property λ ∈ ℓ τ . It is easy to see thatthe ordering of ( λ m ) m ∈ N implies mλ τ m ≤ λ τ + . . . + λ τ m < ∞ X i =1 λ τ i = k λ | ℓ τ k τ < ∞ for any m ∈ N . Hence, there exists some C τ > λ m is bounded from above by C τ · m − r for every r ≤ /τ . Therefore, there is some index such that for every larger m ∈ N we have λ m <
1. We denote the smallest of these indices by m . Similar to the calculationof Novak and Wo´zniakowski in [4, p.180] this leads to ∞ X m = m λ τm ≤ ( p + 1) λ τm + C ττ Z ∞ m + p x − τr dx = ( p + 1) λ τm + C ττ τ r − · m + p ) τr − for every p ∈ N and all τ such that τ r >
1. Thus, in particular, with r = 1 /τ we conclude ∞ X m = m λ τm ≤ ( p + 1) λ τm + 1 /τ /τ − /τ C / (1 /τ − /τ ) τ m + p ! τ (1 /τ − /τ ) for all τ > τ , p ∈ N . δ > τ ≥ τ such that for all τ > τ itis 1 / (1 /τ − /τ ) ∈ ( τ , τ + δ ). Hence, if p ∈ N is sufficiently large then we obtain for all τ > τ ∞ X m = m λ τm ≤ ( p + 1) λ τm + τ + δτ (cid:18) C m + p (cid:19) τ (1 /τ − /τ ) ≤ ( p + 1) λ τm + τ + δτ (cid:18) C m + p (cid:19) τ/ ( τ + δ ) , where we set C = max (cid:8) , C τ + δτ (cid:9) . Finally, since λ m <
1, both the summands tend to zeroas τ approaches infinity. In particular, there need to exist some τ > τ ≥ τ such that ∞ X m = m λ τm ≤ . Step 2 . All the stated assertions can be seen using the second point of Proposition 3.Indeed, for polynomial tractability, it is sufficient to show that X k ∈∇ d λ τd,k ≤ Cd rτ for all d ∈ N (25)and some C, τ > r ≥
0. If this even holds for r = 0 we obtain strongpolynomial tractability.In the case λ < P ∞ m =1 λ τm ) d .Using Step 1 with m = 1 we conclude P k ∈∇ d λ τd,k ≤ − d for some large τ > τ . Hence, theproblem is strongly polynomially tractable in this case.For the proof of the remaining points assume λ = 1. In any case P k ∈∇ λ τ ,k ≤ P ∞ m =1 λ τ m = k λ | ℓ τ k τ < ∞ for all τ ≥ τ , because of λ ∈ ℓ τ . Therefore, we can as-sume d ≥ d = a d + b d ≥ X k =( h,j ) ∈∇ d λ τd,k = X j ∈ N bd λ τb d ,j X h ∈ N ad ,h ≤ ... ≤ h ad λ τa d ,h = ∞ X m =2 λ τm ! b d X h ∈ N ad ,h ≤ ... ≤ h ad λ τa d ,h . (26)If λ < b d is universally bounded then the first factor can be bounded by a constantand the second factor can be estimated using Lemma 4 with V = 0, d replaced by a d and µ λ τ . It follows that if τ is large enough we have X h ∈ N ad ,h ≤ ... ≤ h ad λ τa d ,h ≤ a d X L =1 X j ( L ) ∈ N L , ≤ j ( L )1 ≤ ... ≤ j ( L ) L λ τL,j ( L ) ≤ a d X L =1 ∞ X m =2 λ τm ! L ≤ ∞ X L =1 − L = 2 , where we again used Step 1 and the properties of geometric series. Thus, P k ∈∇ d λ τd,k isuniversally bounded in this case and therefore the problem is strongly polynomially tractable.To prove the last point we argue in the same manner. Now b d ∈ O (ln d ) yields that thefirst factor in the splitting (26) is polynomially bounded in d . For the second factor we againapply Lemma 4, but in this case we set V = m −
2, where m denotes the first index m ∈ N such that λ m <
1. Keep in mind that this index is at least two because of λ = 1. On theother hand it needs to be finite, since λ ∈ ℓ τ . Therefore, the second factor in the splitting(26) is also polynomially bounded in d due to the same arguments as above. All in all, thisproves (25) and the problem is polynomially tractable in this case. (cid:4) We summarize the results obtained for I -symmetric tensor product problems in the fol-lowing theorem. Theorem 2 (Tractability of symmetric problems, absolute error) . Assume S : H → G to be a compact linear operator between two Hilbert spaces and let λ = ( λ m ) m ∈ N denotethe sequence of non-negative eigenvalues of W = S † S w.r.t. a non-increasing ordering.Moreover, for d > ∅ 6 = I d ⊂ { , . . . , d } . Assume S d to be the linear tensor productproblem restricted to the I d -symmetric subspace S I d ( H d ) of the d -fold tensor product space H d , consider the worst case setting w.r.t. the absolute error criterion and let λ ≤ λ ∈ ℓ τ for some τ > • λ <
1, or • λ > λ and ( d − I d ) ∈ O (1).Moreover, the problem is polynomially tractable if and only if λ ∈ ℓ τ for some τ > • λ <
1, or • λ = 1 and ( d − I d ) ∈ O (ln d ). 24 .3 Tractability of symmetric problems (normalized error) Here we briefly focus on the normalized error criterion for the I -symmetric setting. Since( ε init d ) = λ d,ψ (1) = λ d for any kind of symmetric problem, this means that we have toinvestigate the influence of d and 1 /ε ′ on n ( ε ′ · ε init d , d ; S I d ( H d )) = min (cid:8) n ∈ N | λ d,ψ ( n +1) ≤ ( ε ′ ) λ d,ψ (1) (cid:9) = ( k ∈ ∇ d | d Y l =1 (cid:18) λ k l λ (cid:19) > ( ε ′ ) ) for ε ′ ∈ (0 , , d ∈ N . Hence, in fact we have to study the information complexity of a scaled tensor product problem S ′ d : S I d ( H d ) → G d with respect to the absolute error criterion. The squared singular valuesof S ′ equal µ = ( µ m ) m ∈ N with µ m = λ m /λ . Obviously, we always have µ = 1. Furthermore, µ ∈ ℓ τ if and only if λ ∈ ℓ τ . This leads to the following theorem. Theorem 3 (Tractability of symmetric problems, normalized error) . Assume S : H → G to be a compact linear operator between two Hilbert spaces and let λ = ( λ m ) m ∈ N denotethe sequence of non-negative eigenvalues of W = S † S w.r.t. a non-increasing ordering.Moreover, for d > ∅ 6 = I d ⊂ { , . . . , d } . Assume S d to be the linear tensor productproblem restricted to the I d -symmetric subspace S I d ( H d ) of the d -fold tensor product space H d and consider the worst case setting w.r.t. the normalized error criterion.Then the problem is strongly polynomially tractable if and only if λ ∈ ℓ τ for some τ > λ > λ and ( d − I d ) ∈ O (1).Moreover, { S d } is polynomially tractable if and only if λ ∈ ℓ τ for some τ > d − I d ) ∈O (ln d ). We start this subsection with simple sufficient conditions for strong polynomial tractability.
Proposition 6 (Sufficient conditions, antisymmetric case) . Let { S d } be the problem con-sidered in Lemma 3, assume P = A and let λ = ( λ m ) m ∈ N ∈ ℓ τ for some τ ∈ (0 , ∞ ). • If λ < { S d } is strongly polynomially tractable, independent of the number ofantisymmetry conditions. • If λ ≥ τ ≥ τ and d ∈ N such that for the number ofantisymmetric coordinates a d in dimension d it holds thatln ( a d !) d ≥ ln( k λ | ℓ τ k τ ) for all d ≥ d (27)25hen the problem { S d } is also strongly polynomially tractable. Proof.
Like for the symmetric setting, the proof of these sufficient conditions is based on thesecond point of Proposition 3. We show that under the given assumptions ∞ X v =1 λ τd,ψ ( v ) ! /τ ≤ C < ∞ for every d ∈ N and some τ ≥ τ . Once again ψ and ∇ d are given as in (19) and (7), respectively.Since for d = 1 there is no antisymmetry condition we have ψ = id and ∞ X v =1 λ τ ,ψ ( v ) ! /τ = ∞ X v =1 λ τv ! /τ = k λ | ℓ τ k ≤ k λ | ℓ τ k . Therefore, due to the hypothesis λ ∈ ℓ τ the term for d = 1 is finite.Hence, let d ≥ s ∈ N with s ≥ d we define the cubes Q d,s ofmulti-indices similar to (21). With this notation we obtain the representation ∞ X v =1 λ τd,ψ ( v ) = X k ∈∇ d λ τd,k = lim s →∞ X k ∈∇ d ∩ Q d,s λ τd,k . Without loss of generality we may reorder the set of coordinates such that I d = { i , . . . , i a d } = { , . . . , a d } . That is, we assume partial antisymmetry with respect to the first a d coordinates.Furthermore, we define U a d ,s = { j ∈ Q a d ,s | j < j < . . . < j a d } and set b d = d − a d .If b d > ∇ d ∩ Q d,s = U a d ,s × Q b d ,s for all s ≥ d. Because of the product structure of λ d,k ( k ∈ ∇ d ) this implies X k =( j,i ) ∈∇ d ∩ Q d,s λ τd,k = X j ∈ U ad,s a d Y l =1 λ τj l X i ∈ Q bd,s b d Y l =1 λ τi l . Since the sequence λ = ( λ m ) m ∈ N is an element of ℓ τ we can easily estimate the second factorfor every s ≥ d from above by X i ∈ Q bd,s b d Y l =1 λ τi l = b d Y l =1 s X m =1 λ τm = s X m =1 λ τm ! b d ≤ ∞ X m =1 λ τm ! /τ · b d · τ = k λ | ℓ τ k b d · τ . (28)26o handle the first term we need an additional argument. Note that the structure of U a d ,s implies X j ∈ Q ad,s a d Y l =1 λ τj l = X j ∈ Q ad,s ∃ k,m : j k = j m a d Y l =1 λ τj l + a d ! X j ∈ U ad,s a d Y l =1 λ τj l , which leads to the upper bound X j ∈ U ad,s a d Y l =1 λ τj l ≤ a d ! X j ∈ Q ad,s a d Y l =1 λ τj l ≤ a d ! k λ | ℓ τ k a d · τ , where we used the same arguments as in (28). Once again this upper bound does not dependon s ≥ d . Hence, due to d = a d + b d , we conclude ∞ X v =1 λ τd,ψ ( v ) = lim s →∞ X k ∈∇ d ∩ Q d,s λ τd,k ≤ a d ! k λ | ℓ τ k τd for every choice of A I d . Of course, for every 2 ≤ d < d this upper bound is trivially lessthan an absolute constant. Thus, we can assume d ≥ d . Then, due to the hypothesis of thesecond point we have ln( a d !) ≥ ln ( k λ | ℓ τ k τd ), which implies ∞ X v =1 λ τd,ψ ( v ) ! /τ ≤ (cid:18) a d ! k λ | ℓ τ k τd (cid:19) /τ ≤ d ≥ d . Hence, (27) is sufficient for strong polynomial tractability, independent of λ . Therefore itsuffices to show that λ < λ <
1. We know from Step 1 in the proof of Proposition 5 that there exists some τ ≥ τ such that k λ | ℓ τ k τ = ∞ X m =1 λ τm ≤ < . Thus, we see that the right hand side of (27) is negative, whereas the left hand side isnon-negative for every choice of a d . (cid:4) We also briefly comment on this result. First, note that a sequence λ = ( λ m ) m ∈ N thatis not included in any ℓ τ -space, 0 < τ < ∞ , has to converge to zero more slowly than the27nverse of any polynomial, i.e., m − α for α > λ m = 1 / ln( m ) lead to polynomial intractability in the fully antisymmetric setting.Secondly, observe that (27) is quite a weak assumption. For example if we have a d ≥ (cid:24) d ln d α (cid:25) with 0 < α < k λ | ℓ τ k τ )for all sufficiently large d thenln( a d !) d ≥ a d (ln( a d ) − d ≥ α · ln (cid:0) eα · d ln d (cid:1) ln d −→ α > ln( k λ | ℓ τ k τ ) , d → ∞ . If α equals its upper bound, i.e. α = 1 / ln( k λ | ℓ τ k τ ), then the condition (27) does not hold.This also shows that assumptions like a d = (cid:6) d β (cid:7) with β < f ( d ) − λ d,ψ ( v ) where f ( d ) may grow polynomially in ( ε init d ) − with d . We did not use this fact in the proof of thesufficient conditions.Let us now turn to the necessary conditions. We will see that we need a conditionsimilar to (27) in order to conclude polynomial tractability if we deal with slowly decreasingeigenvalues λ . Proposition 7 (Necessary conditions, antisymmetric case) . Let { S d } be the problem con-sidered in Lemma 3 and assume P = A . Furthermore, let { S d } be polynomially tractablewith the constants C, p > q ≥ d tending to infinity, the initial error ε init d tends to zero faster than the inverse ofany polynomial. Furthermore, λ = ( λ m ) m ∈ N ∈ ℓ τ for every τ > p/ δ > d ∈ N such thatln ( k λ | ℓ τ k τ ) − δ ≤ d a d X k =1 ln (cid:18) k λ | ℓ τ k τ λ τk (cid:19) for all d ≥ d . (29)Thus, we have λ < d →∞ a d = ∞ . Proof. Step 1 . For the whole proof assume τ > p/ λ ∈ ℓ τ . Moreover, we again use the notation d = a d + b d , where a d = I d denotes the numberof coordinates with antisymmetry conditions in dimension d . Similar to the symmetric casewe can split the sum of the eigenvalues such that for all d ∈ N X k ∈∇ d λ τd,k = ∞ X m =1 λ τm ! b d X j ∈ N ad , ≤ j <...
0. On the otherhand, due to the general condition λ >
0, the term k λ | ℓ τ k τ /λ τ is strictly larger than one.Thus, it follows that there exists some C > b d k ≤ C ln( d k ) for every k ∈ N . Therefore, we have in particular a d k = d k − b d k → ∞ , for k → ∞ . Moreover, the assumedboundedness of λ d k ,ψ (1) P ( d k ) leads to C P ( d k ) − ≤ λ d k ,ψ (1) ≤ λ C ln( d k )1 · λ · . . . λ a dk = d C ln( λ ) k · λ · . . . λ a dk since λ ≥
1. As we showed in Step 1 of the proof of Proposition 5 the fact λ ∈ ℓ τ yieldsthe existence of some C τ > λ m ≤ C τ m − /τ for every m ∈ N . Indeed, thisholds for C τ = k λ | ℓ τ k , which needs to be larger than one because of λ ≥
1. Hence, λ τ · . . . · λ τa dk ≤ C τa dk τ ( a d k !) − , what implies (cid:16) a d k e (cid:17) a dk ≤ a d k ! ≤ ( C ττ ) a dk P ( d k ) , k ∈ N for some other polynomial P >
0. Thus, if k is sufficiently large we conclude a d k ≤ a d k ln (cid:18) a d k e C ττ (cid:19) ≤ ln( P ( d k )) , since a d k → ∞ implies a d k / ( e C ττ ) ≥ e for k ≥ k . Therefore, the number of antisymmetriccoordinates a d needs to be logarithmically bounded from above for every d out of the sequence29 d k ) k ≥ k . Because also b d k was found to be logarithmically bounded this is a contradictionto the fact d k = a d k + b d k . Thus, the hypothesis λ d k ,ψ (1) P ( d k ) ≥ C > d k ) k . In other words it holds (31). Step 3 . Next we show (29). From the former step we know that there needs to exist some d ∗ ∈ N such that 1 /λ d,ψ (1) ≥ d ≥ d ∗ . Hence, (30) together with τ > p/ (cid:18) k λ | ℓ τ k τ λ τ (cid:19) b d ≤ C (cid:18) d q/p λ d,ψ (1) (cid:19) τ = C d qτ/p λ τb d · λ τ · . . . · λ τa d for d ≥ d ∗ , where we set C = 1 + C + C τ/p ζ (2 τ /p ). Therefore, we conclude1 C d qτ/p k λ | ℓ τ k τd ≤ a d Y k =1 k λ | ℓ τ k τ λ τk for all d ≥ d ∗ , which is equivalent toln ( k λ | ℓ τ k τ ) − ln( C d qτ/p ) d ≤ d a d X k =1 ln (cid:18) k λ | ℓ τ k τ λ τk (cid:19) . (32)Obviously, for given δ >
0, there is some d ∗∗ such that ln( C d qτ/p ) /d < δ for all d ≥ d ∗∗ .Hence, we can choose d = max { d ∗ , d ∗∗ } in order to obtain (29). Step 4 . It remains to show that λ ≥ d →∞ a d is infinite. To this end,note that every summand in (32) is strictly positive. If we assume for a moment the existenceof a subsequence ( d k ) k ∈ N such that a d k is bounded for every k ∈ N then the right hand side of(32) is less than some positive constant divided by d k . Hence, it tends to zero if k approachesinfinity. On the other hand, for large d , the left hand side of (32) is strictly larger than somepositive constant, because of λ ≥ λ >
0. This contradiction completes the proof. (cid:4)
As mentioned before there are examples such that the sufficient condition (27) fromProposition 6 is also necessary (up to some constant factor) in order to conclude polynomialtractability in the antisymmetric setting. Now we are ready to give such an example.
Example 1.
Consider the situation of Lemma 3 for P = A and assume the problem { S d } to be polynomially tractable. In addition, for a fixed τ ∈ (0 , ∞ ), let λ = ( λ m ) m ∈ N ∈ ℓ τ begiven such that λ ≥ m ∈ N such that λ m ≥ k λ | ℓ τ k m α/τ for all m > m and some α > . (33)30hen we claim that for every δ > d ∈ N such that (cid:18) α − δ (cid:19) ln ( k λ | ℓ τ k τ ) ≤ ln( a d !) d for all d ≥ ¯ d. (34)Recall that due to Proposition 6, for the amount of antisymmetry a d , it was sufficient toassume ln ( k λ | ℓ τ k τ ) ≤ ln( a d )! d for every d larger than some fixed d ∈ N in order to conclude strong polynomial tractability.Before we prove the claim it might be useful to give a concrete example where (33) holdstrue. To this end, set λ m = 1 /m , τ = 1, α = 3 and m = 2. Then it is easy to check that k λ | ℓ τ k = ζ (2) = π / λ = 1.To see that the claimed inequality (34) holds true we can use Proposition 7 and, inparticular, inequality (32). Since λ ≥ d a d = ∞ , i.e. a d > m for every d larger than some d ∈ N . Moreover, note that (33) is equivalent toln (cid:18) k λ | ℓ τ k τ λ τm (cid:19) ≤ α ln( m ) for all m > m . Hence, if d ≥ d we can estimate the right hand side of (32) from above by1 d a d X k =1 ln (cid:18) k λ | ℓ τ k τ λ τk (cid:19) ≤ m d · ln (cid:18) k λ | ℓ τ k τ λ τm (cid:19) + αd a d X k = m +1 ln( k ) ≤ C λ d + α ln( a d !) d . Consequently, this leads to1 α ln ( k λ | ℓ τ k τ ) − C λ + ln( C d qτ/p ) α · d ≤ ln( a d !) d for d ≥ max { d ∗ , d } . Now (34) follows easily by choosing ¯ d ≥ max { d ∗ , d } large enough such that the negativeterm on the left is smaller than a given δ > k λ | ℓ τ k τ ).Although there remains a small gap between the necessary and the sufficient conditionsfor the absolute error criterion, the most important cases of antisymmetric tensor productproblems are covered by our results. We summarize the main facts in the next theorem. Theorem 4 (Tractability of antisymmetric problems, absolute error) . Let S : H → G be acompact linear operator between two Hilbert spaces and let λ = ( λ m ) m ∈ N denote the sequence31f non-negative eigenvalues of W = S † S w.r.t. a non-increasing ordering. Moreover, for d > ∅ 6 = I d ⊂ { , . . . , d } . Assume S d to be the linear tensor product problem restrictedto the I d -antisymmetric subspace A I d ( H d ) of the d -fold tensor product space H d and considerthe worst case setting with respect to the absolute error criterion.Then for the case λ < • { S d } is strongly polynomially tractable. • { S d } is polynomially tractable. • There exists a universal constant τ ∈ (0 , ∞ ) such that λ ∈ ℓ τ .Moreover, the same equivalences hold true if λ ≥ I d grows linearly with the dimen-sion d .Finally, before we continue with the normalized error criterion, we want to deduce anexact formula for the complexity in the case of fully antisymmetric functions. Hence, we set I = I d = { , . . . , d } for every d ∈ N and consider S d : A I ( H d ) → G d in the following. In this case the set of parameters ∇ d is given by ∇ d = { k = ( k , . . . , k d ) ∈ N d | k < k < . . . < k d } . Thus, to obtain a worst case error less or equal than a given ε > n asy ( ε, d ) = n ( ε, d ; A ( H d )) = ( k ∈ ∇ d | λ d,k = d Y l =1 λ k l > ε ) linear functionals for the d -variate case.Due to the ordering of ( λ m ) m ∈ N the largest eigenvalue λ d,k with k ∈ ∇ d is given by thesquare of ε init d = √ λ · λ · . . . · λ d . In other words, it is n asy ( ε, d ) = 0 for all ε ≥ ε init d . To calculate the cardinality also for ε < ε init d let us define i d ( δ ) = min (cid:8) i ∈ N | α i = λ i · λ i +1 · . . . · λ i + d − ≤ δ (cid:9) , for δ > d ∈ N . (35)Using this notation we can formulate the following assertion. Again the proof can be foundin the appendix of this paper. 32 roposition 8 (Complexity of fully antisymmetric problems) . Let { S d } be the problemconsidered in Lemma 3 and assume P = A as well as I = I d = { , . . . , d } .Then for every ε > n asy ( ε, d ) = n ent ( ε, d = 1,and n asy ( ε, d ) (36)= i d ( ε ) X l =2 i d − ( ε /λ l − ) X l = l +1 . . . i ( ε / [ λ l − · ... · λ ld − − ]) X l d − = l d − +1 h n ent (cid:16) ε/ q λ l − · . . . · λ l d − − , (cid:17) − l d − + 1 i if d ≥
2. Here the quantities i j , for j = 2 , . . . , d , are defined as in (35). Remark 2.
If we define α ( k ) m = k − Y l =0 λ m + l , m ∈ N , for k ∈ N and a non-increasing sequence λ ≥ λ ≥ . . . >
0, then we can interpret thequantities i k ( δ ) as information complexities of modified univariate problems S ( k )1 . In detail,let S ( k )1 : H → G define a compact linear operator such that W ( k )1 = (cid:16) S ( k )1 (cid:17) † (cid:16) S ( k )1 (cid:17) : H → H possesses the eigenvalues { α ( k ) m | m ∈ N } . Then n ent ( δ,
1) = n ent ( δ, S ( k )1 : H → G ) = i k ( δ ) − δ > . Further, note that for k ≥ i k ( ε / [ λ l − · . . . · λ l d − k − ]) are non-increasingfunctions in l , . . . , l d − k and ε .Out of Proposition 8 we can conclude bounds on the information complexity. If d ≥ ε < ε init d then the sum in (36) contains at least the term with the index l = 2 , l =3 , . . . , l d − = d . That is, for any choice of λ we get the lower bound n asy ( ε, d ) ≥ n ent (cid:16) ε/ p λ · . . . · λ d − , (cid:17) − d + 1 , which can be used to show that we cannot expect the same nice conditions for (strong)polynomial tractability as before if we switch from the absolute to the normalized errorcriterion. We conclude this subsection with a corresponding example.33 xample 2. Assume λ m = m − α for all m ∈ N and some α >
0. Then we need to estimate n ent (cid:16) ε/ p λ · . . . · λ d − , (cid:17) = n ent ( ε · ( d − α , (cid:8) m ∈ N | m − α > ( ε · ( d − α ) (cid:9) = (cid:26) m ∈ N | m < ε /α · ( d − (cid:27) ≥ ε /α · ( d − − . Therefore, n asy ( ε, d ) ≥ ε /α · ( d − − d = d (cid:18) ε /α · d ! − (cid:19) if d ≥ ε < ε init d . Since in this case the initial error ε init d for the d -variate problem equals 1 /d ! α , we need atleast n asy ( ε ′ · ε init d , d ) ≥ d (cid:18) ε ′ ) /α − (cid:19) linear functionals to improve the initial error by a factor ε ′ <
1. Because this bound growslinearly with the dimension the problem is not strongly polynomially tractable with respect tothe normalized error criterion . Nevertheless, the sequence λ is an element of l /α , say, whichimplies strong polynomial tractability for the absolute error criterion due to Theorem 4. Up to now every complexity assertion in this paper was mainly based on Proposition 3which dealt with the general situation of arbitrary compact linear operators between Hilbertspaces and with the absolute error criterion. While investigating tractability properties of I -symmetric problems with respect to the normalized error criterion, we were able to useassertions from the absolute error setting. Since for I -antisymmetric problems the struc-ture of the initial error is more complicated, this approach will not work again. Therefore,we start this subsection with a modified version of another known theorem by Novak andWo´zniakowski [4, Theorem 5.2]. Proposition 9.
Consider a family of compact linear operators { T d : F d → G d | d ∈ N } between Hilbert spaces and the normalized error criterion in the worst case setting. Further-more, for d ∈ N let ( λ d,i ) i ∈ N denote the non-negative sequence of eigenvalues of T d † T d w.r.t.a non-increasing ordering. 34 If { T d } is polynomially tractable with the constants C, p > q ≥ τ > p/ C τ = sup d ∈ N d r ∞ X i = f ( d ) (cid:18) λ d,i λ d, (cid:19) τ /τ < ∞ , (37)where r = 2 q/p and f : N → N with f ( d ) ≡ C ττ ≤ C + C τ/p ζ (2 τ /p ). • If (37) is satisfied for some parameters r ≥ τ > f : N → N such that f ( d ) = ⌈ C d q ⌉ , where C > q ≥
0, then the problem is polynomially tractableand n ( ε ′ · ε init d , d ) ≤ ( C + C ττ ) ( ε ′ ) − τ d max { q,rτ } .Note that this shows that (strong) polynomial tractability is characterized by the bound-edness of the sum over the normalized eigenvalues, were we are allowed to omit the Cd q largest of them. Of course, our results are equivalent to the assertions given by Novak andWo´zniakowski [4], as one can see easily. But now the connection between the different errorcriterions is more obvious. From this point of view Proposition 9 reads more natural than[4, Theorem 5.2]. The key is to apply the same proof technique for both the assertions.Moreover, observe that also the theorem in [4] for the normalized error criterion includesfurther assertions concerning, e.g., the exponent of strong polynomial tractability. Againour proof implies the same results.Similar to the former sections we continue with an application of Proposition 9 to ourantisymmetric tensor product problems. To this end, assume S : H → G to be a compactlinear operator between two Hilbert spaces and let λ = ( λ m ) m ∈ N denote the sequence ofnon-negative eigenvalues of W = S † S w.r.t. a non-increasing ordering. Moreover, for d > ∅ 6 = I d = { , . . . , d } . Assume S d to be the linear tensor product problem restricted to the I d -antisymmetric subspace A I d ( H d ) of the d -fold tensor product space H d and consider theworst case setting w.r.t. the normalized error criterion. Finally, let b d denote the numberof coordinates without antisymmetry conditions in dimension d , i.e. b d = d − a d , where a d = I d for d ∈ N . Proposition 10 (Necessary conditions, antisymmetric case) . Under these assumptions thefact that { S d } is polynomially tractable with the constants C, p > q ≥ λ ∈ ℓ τ for all τ > p/ d tending to infinity, ε init d tends to zero faster than the inverse of any polynomialand b d ∈ O (ln d ). Thus, lim d →∞ a d /d = 1.In addition, if { S d } is strongly polynomially tractable then b d ∈ O (1).35 roof. From Proposition 9 it follows that there is some C > d ∈ N λ τd,ψ (1) X k ∈∇ d λ τd,ψ ( v ) = ∞ X v =1 (cid:18) λ d,ψ ( v ) λ d,ψ (1) (cid:19) τ ≤ C d τq/p , if τ > p/
2. Once more the rearrangement function ψ and the index set ∇ d are given as in (19).Indeed, the proof of Proposition 9 yields that it is sufficient to take C = 1+ C + C τ/p ζ (2 τ /p ).In particular, for d = 1 it is ∇ = N and λ ,k = λ k , for k ∈ N , such that we have ψ = idbecause of the ordering of λ . Hence, we conclude k λ | ℓ τ k τ = ∞ X k =1 λ τk ≤ C λ τ < ∞ . In other words, λ ∈ ℓ τ . Moreover, like with the arguments of Step 1 in the proof ofProposition 7, it follows (cid:18) k λ | ℓ τ k τ λ τ (cid:19) b d ≤ C d τq/p , d ∈ N , (38)since λ d,ψ (1) = λ b d · λ · . . . · λ a d and k λ | ℓ τ k τ > λ τ due to the general assertion λ >
0. Thus,polynomial tractability of { S d } implies b d ≤ C ln( d ) for some C >
0, i.e. b d ∈ O (ln d ).Therefore, obviously, we have1 ≥ a d d = 1 − b d ln d · ln dd ≥ − C · ln dd −→ , d → ∞ . The proof that strong polynomial tractability leads to b d ∈ O (1) can be obtained using(38) with the same arguments as before and q = 0. Finally, we need to show the assertionconcerning ε init d . To this end, we refer to Step 2 in the proof of Proposition 7. (cid:4) During the last decades there has been considerable interest in finding approximations of wave functions , e.g., solutions of the electronic Schr¨odinger equation. Due to the so-called
Pauli principle of quantum physics only functions with certain (anti-) symmetry propertiesare of physical interest.In this last section of the present paper we briefly introduce wave functions and showhow our results allow to handle the approximation problem for such classes of functions. For36 more detailed view, see, e.g, Hamaekers [1], Yserentant [10], or Zeiser [11]. Furthermore,for a comprehensive introduction to the topic, as well as a historical survey, we refer thereader to Hunziker and Sigal [3] and Reed and Simon [7].In particular, the notion of multiple partial antisymmetry with respect to two sets ofcoordinates is useful for describing wave functions Ψ. In computational chemistry such func-tions occur as models which describe quantum states of certain physical d -particle systems.Formally, these functions depend on d blocks of variables y i = ( x ( i ) , s ( i ) ), for i = 1 , . . . , d ,which represent the spacial coordinates x ( i ) = ( x ( i )1 , x ( i )2 , x ( i )3 ) ∈ R and certain additionalintrinsic parameters s ( i ) ∈ C of each particle y within the system. Hence, rearranging thearguments such that x = ( x (1) , . . . , x ( d ) ) and s = ( s (1) , . . . , s ( d ) ) yields thatΨ : ( R ) d × C d → R , ( x, s ) Ψ( x, s ) . In the case of systems of electrons one of the most important parameters is called spin andit can take only two values, i.e., s ( i ) ∈ C = {− , + } . Due to the Pauli principle the onlywavefunctions Ψ that are physically admissible are those which are antisymmetric in thesense that for I ⊂ { , . . . , d } and I C = { , . . . , d } \ I Ψ( π ( x ) , π ( s ))) = ( − | π | Ψ( x, s ) for all π ∈ S I ∪ S I C . Thus, Ψ changes its sign if we replace any particles y i and y j by each other which possesthe same spin, i.e. s ( i ) = s ( j ) . So, the set of particles, and therefore also the set of spacialcoordinates, naturally split into two groups I + and I − . In detail, for wave functions of d particles y i we can (without loss of generality) assume that the first I + indices i belongto the group of positive spin, whereas the rest of them possess negative spin, i.e. I + = { , . . . , I + } and I − = { I + + 1 , . . . , d } .In physics it is well-known that some problems, e.g., the electronic Schr¨odinger equation,which involve (general) wave functions can be reduced to a bunch of similar problems, whereeach of them only acts on functions Ψ s out of a certain Hilbert space F d = F d ( s ). That is,Ψ s = Ψ( · , s ) ∈ F d = { f : ( R ) d → R } with a given fixed spin configuration s ∈ C d . Of course, every possible spin configuration s corresponds to exactly one choice I + ⊂ { , . . . , d } of indices. Moreover, it is known that F d is a Hilbert space which possesses a tensor product structure. Therefore, we can modelwave functions as elements of certain classes of smoothness, e.g., F d ⊂ H d = W (1 ,..., ( R d ),as Yserentant [10] recently did, and incorporate spin properties by using the projections ofthe type A = A I + ◦ A I − , as defined in Section 2.In particular, Lemma 2 yields F d = A ( H d ) = A I + ( H I + ) ⊗ A I − ( H I − )37nd the system of all e ξ k = p S I + · S I − · A ( η k ) , k ∈ e ∇ d , with e ∇ d = { k = ( i, j ) ∈ N I + + I − | i < i < . . . < i I + and j < . . . < j I − } builds an orthonormal basis of F d = A ( H d ), where the set { η k | k = ( k , . . . , k d ) ∈ N d } is onceagain assumed to be an orthonormal tensor product basis of H d = H ⊗ . . . ⊗ H constructedwith the help of { η i | i ∈ N } , an arbitrary orthonormal basis of H .Note that in the former sections the underlying Hilbert space H always consists ofunivariate functions. In contrast wave functions of one particle depend on three variables,but we want to stress the point that this is just a formal issue. However, this approachradically decreases the degrees of freedom and improves the solvability of certain problems S d like the approximation problem, i.e. S = id : H → G , considered in connection withthe electronic Schr¨odinger equation.Theorem 1 then provides an algorithm which is optimal for the G d -approximation of d -particle wave functions in F d with respect to all linear algorithms that use at most n continuous linear functionals. Moreover, the error can be calculated exactly in terms of thesquared singular values λ = ( λ m ) m ∈ N of S .Furthermore, it is possible to prove a modification of Theorem 4 for problems dealing withwave functions. In fact, for the mentioned approximation problem polynomial tractabilityas well as strong polynomial tractability are equivalent to the fact that the sequence λ ofthe squared singular values of the univariate problem belong to some ℓ τ -space if we considerthe absolute error criterion. The reason is that all the assertions in Section 4.4 can be easilyextended to the multiple partially antisymmetric case. In detail, if we denote the numberof antisymmetric coordinates x ( i ) within each antisymmetry group I md ⊂ { , . . . , d } by a d,m , m = 1 , . . . , M , then the constraint a d + b d = d extends to a d, + . . . + a d,M + b d = d . Here b d again denotes the number of coordinates without any antisymmetry condition. In conclusion,the sufficient condition (27) in Proposition 6 transfers to1 d M X m =1 ln( a d,m !) ≥ k λ | ℓ τ k τ , for all d ≥ d , which is always satisfied in the case of wave functions, since then M = 2 and at leastthe cardinality a d,m of one of the groups of the same spin needs to grow linearly with thedimension d . 38 cknowledgments The author thanks E. Novak and H. Wo´zniakowski for their valuable comments on this paper.
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Proof of Lemma 1 in Section 2
We show Lemma 1 in the case of function spaces.
Proof.
Obviously, P ∈ { S I , A I } is well-defined due to the assumptions (A1) and (A2). Thelinearity directly follows from the definition and, using (A3), the operator norm is boundedby max { c π | π ∈ S I } .To show that the operators are idempotent, i.e. P = P , we first prove that A I ( f )satisfies (2) for every f ∈ F . Therefore, we use the representation( A I ( f ))( π ( x )) = X σ ∈S I ( − | σ | f ( σ ( π ( x ))) = X λ ∈S I ( − | λ | + | π | f ( λ ( x )) = ( − | π | ( A I ( f ))( x )for every fixed π ∈ S I . Here we imposed λ = σ ◦ π ∈ S I and used (cid:12)(cid:12) λ ◦ π − (cid:12)(cid:12) = | λ | + (cid:12)(cid:12) π − (cid:12)(cid:12) = | λ | + | π | . Hence, we have A I ( F ) ⊂ { f ∈ F | f satisfies (2) } . In a second step, it is easy to check thatfor every function g ∈ F which satisfies (2) it is A I ( g ) = g . Thus, { f ∈ F | f satisfies (2) } ⊂ A I ( F ) and A I is a projector onto A I ( F ).Since the same arguments also apply for the symmetrizer S I this shows (3), as well as P = P for P ∈ { S I , A I } . Because of the boundedness of the operators the subsets S I ( F )and A I ( F ) are closed linear subspaces of F and we obtain the orthogonal decompositions F = S I ( F ) ⊕ ( S I ( F )) ⊥ = A I ( F ) ⊕ ( A I ( F )) ⊥ , where the ⊥ denotes the orthogonal complement with respect to h· , ·i F , i.e. the image of theprojectors (id − S I ) and (id − A I ), respectively. (cid:4) The proof of Lemma 1 in the case of arbitrary tensor product Hilbert spaces works exactlyin the same way.
Proof of Lemma 2 in Section 2
We prove Lemma 2 in the case of function spaces. For the case of arbitrary tensor prod-uct Hilbert spaces only slight modifications are needed. Indeed, the only difference is theconclusion of formula (39) in Step 2. In the general setting this simply follows from ourdefinitions. 41 roof. Step 1 . We start by proving orthonormality. Therefore, let us recall (6). To abbreviatethe notation further, we suppress the index H d at the inner products h· , ·i H d in this proof.For P I = A I and j, k ∈ ∇ d easy calculus yields h ξ j , ξ k i = S I p M I ( j )! · M I ( k )! h A I ( η d,j ) , A I ( η d,k ) i = 1 S I p M I ( j )! · M I ( k )! X π,σ ∈S I ( − | π | + | σ | (cid:10) η d,π ( j ) , η d,σ ( k ) (cid:11) . Of course, up to the factor controlling the sign, the same is true for the case P I = S I .Assume now there exists m ∈ { , . . . , d } such that j m = k m . Then the ordering of j, k ∈ ∇ d implies that π ( j ) = σ ( k ) for all σ, π ∈ S I , since π and σ leave the coordinates m ∈ I C fixed.Hence, we conclude that we have π ( j ) = σ ( k ) only if j = k .At this point we have to distinguish the antisymmetric and the symmetric case. For P = A I the only way to conclude π ( j ) = σ ( k ) is to claim j = k and π = σ . Furthermore, wesee that in the antisymmetric case we have M I ( j )! = 1 for all j ∈ ∇ d , since all coordinates j l ,where l ∈ I , differ. Therefore, in this case the last inner product coincides with δ j,k · δ π,σ because of the mutual orthonormality of { η d,j | j ∈ N d } . Hence, h ξ j , ξ k i = 1 S I X π ∈S I ( − | π | δ j,k = δ j,k for all j, k ∈ ∇ d as claimed.So, let us consider the case P I = S I and j = k ∈ ∇ d , because we already saw thatotherwise h ξ j , ξ k i equals zero. Then for fixed σ ∈ S I there are M I ( j )! different permutations π ∈ S I such that π ( j ) = σ ( j ). This leads to h ξ j , ξ j i = 1 S I · M I ( j )! X σ ∈S I M I ( j )! = 1and completes the proof of orthonormality. Step 2 . It remains to show that the span of { ξ k | k ∈ ∇ d } is dense in P I ( H d ) for P ∈{ S I , A I } . To this end, note that every multi-index j ∈ N d can be represented by a uniquelydefined multi-index k ∈ ∇ d and exactly M I ( k )! different permutations π ∈ S I such that j = π ( k ).Now assume f ∈ A I ( H d ), i.e. f ∈ H d satisfies (2). Then the expansion of f ( π ( x )) withrespect to the basis functions { η d,j | j ∈ N d } in H d yields for π ∈ S I ( − | π | X j ∈ N d h f, η d,j i · η d,j ( x ) = X k ∈ N d h f, η d,k i · η d,k ( π ( x )) for every x ∈ D d . η d,k ( π ( x )) = η d,σ ( k ) ( x ) with σ = π − . Therefore,we conclude for x ∈ D d X j ∈ N d (cid:0) ( − | π | · h f, η d,j i (cid:1) · η d,j ( x ) = X k ∈ N d (cid:10) f, η d,π ( σ ( k )) (cid:11) · η d,σ ( k ) ( x ) = X j ∈ N d (cid:10) f, η d,π ( j ) (cid:11) · η d,j ( x ) . Because the expansion is uniquely defined we get( − | π | · h f, η d,j i = (cid:10) f, η d,π ( j ) (cid:11) for all j ∈ N d and π ∈ S I . (39)Using the observations from the beginning of this step we can decompose the basis expansionof f ∈ A I ( H d ) ⊂ H d and use the derived formula (39) to get f = X j ∈ N d h f, η d,j i η d,j = X k ∈∇ d X σ ∈S I (cid:10) f, η d,σ ( k ) (cid:11) η d,σ ( k ) M I ( k )! = X k ∈∇ d M I ( k )! X σ ∈S I ( − | σ | h f, η d,k i η d,σ ( k ) . Now (6) yields that f = X k ∈∇ d s S I M I ( k )! · h f, η d,k i · s S I M I ( k )! · A I ( η d,k ) . Furthermore, summing up (39) with respect to π leads to h f, η d,k i = 1 S I X π ∈S I ( − | π | (cid:10) f, η d,π ( k ) (cid:11) = * f, S I X π ∈S I ( − | π | η d,π ( k ) + = h f, A I ( η d,k ) i , for k ∈ ∇ d , such that finally f ∈ A I ( H d ) possesses the following representation f = X k ∈∇ d h f, ξ k i · ξ k , since ξ k = p S I /M I ( k )! · A I ( η d,k ) per definition. This proves the assertion for the case P I = A I . The remaining case P I = S I can be treated in the same way. (cid:4) Proof of Proposition 1 in Section 3.2
Proof.
The proof is organized as follows. First we show that the problem operator S d andthe (anti-) symmetrizer P I commute on H d , i.e. it holds (14). In a second step we conclude4315) out of this. The (anti-) symmetry of A ∗ f for an optimal algorithm A ∗ then followsimmediately. Step 1 . Assume E d = { η d,j | j ∈ N d } to be an arbitrary tensor product ONB of H d , asdefined in (9). Then, for fixed j ∈ N d , formula (6) and the structure of the linear tensorproduct operator S d = S ⊗ . . . ⊗ S yields in the case P I = A I S d ( A HI ( η d,j )) = S d S I X π ∈S I ( − | π | d O l =1 η j π ( l ) ! = 1 S I X π ∈S I ( − | π | d O l =1 S ( η j π ( l ) ) = A GI ( S d ( η d,j )) . Obviously, the same is true for P I = S I . Hence, it holds (14) at least on the set of basiselements E d of H d . Because of the representation g = X j ∈ N d h g, η d,j i H d · η d,j , g ∈ H d , as well as the linearity and boundedness of the operators P HI , P GI and S d we can extend therelation (14) from E d to the whole space H d . Step 2 . Now let f ∈ P HI ( H d ) and let Af denote an arbitrary approximation of S d f . Then S d f = S d ( P HI f ) = P GI ( S d f ), due to Step 1. Using the fact that P GI provides an orthogonalprojection onto P GI ( G d ), see (4), we obtain (15), k S d f − Af | G d k = (cid:13)(cid:13) P GI ( S d f ) − [ P GI ( Af ) + (id G − P GI )( Af )] | G d (cid:13)(cid:13) = (cid:13)(cid:13) P GI ( S d f − Af ) | G d (cid:13)(cid:13) + (cid:13)(cid:13) (id G − P GI )( Af ) | G d (cid:13)(cid:13) = (cid:13)(cid:13) S d f − P GI ( Af ) | G d (cid:13)(cid:13) + (cid:13)(cid:13) Af − P GI ( Af ) | G d (cid:13)(cid:13) , as claimed. (cid:4) A self-contained proof of Theorem 1 in Section 3
In order to deduce a lower bound on the n -th minimal error of approximating S d on (anti-)symmetric subspaces P I ( H d ), where P ∈ { A , S } , let us define the classes of algorithms underconsideration.An algorithm A n,d for S d : F d = P I ( H d ) → G d which uses n pieces of information ismodeled as a mapping φ : R n → G d and a function N : F d → R n such that A n,d = φ ◦ N . Indetail, the information map N is given by N ( f ) = ( L ( f ) , L ( f ) , . . . , L n ( f )) , f ∈ F d , (40)44here L j ∈ Λ. Here we distinguish certain classes of information operations Λ. In one casewe assume that we can compute continuous linear functionals. Then Λ = Λ all coincideswith F ∗ d , the dual space of F d . If L v depends continuously on f but is not necessarily linearthe class is denoted by Λ cont . Note that in both the cases also N is continuous and weobviously have Λ all ⊂ Λ cont .Furthermore, we distinguish between adaptive and non-adaptive algorithms. The lattercase is described above in formula (40), where L v does not depend on the previously computedvalues L ( f ) , . . . , L v − ( f ). In contrast, we also discuss algorithms of the form A n,d = φ ◦ N with N ( f ) = ( L ( f ) , L ( f ; y ) , . . . , L n ( f ; y , . . . , y n − )) , f ∈ F d , where y = L ( f ) and y v = L v ( f ; y , . . . , y v − ) for v = 2 , , . . . , n . If N is adaptive we restrictourselves to the case where L v depends linearly on f , i.e. L v ( · ; y , . . . , y v − ) ∈ Λ all .In all cases of information maps, the mapping φ can be chosen arbitrarily and is notnecessarily linear or continuous. The smallest class of algorithms under consideration is theclass of linear, non-adaptive algorithms of the form A n,d f = n X v =1 L v ( f ) · g v , with some g v ∈ G d and L v ∈ Λ all . We denote the class of all such algorithms by A lin n . On theother hand, the most general classes consist of algorithms A n,d = φ ◦ N , where φ is arbitraryand N either uses non-adaptive continuous or adaptive linear information. We denote therespective classes by A cont n and A adapt n .For the proof that the upper bound given in Proposition 2 is sharp we use a generalizationof Lemma 1 in W. [9]. Lemma 5.
Suppose S to be a homogeneous operator between linear normed spaces X and Y , i.e. S ( αx ) = αS ( x ) for all x ∈ X and α ∈ R . Furthermore, assume that V ⊂ X is alinear subspace with dimension m and there exists a constant a ≥ a · k f | X k ≤ k S ( f ) | Y k for all f ∈ V. Then for every n < m and every algorithm A n ∈ A cont n ∪ A adapt n e wor ( A n ; S, X ) = sup f ∈B ( X ) k S ( f ) − A n ( f ) | Y k ≥ a. roof. It is well-known that for A n = φ ◦ N with n < m there exists f ∗ ∈ V such that N ( f ∗ ) = N ( − f ∗ ) and k f | X k = 1. Thus, A n ( f ∗ ) = A n ( − f ∗ ). For a more detailed view, see,W. [9, Lemma 1] and the references in there. Using the triangle inequality for Y we obtain e wor ( A n ; X ) ≥ max {k S ( ± f ∗ ) − A n ( ± f ∗ ) | Y k} = max {k S ( f ∗ ) ± A n ( f ∗ ) | Y k}≥
12 ( k S ( f ∗ ) + A n ( f ∗ ) | Y k + k S ( f ∗ ) − A n ( f ∗ ) | Y k ) ≥ k · S ( f ∗ ) | Y k ≥ a k f ∗ | X k = a and the proof is complete. (cid:4) Now let X = P I ( H d ) and Y = G d . Furthermore, for a given n ∈ N , define a = p λ d,ψ ( n +1) and consider V = span (cid:8) ξ ψ (1) , . . . , ξ ψ ( n +1) (cid:9) ⊂ P I ( H d ). Then, obviously, dim V = n + 1 = m > n . With the representation (16) and formula (17) from the proof of Proposition 2 weconclude k S d f | G d k = n +1 X v =1 (cid:10) f, ξ ψ ( v ) (cid:11) · λ d,ψ ( v ) ≥ λ d,ψ ( n +1) n +1 X v =1 (cid:10) f, ξ ψ ( v ) (cid:11) = a k f | H d k , f ∈ V, where we used the monotonicity of { λ d,ψ ( v ) } v ∈ N and Parseval’s identity. This leads to thedesired lower bound result: Proposition 11 (Lower bound) . Under the assumptions of Theorem 1 the n -th minimalerror with respect to the class A cont n ∪ A adapt n is bounded from below by e ( n, d ; P I ( H d )) = inf A n,d e wor ( A n,d ; P I ( H d )) ≥ q λ d,ψ ( n +1) for all d ∈ N , n ∈ N . Hence, this together with Proposition 2 shows that A ∗ n,d given in (12) is n -th optimalwith respect to the class A cont n ∪ A adapt n as claimed in Theorem 1. Proof of Proposition 3 in Section 4.1
Proof.
If the problem is polynomially tractable then there exist constants
C, p > q ≥ d ∈ N and ε ∈ (0 , n ( ε, d ) = n ( ε, d ; F d ) ≤ C · ε − p · d q . ε init d = e (0 , d ) = p λ d, > T d . Since e ( n, d ) = p λ d,n +1 itis n ( ε, d ) = { i ∈ N | λ d,i > ε } and therefore λ d,n ( ε,d )+1 ≤ ε . The non-increasing orderingof ( λ d,i ) i ∈ N implies λ d, ⌊ Cε − p d q ⌋ +1 ≤ ε . If we set i = ⌊ C · ε − p · d q ⌋ + 1 and vary ε ∈ (0 ,
1] then i takes the values ⌊ C · d q ⌋ + 1, ⌊ C · d q ⌋ + 2, and so forth. On the other hand, we have i ≤ Cε − p d q + 1, which is equivalentto ε ≤ ( Cd q / ( i − /p if i ≥
2. Thus, λ d,i ≤ λ d,n ( ε,d )+1 ≤ ε ≤ (cid:18) Cd q i − (cid:19) /p for all i ≥ max { , ⌊ C · d q ⌋ + 1 } . Choosing τ ≥ f ( d ) = ⌈ (1 + C ) · d q ⌉ ≥ max { , ⌊ C · d q ⌋ + 1 } we conclude ∞ X i = f ( d ) λ τd,i ≤ ∞ X i = f ( d ) (cid:18) Cd q i − (cid:19) τ/p = ( Cd q ) τ/p ∞ X i = f ( d ) − i τ/p ≤ ( Cd q ) τ/p · ζ (cid:18) τp (cid:19) , where ζ denotes the Riemann zeta function. In other words, if τ > p/ > d q/p ∞ X i = f ( d ) λ τd,i /τ ≤ C /p · ζ (cid:18) τp (cid:19) /τ < ∞ for every d ∈ N . Setting r = 2 q/p proves the assertion, as well as the claimed bound on C τ .Conversely, assume now that (18) holds with f ( d ) = l C · (cid:0) min (cid:8) ε init d , (cid:9)(cid:1) − p · d q m where C > p, q ≥ . That is, for some r ≥ τ > < C = sup d ∈ N d r ∞ X i = f ( d ) λ τd,i /τ < ∞ . For n ≥ f ( d ), the ordering of ( λ d,i ) i ∈ N implies P ni = f ( d ) λ τd,i ≥ λ τd,n · ( n − f ( d ) + 1). Hence, λ d,n · ( n − f ( d ) + 1) /τ ≤ n X i = f ( d ) λ τd,i /τ ≤ ∞ X i = f ( d ) λ τd,i /τ ≤ C d r , λ d,n +1 ≤ C d r · (( n + 1) − f ( d ) + 1) − /τ , for all n ≥ f ( d ) −
1. Note that for ε ∈ (0 , min (cid:8) ε init d , (cid:9) ] we have C d r · (( n + 1) − f ( d ) + 1) − /τ ≤ ε if and only if n ≥ n ∗ = (cid:24)(cid:18) C d r ε (cid:19) τ (cid:25) + f ( d ) − . In particular, it is λ d,n +1 ≤ ε at least for n ≥ max { n ∗ , f ( d ) − } . Therefore, for every d ∈ N and for all ε ∈ (0 , min (cid:8) ε init d , (cid:9) ] it is n ( ε, d ; F d ) ≤ max { n ∗ , f ( d ) − } ≤ f ( d ) − (cid:18) C d r ε (cid:19) τ ≤ C · (cid:0) min (cid:8) ε init d , (cid:9)(cid:1) − p · d q + C τ ε − τ d rτ ≤ ( C + C τ ) · ε − max { p, τ } · d max { q,rτ } . Hence, the problem is polynomially tractable. (cid:4)
An explicit proof of Lemma 4 in Section 4.2
Proof. Step 1 . By induction on s we first show for every fixed m ∈ N X k ∈ N s ,m ≤ k ≤ ... ≤ k s µ s,k = µ sm + s X l =1 µ s − lm X j ( l ) ∈ N l ,m +1 ≤ j ( l )1 ≤ ... ≤ j ( l ) l µ l,j ( l ) for all s ∈ N . (41)Easy calculus shows that this holds at least for the initial step s = 1. Therefore, assume (41)to be true for some s ∈ N . Then X k ∈ N s +1 ,m ≤ k ≤ ... ≤ k s +1 µ s +1 ,k = ∞ X k = m µ k X h ∈ N s ,k ≤ h ≤ ... ≤ h s µ s,h = µ m X h ∈ N s ,m ≤ h ≤ ... ≤ h s µ s,h + ∞ X k = m +1 µ k X h ∈ N s ,k ≤ h ≤ ... ≤ h s µ s,h = µ m X h ∈ N s ,m ≤ h ≤ ... ≤ h s µ s,h + X k ∈ N s +1 ,m +1 ≤ k ≤ ... ≤ k s +1 µ s +1 ,k Now, by inserting the induction hypothesis for the first sum and renaming k to j ( s +1) in theremaining sum, we conclude X k ∈ N s +1 ,m ≤ k ≤ ... ≤ k s +1 µ s +1 ,k = µ s +1 m + s X l =1 µ s +1 − lm X j ( l ) ∈ N l ,m +1 ≤ j ( l )1 ≤ ... ≤ j ( l ) l µ l,j ( l ) + X j ( s +1) ∈ N s +1 ,m +1 ≤ j ( s +1)1 ≤ ... ≤ j ( s +1) s +1 µ s +1 ,j ( s +1) . s + 1 and the induction is complete. Step 2 . Here we prove (24) via another induction on V ∈ N . Therefore, let d ∈ N befixed arbitrarily. The initial step, V = 0, corresponds to (41) for s = d and m = 1. Thus,assume (24) to be true for some fixed V ∈ N . Then it is X k ∈ N d , ≤ k ≤ ... ≤ k d µ d,k ≤ µ d d V V + d X L =1 µ − L X j ( L ) ∈ N L ,V +2 ≤ j ( L )1 ≤ ... ≤ j ( L ) L µ L,j ( L ) = µ d d V V + d X L =1 µ − L µ LV +2 + L X l =1 µ L − lV +2 X j ( l ) ∈ N l , ( V +2)+1 ≤ j ( l )1 ≤ ... ≤ j ( l ) l µ l,j ( l ) , using (41) for s = L and m = V + 2. Now we estimate 1 + V by d (1 + V ), take advantageof the non-increasing ordering of ( µ m ) m ∈ N and extend the inner sum from L to d in order toobtain X k ∈ N d , ≤ k ≤ ... ≤ k d µ d,k ≤ µ d d V +1 V + 1) + d X l =1 µ − l X j ( l ) ∈ N l , ( V +1)+2 ≤ j ( l )1 ≤ ... ≤ j ( l ) l µ l,j ( l ) . Since this estimate corresponds to (24) for V + 1 the claim is proven. (cid:4) Proof of Proposition 8 in Section 4.4
Proof.
Note that due to lim m →∞ λ m = 0 the quantity i d is well defined, because ( α i ) i ∈ N isa non-increasing sequence which tends to zero for i tending to infinity, and i d ( δ ) > δ < ε init d . Furthermore, we have i ( δ ) = { m ∈ N | λ m > δ } + 1 = n ent ( δ,
1) + 1 = n asy ( δ,
1) + 1and if d ≥ i d to obtain i d ( δ ) = min (cid:26) i ∈ N | λ i +1 · . . . · λ i + d − ≤ λ i δ (cid:27) . k = ( k , . . . , k d − ) ∈ ∇ d − with k > i d ( δ ) it is λ d − ,k = λ k · . . . · λ k d − ≤ λ i d ( δ )+1 · . . . · λ i d ( δ )+ d − ≤ λ i d ( δ ) δ or, equivalently, (cid:26) k ∈ ∇ d − | i < k and λ d − ,k > λ i δ (cid:27) = ∅ for all i ≥ i d ( δ ) . This leads to the disjoint decomposition of { j ∈ ∇ d | λ d,j > δ } = (cid:26) j = ( i, k ) ∈ N × ∇ d − | i < k and λ d − ,k > λ i δ (cid:27) = i d ( δ ) − [ i =1 (cid:26) ( i, k ) | k ∈ ∇ d − such that i < k and λ d − ,k > λ i δ (cid:27) . Therefore, the information complexity of the d -variate problem is given by n asy ( ε, d ) = { j ∈ ∇ d | λ d,j > ε } = i d ( ε ) − X i =1 (cid:26) k ∈ ∇ d − | i < k and λ d − ,k > λ i ε (cid:27) = i d ( ε ) X l =2 (cid:26) k ∈ ∇ d − | l ≤ k and λ d − ,k > λ l − ε (cid:27) . Obviously, for fixed l ∈ { , . . . , i d ( ε ) } , we can repeat this procedure and obtain { j ∈ ∇ d − | l ≤ j and λ d − ,j > δ } = i d − ( δ ) X l = l +1 (cid:26) k ∈ ∇ d − | l ≤ k and λ d − ,k > λ l − δ (cid:27) , if d > δ = ε /λ l − . Note that ε < ε init d implies i d − ( δ ) ≥ l + 1 such that { l +1 , . . . , i d − ( δ ) } 6 = ∅ . Iterating the argument we get n asy ( ε, d )= i d ( ε ) X l =2 i d − ( ε /λ l − ) X l = l +1 . . . i ( ε / [ λ l − · ... · λ ld − − ]) X l d − = l d − +1 (cid:26) k ∈ ∇ | l d − ≤ k and λ ,k > λ l d − − δ (cid:27) δ = ε / [ λ l − · . . . · λ l d − − ]. It remains to calculate the cardinality of the last set. Ofcourse, we have (cid:26) k ∈ ∇ | l d − ≤ k and λ ,k > λ l d − − δ (cid:27) = (cid:26) k ∈ N | l d − ≤ k and λ k > λ l d − − δ (cid:27) = (cid:26) k ∈ N | λ k > λ l d − − δ (cid:27) \ (cid:26) k ∈ { , . . . , l d − − } | λ k > λ l d − − δ (cid:27) . The first of these sets in the last line contains exactly n ent ( δ/ p λ l d − − ,
1) elements. On theother hand, if k ≤ l d − ≤ i ( δ ) then λ k λ l d − − ≥ λ i ( δ ) λ i ( δ ) − > δ , where the last inequality holds due to the definition of i ( δ ). Therefore, the last set coincideswith { , . . . , l d − − } and its cardinality is equal to l d − −
1. Furthermore, note that theestimate also shows that n ent ( δ/ p λ l d − − ,
1) is at least equal to l d − . Thus, (cid:26) k ∈ ∇ | l d − ≤ k and λ ,k > λ l d − − δ (cid:27) = n ent (cid:16) δ/ q λ l d − − , (cid:17) − l d − + 1 ≥ (cid:4) Proof of Proposition 9 in Section 4.5
One possibility to prove the second point of Proposition 9 is to apply Proposition 3 to ascaled problem { e T d } such that f W d = e T † d e T d possesses the eigenvalues e λ d,i = λ d,i /λ d, for i ∈ N . Then the initial error of e T d equals 1 such that f in Proposition 3 does not dependon p . That is, we can choose f ( d ) = ⌈ C d q ⌉ + 1 for some q ≥ f ( d ) ≡ Proof. If { T d } is polynomially tractable with respect to the normalized error criterion thenthere exist constants C, p > q ≥ d ∈ N and ε ′ ∈ (0 , n ( ε ′ · ε init d , d ) = n ( ε ′ · ε init d , d ; F d ) ≤ C · ( ε ′ ) − p · d q . As before the quantity ε init d = p λ d, > T d and ε ′ is the (mul-tiplicative) improvement of it. Since e ( n, d ) = p λ d,n +1 it is n ( ε, d ) = { i ∈ N | λ d,i > ε } ε = ε ′ · ε init d . Therefore, λ d,n ( ε ′ · ε init d ,d )+1 ≤ ( ε ′ ) · λ d, . Hence, the non-increasing orderingof ( λ d,i ) i ∈ N implies in this setting λ d, ⌊ C ( ε ′ ) − p d q ⌋ +1 ≤ ( ε ′ ) · λ d, . If we set i = (cid:4) C ( ε ′ ) − p d q (cid:5) + 1 and vary ε ′ ∈ (0 ,
1] then i takes the values ⌊ Cd q ⌋ + 1, ⌊ Cd q ⌋ + 2and so on. Again we have 1 ≤ i ≤ C ( ε ′ ) − p d q + 1 on the other hand, which is equivalent to( ε ′ ) ≤ ( Cd q / ( i − /p if i ≥
2. Thus, λ d,i ≤ λ d,n ( ε ′ · ε init d ,d )+1 ≤ ( ε ′ ) · λ d, ≤ (cid:18) Cd q i − (cid:19) /p · λ d, for all i ≥ max { , ⌊ Cd q ⌋ + 1 } . Choosing τ ≥ f ∗ ( d ) = ⌈ (1 + C ) d q ⌉ ≥ max { , ⌊ Cd q ⌋ + 1 } we conclude here ∞ X i = f ∗ ( d ) (cid:18) λ d,i λ d, (cid:19) τ ≤ ∞ X i = f ∗ ( d ) (cid:18) Cd q i − (cid:19) τ/p ≤ ( Cd q ) τ/p · ζ (cid:18) τp (cid:19) , where ζ again is the Riemann zeta function. On the other hand, it is obvious that f ∗ ( d ) − X i =1 (cid:18) λ d,i λ d, (cid:19) τ ≤ f ∗ ( d ) − ≤ (1 + C ) d q · τ/p , because λ d,i ≤ λ d, for all i ∈ N . Therefore, if τ > p/ d τq/p ∞ X i =1 (cid:18) λ d,i λ d, (cid:19) τ ≤ C + C τ/p · ζ (cid:18) τp (cid:19) < ∞ for all d ∈ N . This proves the assertion setting r ≥ q/p .The proof of the second point again works like for Proposition 3. Assume that (37) holdswith f ( d ) = ⌈ C d q ⌉ , where C > q ≥
0. That is, for some r ≥ τ > C τ = sup d ∈ N d r ∞ X i = f ( d ) (cid:18) λ d,i λ d, (cid:19) τ /τ < ∞ . Since ( λ d,i ) i ∈ N is assumed to be non-increasing the same also holds for the rescaled sequence( λ d,i /λ d, ) i ∈ N such that P ni = f ( d ) ( λ d,i /λ d, ) τ ≥ ( λ d,n /λ d, ) τ · ( n − f ( d ) + 1) for n ≥ f ( d ). Hence, λ d,n λ d, · ( n − f ( d ) + 1) /τ ≤ ∞ X i = f ( d ) (cid:18) λ d,i λ d, (cid:19) τ /τ ≤ C τ d r , λ d,n +1 ≤ C τ d r · (( n + 1) − f ( d ) + 1) − /τ · λ d, for all n ≥ f ( d ) −
1. As beforewe have C τ d r · (( n + 1) − f ( d ) + 1) − /τ ≤ ( ε ′ ) , for ε ′ ∈ (0 , n ≥ n ∗ = (cid:24)(cid:18) C τ d r ( ε ′ ) (cid:19) τ (cid:25) + f ( d ) − . In particular, λ d,n +1 ≤ ( ε ′ ) · λ d, at least for n ≥ max { n ∗ , f ( d ) − } . Therefore, we concludein this setting for all ε ′ ∈ (0 ,
1] and every d ∈ N n ( ε ′ · ε init d , d ; F d ) ≤ max { n ∗ , f ( d ) − } ≤ f ( d ) − (cid:18) C τ d r ( ε ′ ) (cid:19) τ ≤ C d q + C ττ ( ε ′ ) − τ d rτ ≤ ( C + C ττ ) · ( ε ′ ) − τ · d max { q,rτ } . Hence, the problem is polynomially tractable. Furthermore, strong polynomial tractabilityholds if r = q = 0. (cid:4)(cid:4)