    A variant of Schur's product theorem and its applications

We show the following version of the Schur's product theorem. If M=(M_j,k)_j,k=1^n∈R^n× n is a positive semidefinite matrix with all entries on the diagonal equal to one, then the matrix N=(N_j,k)_j,k=1^n with the entries N_j,k=M_j,k^2-1/n is positive semidefinite. As a corollary of this result, we prove the conjecture of E. Novak on intractability of numerical integration on a space of trigonometric polynomials of degree at most one in each variable. Finally, we discuss also some consequences for Bochner's theorem, covariance matrices of χ^2-variables, and mean absolute values of trigonometric polynomials.

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1 Introduction

Over twenty years ago, motivated by tractability studies of numerical integration, E. Novak  made the following conjecture:

Conjecture 1 (E. Novak).

The matrix

 {d∏i=11+cos(xj,i−xk,i)2−1n}nj,k=1

is positive semidefinite for all and all choices of

Erich Novak published this conjecture also in NA Digest in November 1997 and tested it numerically. It also appeared as Open Problem 3 in . Nevertheless, it seems that up to now the problem remained unsolved.

Further extensive numerical tests were provided by A. Hinrichs and the author in , all supporting the belief that Conjecture 1 is true. The main difficulty in actually proving this conjecture turned out to be to identify the important properties of the function , which play a role in this question. Led by other numerical tests,  conjectures that the same property is true for all positive positive-definite functions with the value at zero equal to one. Unfortunately, even in this form, the conjecture remained unsolved.

Our proof of this conjecture, which we present in this note, is based on a certain simple but rather unexpected and apparently unknown property of the Hadamard product. To state it we need few simple notations. If is symmetric, we say that it is positive semidefinite, if for all Similarly, is called positive semidefinite, if it is selfadjoint and for all . If are two matrices, we denote by their Hadamard product , i.e. a matrix with entries for all Furthermore, the partial ordering means that is positive semidefinite. Finally, is a matrix with all entries equal to one.

Using this notation, the main result of this paper then reads as follows.

Theorem 1.

Let be a positive semidefinite matrix with for all . Then

 M∘M⪰1n⋅En.

We give the proof of Theorem 1 in Section 2. Let us note that it actually resembles the original work of Schur . This section also includes a complex version of Theorem 1 as well as a variant for matrices with general diagonal. Sections 3 and 4 discuss the connections with Bochner’s theorem and with covariance matrices of multivariate random variables. Finally, Section 5 gives an account on the numerical integration, which was the original motivation of E. Novak, and shows, how Theorem 1 implies Conjecture 1.

2 Proof of Theorem 1 and its variants

Proof of Theorem 1.

Using the singular value decomposition of

, we can write , where . We denote the rows of by , Then and for every .

We need to show that

 n∑j,k=1cjckM2j,k≥1n(n∑j=1cj)2 (1)

for every . We start with a reformulation of the left-hand side of (1)

 n∑j,k=1cjckM2j,k =n∑j,k=1cjck(AAT)2j,k=n∑j,k=1cjck⟨Aj,Ak⟩2 =n∑j,k=1cjck(n∑u=1Aj,uAk,u)2=n∑j,k=1cjckn∑u,v=1Aj,uAk,uAj,vAk,v (2)

We leave out the terms with and apply a variant of the inequality between arithmetic and quadratic mean

 n∑u=1ξ2u≥1n(n∑u=1ξu)2

for

 ξu=n∑j=1cjA2j,u.

In this way, we extend (2) and finish the proof of (1) by using

 n∑j,k=1cjckM2j,k≥n∑u=1(n∑j=1cjA2j,u)2≥1n(n∑u=1n∑j=1cjA2j,u)2=1n(n∑j=1cj)2. (3)

Theorem 1 can be extended to the setting, where the entries on the diagonal are not identically equal to one. For this sake, we denote by the diagonal entries of whenever .

Theorem 2.

Let be a positive semidefinite matrix. Then

 M∘M⪰1n(diag M)(diag M)T.
Proof.

The proof follows in the same manner as the proof of Theorem 1. Indeed, writing again , (2) and (3) becomes

 n∑j,k=1cjckM2j,k ≥1n(n∑u=1n∑j=1cjA2j,u)2=1n(n∑j=1cj⋅∥Aj∥22)2 =1n(n∑j=1cjMj,j)2=1n⟨c,diag M⟩2.

Without much additional effort, Theorem 1 allows also a complex version.

Theorem 3.

Let be a positive semidefinite Hermitian matrix with for all . Let be a matrix with entries . Then

 N⪰1n⋅En.
Proof.

Let with the rows of again denoted by , Then . Furthermore, let be arbitrary. Then the analogue of (2) becomes

 n∑j,k=1cj¯¯¯¯¯ck|Mj,k|2 =n∑j,k=1cj¯¯¯¯¯ck∣∣n∑u=1Aj,u¯¯¯¯¯¯¯¯¯¯Ak,u∣∣2=n∑j,k=1cj¯¯¯¯¯ckn∑u,v=1Aj,u¯¯¯¯¯¯¯¯¯Aj,v¯¯¯¯¯¯¯¯¯¯Ak,uAk,v =n∑u,v=1∣∣n∑j=1cjAj,u¯¯¯¯¯¯¯¯¯Aj,v∣∣2≥n∑u=1∣∣n∑j=1cj|Aj,u|2∣∣2 ≥1n∣∣n∑u=1n∑j=1cj|Aj,u|2∣∣2=1n∣∣n∑j=1cj∣∣2=c∗Encn.

Remark 1.

Let us point out that Theorem 3 fails if one replaces the condition by (which we checked by numerical simulations).

3 Bochner’s theorem

If is a finite Borel measure on

, then its Fourier transform is given by

 g(ξ)=^μ(ξ)=∫Rde−2πiξ⋅xdμ(x),ξ∈Rd,

where is the inner product of and Classical Bochner’s theorem (actually its easy part), cf. [1, 2], states that the matrix

 (g(ξj−ξk))nj,k=1

is positive semidefinite for every choice of .

The proof follows by a simple calculation, as we have for every

 n∑j,k=1cj¯¯¯¯¯ckg(ξj−ξk) =n∑j,k=1cj¯¯¯¯¯ck∫Rde−2πi(ξj−ξk)⋅xdμ(x) =∫Rdn∑j,k=1cj¯¯¯¯¯cke−2πiξj⋅xe2πiξk⋅xdμ(x) (4) =∫Rd∣∣n∑j=1cje−2πiξj⋅x∣∣2dμ(x)≥0.

We refer to  for a classical overview of positive definite functions. Theorem 1 leads to the following modification of one part of Bochner’s Theorem.

Theorem 4.

Let be a finite Borel measure on and let be its Fourier transform. Then

 (|g(ξj−ξk)|2)nj,k=1⪰g2(0)n⋅En

for every choice of

Proof.

By the easy part of Bochner’s theorem, cf. (4), we know that the matrix with is positive semidefinite. Furthermore, is real for every . The result then follows by invoking Theorem 3. ∎

Theorem 4 can be generalized to other locally compact abelian groups by just applying the corresponding version of Bochner’s theorem. In this way, one can prove for example the following version for torus , cf. Conjecture 3 in .

Theorem 5.

Let be a non-negative summable sequence, i.e. for every and . Let for every . Then

 (|g(ξj−ξk)|2)nj,k=1⪰g2(0)n⋅En

for every choice of

Theorem 4 allows an interesting reformulation in a language of independent random variables.

Theorem 6.

Let

be a Borel probability measure on

. Let and

be two independent random vectors, both distributed with respect to

. Then

 E∣∣n∑j=1cje−2πixj⋅(ω1−ω2)∣∣2≥1n∣∣n∑j=1cj∣∣2 (5)

for every choice of and

Proof.

We denote again by the Fourier transform of . The proof follows from Theorem 4 and the following direct calculation

 E∣∣n∑j=1cje−2πixj⋅(ω1−ω2)∣∣2 =En∑j,k=1cj¯¯¯¯¯cke−2πi(xj−xk)⋅(ω1−ω2) =n∑j,k=1cj¯¯¯¯¯ck[Ee−2πi(xj−xk)⋅ω1]⋅[Ee2πi(xj−xk)⋅ω2] =n∑j,k=1cj¯¯¯¯¯ck|g(xj−xk)|2.

Remark 2.

If and are i.i.d. standard normal variables, then is again a normal variable. After rescaling, (5) gives for every and every

 E∣∣n∑j=1cjeiyjω∣∣2≥1n∣∣n∑j=1cj∣∣2. (6)

4 Covariance matrices

Theorem 1 has an interesting consequence for covariance matrices of multivariate

-distributions (with one degree of freedom). Let

be a positive semidefinite matrix with ones on the diagonal and let

be Gaussian random variables with zero mean, unit variance and

for all Using Wick’s theorem, cf. [5, 6, 11], we observe that

 Cov(X2j,X2k)=E[X2jX2k]−E[X2j]⋅E[X2k]=2(E[XjXk])2=2M2j,k.

Applying Theorem 1, we obtain

 (Cov(X2j,X2k))nj,k=1=(2M2j,k)nj,k=1⪰2n⋅En.

Thus we have proven

Theorem 7.

Let be a vector of standard normal variables. Then

 (Cov(X2j,X2k))nj,k=1⪰2n⋅En.

5 Numerical integration

In this section, we summarize the approach of , where E. Novak studied, how well the quadrature formulas

 Qn(f)=n∑i=1cif(xi),ci∈R,xi∈[0,1]d (7)

approximate the integral Here belongs to a unit ball of a Hilbert space , which is defined as a

-fold tensor product of a space

, which in turn is a three dimensional Hilbert space with an orthonormal basis given by the functions

 e1(x)=1,e2(x)=cos(2πx),e3(x)=sin(2πx),x∈[0,1].

Hence is a -dimensional Hilbert space. The point evaluation may be written in the form

 f(x)=⟨f,δx⟩Fdwithδx(z)=d∏j=1[1+cos(2π(xj−zj))].

In this way, becomes a reproducing kernel Hilbert space with the kernel

 Kd(x,y)=⟨δx,δy⟩Fd=d∏j=1[1+cos(2π(xj−yj))],x,y∈[0,1]d.

This allows to compute the worst-case error of given by (7) as

 ewor(Qn)2 =1−2n∑j=1cj+n∑j,k=1cjckKd(xj,xk).

If all ’s are positive, we can use the positivity of and obtain

 ewor(Qn)2≥1−2n∑j=1cj+n∑j=1c2j2d.

For the optimal choice this becomes

 ewor(Qn)2≥max(1−n2−d,0). (8)

This estimate shows the intractability of numerical integration on

with quadrature formulas with positive weights since for a fixed error the number of sample points needs to grow exponentially with the dimension .

To estimate from below for quadrature rules with general weights , we use the fact that the projection of any onto the ray generated by is given by and its norm is equal to . In this way we obtain

 infcj,xj∥∥∥1−n∑j=1cjδxj∥∥∥2Fd =infcj,xjinfα∈R∥∥∥1−αn∑j=1cjδxj∥∥∥2Fd =infcj,xj⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩∥1∥2Fd−∣∣⟨1,n∑j=1cjδxj⟩Fd∣∣2∣∣⟨n∑j=1cjδxj,n∑j=1cjδxj⟩Fd∣∣⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎬⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎭ (9) =1−supcj,xj(n∑j=1cj)2n∑j,k=1cjckKd(xj,xk).

Erich Novak conjectured, that the estimate (8) applies also for quadrature formulas (7) with general weights, which is by (9) equivalent to Conjecture 1.

Finally, let us show how Theorem 1 implies the positive answer to Conjecture 1. We define matrices by

 Mij,k=cos(xj,i−xk,i2),i=1,…,d,j,k=1,…,n.

By Bochner’s theorem, the matrices are all positive semidefinite and so is their Hadamard product Obviously, has all its diagonal elements equal to one. Finally, Theorem 1 shows that the matrix with entries

 M2j,k−1n=d∏i=1cos2(xj,i−xk,i2)−1n=d∏i=11+cos(xj,i−xk,i)2−1n

is also positive semidefinite.

Hence, the integration problem on is intractable even when we allow negative weights ’s in the quadrature formula (7).

Remark 3.

Theorem 1 opens an interesting and (most likely) largely unexplored area of research of the partial matrix ordering given by . For example, the classical calculation (4

) can be reformulated as the statement that the zero matrix is a lower bound of the set

 {(^μ(ξj−ξk))nj,k=1:μ is a finite Borel % measure},

where Theorem 4 states that the matrix is a lower bound of the set

 {(|^μ(ξj−ξk)|2)nj,k=1:μ is a probability % Borel measure}.

In general, one may try to prove other non-trivial bounds for the infimum of other classes of matrices.

Other interesting questions in connection with Theorem 1 include

• Is there a version of Theorem 1, which would deal with the Hadamard product of two different matrices, i.e. with instead of ?

• Is there a variant of Theorem 1 for higher Hadamard powers of ?

• Does Theorem 4 hold also when gets replaced by a general positive positive-definite function?

• And more generally, is there some converse of Theorem 4?

Acknowledgement: The author would like to than Aicke Hinrichs (JKU Linz) and Erich Novak (FSU Jena) for fruitful discussions.

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