Hadamard products and moments of random vectors

TL;DR

This paper introduces new inequalities relating weak and strong moments of random vector norms, with applications to concentration phenomena and operator norm bounds, advancing understanding in high-dimensional probability.

Contribution

It provides novel comparison inequalities between moments of random vectors with near-optimal constants, extending tools for analyzing concentration and operator norms.

Findings

Derived comparison inequalities with universal constants.

Applied inequalities to concentration of log-concave vectors.

Provided bounds on p-summing norms of finite rank operators.

Abstract

We derive new comparison inequalities between weak and strong moments of norms of random vectors with optimal (up to an universal factor) constants. We discuss applications to the concentration of log-concave random vectors and bounds on $p$-summing norms of finite rank operators.

Equations

$${\left(\mathbb{E}\underset{t\in T}{\sup }|⟨t,X⟩{|}^p\right)}^{1/p}\le 2\sqrt{e}\sqrt{\frac{n+p}{p}}\underset{t\in T}{\sup }{\left(\mathbb{E}|⟨t,X⟩{|}^p\right)}^{1/p}\text{mbox}forp\ge 2.$$

$$(\mathbb{E}\Vert X{\Vert }^p{)}^{1/p}\le 2\sqrt{e}\sqrt{\frac{n+p}{p}}\underset{\Vert t{\Vert }_{\ast }\le 1}{\sup }{\left(\mathbb{E}|⟨t,X⟩{|}^p\right)}^{1/p}\text{mbox}forp\ge 2.$$

$${\left(\mathbb{E}\Vert X{\Vert }_{{\mathcal{Z}}_p(X)}^p\right)}^{1/p}\le 2\sqrt{e}\sqrt{\frac{n+p}{p}},$$

$$\Vert s{\Vert }_{{\mathcal{Z}}_p(X)}:=\sup \{|⟨t,s⟩|:\mathbb{E}|⟨t,X⟩{|}^p\le 1\}.$$

$${\left(\mathbb{E}\Vert X{\Vert }_{{\mathcal{Z}}_p(X)}^p\right)}^{1/p}={\left(\mathbb{E}|U_1{|}^p\right)}^{-1/p}\sim \sqrt{\frac{n+p}{p}},$$

$$\mathbb{E}|⟨t,X⟩{|}^p=\frac{1}{n}\sum _{i=1}^n|t_i{|}^p,\Vert s{\Vert }_{{\mathcal{Z}}_p(X)}=n^{1/p}{\left(\sum _{i=1}^n|s_i{|}^q\right)}^{1/q},$$

$${\left(\mathbb{E}\Vert X{\Vert }_{{\mathcal{Z}}_p(X)}^p\right)}^{1/p}=n^{1/p}\ll \sqrt{\frac{n+p}{p}}.$$

$${\left(\mathbb{E}\underset{t\in T}{\sup }|⟨t,X⟩{|}^p\right)}^{1/p}\le C_{n,p}\underset{t\in T}{\sup }{\left(\mathbb{E}|⟨t,X⟩{|}^p\right)}^{1/p}.$$

$${\left(\sum _{j=1}^l\underset{1\le i\le k}{\sup }|⟨t_i,x_j⟩{|}^p\right)}^{1/p}\le C_{n,p}\underset{1\le i\le k}{\sup }{\left(\sum _{j=1}^l|⟨t_i,x_j⟩{|}^p\right)}^{1/p}.$$

$${\left(\sum _{j=1}^l\underset{1\le i\le k}{\sup }|a_{ij}{|}^p\right)}^{1/p}\le C_{n,p}\underset{1\le i\le k}{\sup }{\left(\sum _{j=1}^l|a_{ij}{|}^p\right)}^{1/p}.$$

$$\mathbb{E}\underset{t\in T}{\sup }|⟨t,X⟩{|}^2$$

$$=\mathbb{E}\sup \{|⟨C^{1/2}s,C^{-1/2}X⟩{|}^2:|C^{1/2}s{|}^2\le \alpha \}=\alpha \mathbb{E}|C^{-1/2}X{|}^2=\alpha n.$$

$$A^{\circ m}={\left(\sum _{r=1}^n{u}_r\otimes v_r\right)}^{\circ m}$$

$$=\sum _{r_1,\dots ,r_m=1}^n\left(u_{r_1}\circ \dots \circ u_{r_m}\right)\otimes \left(v_{r_1}\circ \dots \circ v_{r_m}\right).$$

$$\sum _{j=1}^l\underset{1\le i\le k}{\sup }$$

$$=C_{n,2m}^{2m}\underset{1\le i\le k}{\sup }\sum _{j=1}^l|⟨t_i,x_j⟩{|}^{2m}{p}_j$$

$$\le C_{n,2m}^{2m}\underset{1\le i\le k}{\sup }{\left(\sum _{j=1}^l|⟨t_i,x_j⟩{|}^p\right)}^{2m/p}{\left(\sum _{j=1}^l{p}_j^{p/(p-2m)}\right)}^{(p-2m)/p}$$

$$=C_{n,2m}^{2m}\underset{1\le i\le k}{\sup }{\left(\sum _{j=1}^l|⟨t_i,x_j⟩{|}^p\right)}^{2m/p}{\left(\sum _{j=1}^l\underset{1\le i\le k}{\sup }|⟨t_i,x_j⟩{|}^p\right)}^{(p-2m)/p},$$

$$C_{n,2m}^2\le \frac{e(n+m-1)}{m}\le e\frac{n+p/2}{p/4}\le 4e\frac{n+p}{p}.$$

$${\forall }_{A\in \mathcal{B}({\mathbb{R}}^n)}{\forall }_{p>0}{\nu }^n(A)\ge \frac{1}{2}\Rightarrow 1-{\nu }^n(A+C\sqrt{p}{B}_2^n+Cp{B}_1^n)\le e^{-p}(1-{\nu }^n(A)).$$

$${\forall }_{A\in \mathcal{B}({\mathbb{R}}^n)}{\forall }_{p>0}{\gamma }_n(A)\ge \frac{1}{2}\Rightarrow 1-{\gamma }_n(A+C\sqrt{p}{B}_2^n)\le e^{-p}(1-{\gamma }_n(A)),$$

$${\mathcal{Z}}_p(\mu )={\mathcal{Z}}_p(X)=\{t\in {\mathbb{R}}^n:\Vert t{\Vert }_{{\mathcal{Z}}_p(X)}\le 1\}.$$

$$\mu (A)\ge \frac{1}{2}⟹1-\mu (A+\beta {\mathcal{Z}}_p(\mu ))\le e^{-p}(1-\mu (A)),p\ge 2.$$

$${\forall }_{p\ge 2}{\forall }_{A\in \mathcal{B}({\mathbb{R}}^n)}\mu (A+\beta {\mathcal{Z}}_p(\mu ))\ge \min \left\{\frac{1}{2},e^p\mu (A)\right\}.$$

$$\mu (A+C{n}^{5/12}{\mathcal{Z}}_p(\mu ))$$

$$\mu \left(2e^{3/2}\sqrt{\frac{n+p}{p}}{\mathcal{Z}}_p(\mu )\right)\ge 1-e^{-p}.$$

$$\mu (A+C{n}^{5/12}{\mathcal{Z}}_p(\mu ))\ge \mu \left(A+18e^{3/2}\sqrt{\frac{n+p}{p}}{\mathcal{Z}}_p(\mu )\right)\ge \min \left\{\frac{1}{2},e^p\mu (A)\right\}.$$

$${\forall }_{x_1,\dots x_m\in F_1}{\left(\sum _{i=1}^m\Vert T{x}_i{\Vert }^p\right)}^{1/p}\le \alpha \underset{x^{\ast }\in F_1^{\ast },\Vert x^{\ast }\Vert \le 1}{\sup }{\left(\sum _{i=1}^m|x^{\ast }(x_i){|}^p\right)}^{1/p}.$$

$${\pi }_p({\ell }_2^n)={\left(\mathbb{E}|U_1{|}^p\right)}^{-1/p}\sim \sqrt{\frac{n+p}{p}}.$$

$${\pi }_p(F)\le 2\sqrt{2}\sqrt{\frac{\dim F+p}{p}}\le C{\pi }_p({\ell }_2^{\dim F}).$$

$${\pi }_p(T)\le 2\sqrt{e}\sqrt{\frac{\mathrm{rk}(T)+p}{p}}\Vert T\Vert .$$

$${\pi }_p(T)$$

$$=2\sqrt{e}\sqrt{\frac{\mathrm{rk}(T)+p}{p}}\Vert T\Vert .$$

Full text

The authors were supported by the National Science Centre, Poland, grant 2015/18/A/ST1/00553. P.N. was supported in part by the National Science Centre, Poland, grant 2018/31/D/ST1/01355.

1. Introduction

The study of moments of random variables is an essential issue of probability theory, one of the reasons being the fact that tail estimates for random variables are related to bounds for their moments via the Markov inequality. In probabilistic convex geometry and concentration of measure theory one is often interested in bounding $p$th moments of random vectors.

To be more precise, the $p$th strong moment of a random vector $X$ in ${\mathbb{R}}^{n}$ with respect to a given norm structure $({\mathbb{R}}^{n},\|\cdot\|)$ is defined as $M_{p}(X)=({\mathbb{E}}\|X\|^{p})^{1/p}$. Another related quantity is the so-called weak $p$th moment defined as $\sigma_{p}(X)=\sup_{\|t\|_{\ast}\leq 1}({\mathbb{E}}|\left\langle t,X\right\rangle|^{p})^{1/p}$, where $\|\cdot\|_{\ast}$ denotes the dual norm. Weak moments are usually much easier to compute or estimate, and so comparison inequalities between weak and strong moments are of interest in convex geometry, see e.g. [5]. While the one-sided estimate $\sigma_{p}(X)\leq M_{p}(X)$ follows trivially from the fact that $\|x\|=\sup_{\|t\|_{\ast}\leq 1}|\left\langle t,x\right\rangle|$, obtaining the reverse bounds turns out to be much more challenging. As an example let us mention the Paouris inequality $M_{p}(X)\leq C(M_{1}(X)+\sigma_{p}(X))$ valid for the standard Euclidean norm and arbitrary log-concave random vector $X$ in ${\mathbb{R}}^{n}$, see [18] and [1] (see also [14] for an extension of this result to a larger class of norms). Here and in the sequel $C$ denotes an absolute constant, whose value may change at each occurrence.

Usually, to compare weak and strong moments one applies the concentration of measure theory [16] or the chaining method [22]. What is crucial for such proofs is the regularity of the random vector $X$ and/or the special form of the norm.

One may however wonder what is the minimal constant $C_{n,p}$ such that the inequality $M_{p}(X)\leq C_{n,p}\sigma_{p}(X)$ holds for any $n$-dimensional random vector $X$ and any norm on ${\mathbb{R}}^{n}$. First attempt to this problem was discussed in [12], where it was proved that for unconditional random vectors such comparison holds with $C_{n,p}=C\sqrt{(n+p)/p}$. Our main result shows that this inequality is valid without any symmetry assumptions.

Theorem 1**.**

For any $n$-dimensional random vector $X$ and any nonempty set $T$ in ${\mathbb{R}}^{n}$ we have

[TABLE]

In particular, for any normed space $({\mathbb{R}}^{n},\|\cdot\|)$ we have

[TABLE]

The proof of the main result uses elementary linear algebra, namely a Hadamard power trick inspired by the proof of the so-called Welch bound (see [23]) given in [6].

Remark 2*.*

Let us notice that by homogeneity one can always assume that the supremum on the right hand side of (1) is one. Then by enlarging the set $T$ we may assume that $T$ is the set of all vectors $t$ satisfying ${\mathbb{E}}|\left\langle t,X\right\rangle|^{p}\leq 1$. Thus, inequality (1) may be equivalently stated as

[TABLE]

where

[TABLE]

This has been conjectured (with a universal constant in place of $2\sqrt{e}$) by the second named author in [12] (see Problem 1 therein). Thus Theorem 1 positively resolves this conjecture.

Remark 3*.*

It is not hard to check (see [12] for details) that for rotationally invariant $n$-dimensional random vector $X$ with finite $p$th moment one has

[TABLE]

where $U_{1}$ is the first coordinate of the random vector $U$ uniformly distributed on the unit sphere. Therefore the bound in Theorem 1 is sharp up to a universal multiplicative factor.

Remark 4*.*

In general one cannot reverse bound (2) for $2\ll p\ll n$. Indeed, let $e_{1},\ldots,e_{n}$ be the canonical basis of ${\mathbb{R}}^{n}$ and ${\mathbb{P}}(X=\pm e_{i})=1/(2n)$ for $1\leq i\leq n$. Then for $s,t\in{\mathbb{R}}^{n}$,

[TABLE]

where $q$ denotes the Hölder dual to $p$. Thus for $2\ll p\ll n$,

[TABLE]

We think however that (2) may be reversed in the case of symmetric log-concave vectors (see the discussion in Section 4).

Inequality (1) arose as a result of investigating optimal concentration of measure inequalities. The discussion of this and other applications will be given is Section 3. Section 2 is devoted to the proof of Theorem 1. We conclude with reviewing some related open problems in Section 4.

2. Proof of the main result

The following Lemma gives three equivalent formulations of our main inequality.

Lemma 5**.**

*Let $1\leq p<\infty$ and $n$ be a positive integer. The following conditions are equivalent:

i) For any $n$-dimensional random vector $X$ and any nonempty set $T$ in ${\mathbb{R}}^{n}$ we have*

[TABLE]

ii) For any $k,l\geq 1$ and any vectors $t_{1},\ldots,t_{k}$ and $x_{1},\ldots,x_{l}$ in ${\mathbb{R}}^{n}$ we have

[TABLE]

iii) For any $k,l\geq 1$ and any matrix $(a_{ij})_{i\leq k,j\leq l}$ of rank at most $n$ we have

[TABLE]

Proof.

i)$\iff$ ii) Simple approximation argument shows that it is enough to show i) for a finite set $T$ and a random vector $X$ equidistributed over a finite subset of ${\mathbb{R}}^{n}$.

ii)$\iff$iii) Matrix $A:=(\langle t_{i},x_{j}\rangle)_{i\leq k,j\leq l}$ is a product $TX$, where $T$ is a $k\times n$ matrix whose rows are the vectors $t_{i}$ and $X$ is an $n\times l$ matrix whose columns are the vectors $x_{j}$. Thus it is a matrix of a linear map ${\mathbb{R}}^{l}\to{\mathbb{R}}^{n}\to{\mathbb{R}}^{k}$ and so it has rank at most $n$. Moreover any $k\times l$ matrix of rank at most $n$ may be represented as such a product $TX$ due to the rank factorization theorem. ∎

The proof of Theorem 1 is divided into three steps.

Proposition 6**.**

Conditions i)-iii) hold

• (a)

for $p=2$ with $C_{n,2}=\sqrt{n}$,

• (b)

for $p=2m$, $m=1,2,\ldots$ with $C_{n,2m}=\binom{n+m-1}{m}^{1/(2m)}$,

• (c)

for $p\in(2m,2m+2]$, $m=1,2,\ldots$ with $C_{n,p}=C_{n,2m}=\binom{n+m-1}{m}^{1/(2m)}\leq 2\sqrt{e}\sqrt{\frac{n+p}{p}}$.

Proof.

(a) We will show condition i). Without loss of generality we may assume that a vector $X$ is symmetric, bounded and has a nondegenerate covariance matrix $C$. Let $\alpha:=\sup_{t\in T}{\mathbb{E}}|\langle t,X\rangle|^{2}=\sup_{t\in T}\langle Ct,t\rangle$. Then

[TABLE]

(b) We will establish condition iii). Let $A$ be an $k\times l$ matrix of rank at most $n$. Then by the rank factorization theorem $A$ can be written in the form $A=TX$, where $T$ is $k\times n$ and $X$ is $n\times l$. Then $A=\sum_{r=1}^{n}u_{r}\otimes v_{r}$, where $u_{r}$ in ${\mathbb{R}}^{k}$ are column vectors of $T$ and $v_{r}$ in ${\mathbb{R}}^{l}$ are row vectors of $X$, $r=1,\ldots,n$. Then the $m$th Hadamard power of $A$ equals

[TABLE]

Observe that Hadamard product is commutative so we may restrict the sum to nondecreasing $m$-tuples $(r_{1},\ldots,r_{m})$ with values in $\{1,\ldots,n\}$. Number of such tuples is $\binom{n+m-1}{m}$. Thus matrix $A^{\circ m}$ has rank at most $\binom{n+m-1}{m}$ (see [19] for a similar arguments for Hadamard powers of Gram matrices). We conclude by using part (a) for matrix $A^{\circ m}$.

(c) We will verify condition ii). Let $p_{j}=\max_{i\leq k}|\langle t_{i},x_{j}\rangle|^{p-2m}$, by the homogenity we may assume that $\sum_{j=1}^{l}p_{j}=1$. Let $X$ be a random variable such that ${\mathbb{P}}(X=x_{j})=p_{j}$, Lemma 6 yields

[TABLE]

where the second inequality follows by the Hölder inequality. After rearranging we get the desired bound.

To conclude observe that

[TABLE]

∎

3. Applications

Concentration inequalities

Let $\nu$ be a symmetric exponential measure with parameter $1$, i.e. the measure on the real line with the density $\frac{1}{2}e^{-|x|}$. Talagrand [21] showed that the product measure $\nu^{n}$ satisfies the following two-sided concentration inequality

[TABLE]

where by a letter $C$ here and in the sequel we denote universal constants.

This is a remarkably strong concentration result implying, for example, the celebrated concentration of measure phenomenon for the canonical Gaussian measure $\gamma_{n}$ on ${\mathbb{R}}^{n}$:

[TABLE]

discovered (in the sharp isoperimetric form) by Sudakov and Tsirelson in [20], and independently by Borell in [3].

It is not hard to check that $\mathcal{Z}_{p}(\nu^{n})\sim\sqrt{p}B_{2}^{n}+pB_{1}^{n}$ and $\mathcal{Z}_{p}(\gamma_{n})\sim\sqrt{p}B_{2}^{n}$ for $p\geq 2$, where for a probability measure $\mu$ on ${\mathbb{R}}^{n}$ and a random vector $X$ distributed according to $\mu$ we set

[TABLE]

In the context of convex geometry it is natural to ask if similar inequalities hold for other log-concave measures, namely measures with densities of the form $e^{-V}$, where $V:{\mathbb{R}}^{n}\to R$ is convex. An easy observation from [15] shows that if $\mu$ is a symmetric log-concave probability measure and $K$ is a convex set such that for any halfspace $A$ satisfying $\mu(A)\geq\frac{1}{2}$ we have $\mu(A+K)\geq 1-\frac{1}{2}e^{-p}$, then necesarily $K\supset c\mathcal{Z}_{p}$. This motivates the following definiton proposed in [15] by Wojtaszczyk and the second named author.

Definition 7**.**

We say that a measure $\mu$ satisfies the optimal concentration inequality with constant $\beta$ (​ $\mathrm{CI}(\beta)$ in short) if for any Borel set $A$ we have

[TABLE]

All centered product log-concave measures satisfy the optimal concentration inequality with a universal constant $\beta$ ([15]). A natural conjecture (discussed in [15, 13]) states that this is true also for nonproduct measures. However, one has to mention that it would imply (see Corollary 3.14. in [15]) the celebrated KLS conjecture (proposed in [11] as a tool for proving efficiency of certain Metropolis type algorithms for computing volumes of convex sets) on the boundedness of the Cheeger constant for isotropic log-concave measures . It was shown in [13] that every log-concave measure on ${\mathbb{R}}^{n}$ satisfies $CI(c\sqrt{n})$ with a universal constant $c$. The following corollary improves upon this bound.

Corollary 8**.**

Every centered log-concave probability measure on ${\mathbb{R}}^{n}$ satisfies the optimal concentration inequality with constant $\beta\leq Cn^{5/12}$.

Proof.

We follow the ideas expained after the proof of Proposition 7 in [13], but instead of Eldan’s bound on the Cheeger constant [8] we use the recent result of Lee and Vempala [17].

Since the concentration inequality is invariant with respect to linear transformation we may assume that $\mu$ is isotropic. Then in particular $\mathcal{Z}_{p}(\mu)\supset\mathcal{Z}_{2}(\mu)=B_{2}^{n}$.

By Proposition 2.7 in [15] $\mathrm{CI}(\beta)$ may be equivalently stated as

[TABLE]

To show the above bound with $\beta=Cn^{5/12}$ we consider two cases.

i) If $2\leq p\leq n^{1/6}$ then

[TABLE]

where the last inequality follows by the Lee-Vempala [17] $Cn^{1/4}$ bound on the Cheeger constant.

ii) If $p\geq\max\{2,n^{1/6}\}$ then observe first that Theorem 1 (see Remark 2) yields

[TABLE]

Therefore Lemma 9 in [13] gives

[TABLE]

∎

p-summing norms of finite rank operators

The theory of absolutely summing operators is an important part of the modern Banach space theory and found numerous powerful applications in harmonic analysis, approximation theory, probability theory and operator theory [7].

Recall that a linear operator $T$ between Banach spaces $F_{1}$ and $F_{2}$ is $p$-summing if there exists a constant $\alpha<\infty$, such that

[TABLE]

The smallest constant $\alpha$ in the above inequality is called the $p$-summing norm of $T$ and denoted by $\pi_{p}(T)$. For a Banach space $F$ by $\pi_{p}(F)$ we denote the $p$-summing constant of the identity map of $F$.

It is well known that $\pi_{p}(F)<\infty$ if and only if $F$ is finite dimensional. Moreover $\pi_{2}(F)=\sqrt{\dim F}$ (see [9, Theorem 16.12.3]). Summing constants of some finite dimensional spaces were computed by Gordon in [10]. In particular he showed that

[TABLE]

Immediate consequence of our main result is that up to a universal constant Hilbert space has the largest $p$-summing constant among all normed spaces of fixed dimension.

Corollary 9**.**

For any finite dimensional Banach space $F$ and $p\geq 2$ we have

[TABLE]

Proof.

We apply Theorem 1 for random vectors uniformly distributed on finite subsets of $F$ and $T$ the unit ball in $F^{*}$. ∎

Using the ideal properties we get a bound for $p$-summing constant of finite rank operators.

Corollary 10**.**

Let $T$ be a finite rank linear operator between Banach spaces $F_{1}$ and $F_{2}$. Then the $p$-absolutely summing constant of $T$ satisfies

[TABLE]

Proof.

Let $F_{3}:=T(F_{1})$, then $F_{3}$ is a subspace of $F_{2}$ of dimension $\mathrm{rk}(T)$. Observe that $T=i\circ I\circ\tilde{T}$ where $\tilde{T}$ is $T$ considered as an operator between $F_{1}$ and $F_{3}$, $I$ is the identity map on $F_{3}$ and $i$ is the embedding of $F_{3}$ into $F_{1}$. The operator ideal properties of the $p$-summing norm and Corollary 9 imply

[TABLE]

∎

4. Open Questions

Corollary 6 in [12] states that for unconditional log-concave vectors in ${\mathbb{R}}^{n}$ and $2\leq p\leq n$ we have

[TABLE]

We do not know whether such bounds holds without unconditionality assumptions. We are only able to show the following weaker lower bound. Recall that the isotropic constant of a centered logconcave vector $X$ with density $g$ is defined as

[TABLE]

It is known that for all log-concave vectors $L_{X}\geq 1/C$, the famous open conjecture, due to Bourgain [4], states that $L_{X}\leq C$ (see [2, 5] for more details and discussions of known upper bounds).

Proposition 11**.**

For any centered log-concave $n$-dimensional random vector with nondegenerate covariance matrix we have

[TABLE]

where $L_{X}$ is the isotropic constant of $X$.

Proof.

Since the assertion is linearly invariant we may and will assume that $X$ is isotropic, i.e. it has the identity covariance matrix. The density of $X$ is then bounded by $L_{X}^{n}$, hence

[TABLE]

where the last estimate follows by the Paouris [18] bound on the volume of $\mathcal{Z}_{p}$-bodies (see also [5, Theorem 5.1.17]).

Thus

[TABLE]

∎

In the last years it was showed that various constants related to the $n$-dimensional log-concave measures (isotropic constant, Cheeger constant, thin-shell constant) are bounded by $Cn^{1/4}$. We think that the same should be true for the $\mathrm{CI}$ constant.

Finally, we do not know whether the optimal constants in (2) are attained for rotationally invariant random vectors or, in other words, whether conditions in Lemma 5 hold with $C_{n,p}=({\mathbb{E}}|U_{1}|^{p})^{1/p}$, where $U_{1}$ is the first coordinate of the random vector $U$ uniformly distributed on the unit sphere. Equivalently one may ask whether for any finite dimensional Banach space $F$ one has $\pi_{p}(F)\leq\pi_{p}(\ell_{2}^{\dim F})$.