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On Moment-Entropy inequalities in the space of matrices

Dylan Langharst

TL;DR

The paper generalizes affine information theory to the space of matrices by coupling a matrix-valued random vector with a vector through an origin-symmetric body $Q$ and the associated pseudo-norm $h_Q$, forming a matrix analogue of moment-entropy inequalities. Central to the approach is the introduction of the $(L^p,Q)$-polar projection body $\Pi_{Q,p}^{\circ}$ and the Ball-body framework, enabling a sphere-based Blaschke–Santaló analysis that yields a sharp matrix inequality for $\mathbb{E}[h_Q(\mathfrak{X}^t\mathcal{Y})^p]$ in terms of $N_\lambda$ functionals and the volume of $\Pi_{Q,p}^{\circ}B_2^{n}$. Equality cases force $\mathfrak{X}$ and $\mathcal{Y}$ to be generalized Gaussians associated with $(p,\lambda)$ up to affine transformations, extending the classical Lutwak–Yang–Zhang framework to matrix spaces. The work further develops a matrix-oriented $(L^p,Q)$ Santaló theory and derives affine Fisher-information-type inequalities, unifying moment-entropy and isoperimetric-type results in a matrix setting with sharp constants.

Abstract

In a series of works, Lutwak, Yang and Zhang established what could be called affine information theory, which is the study of moment-entropy and Fisher-information-type inequalities that are invariant with respect to affine transformations for random vectors. Their set of tools stemmed from sharp affine isoperimetric inequalities in the $L^p$ Brunn-Minkowski theory of convex geometry they had established. In this work, we generalize the affine information theory to the setting of matrices. These inequalities on the space of $n\times m$ matrices are induced by the interaction between $\mathbb{R}^n$ with its Euclidean structure and $\mathbb{R}^m$ equipped with a pseudo-norm.

On Moment-Entropy inequalities in the space of matrices

TL;DR

The paper generalizes affine information theory to the space of matrices by coupling a matrix-valued random vector with a vector through an origin-symmetric body and the associated pseudo-norm , forming a matrix analogue of moment-entropy inequalities. Central to the approach is the introduction of the -polar projection body and the Ball-body framework, enabling a sphere-based Blaschke–Santaló analysis that yields a sharp matrix inequality for in terms of functionals and the volume of . Equality cases force and to be generalized Gaussians associated with up to affine transformations, extending the classical Lutwak–Yang–Zhang framework to matrix spaces. The work further develops a matrix-oriented Santaló theory and derives affine Fisher-information-type inequalities, unifying moment-entropy and isoperimetric-type results in a matrix setting with sharp constants.

Abstract

In a series of works, Lutwak, Yang and Zhang established what could be called affine information theory, which is the study of moment-entropy and Fisher-information-type inequalities that are invariant with respect to affine transformations for random vectors. Their set of tools stemmed from sharp affine isoperimetric inequalities in the Brunn-Minkowski theory of convex geometry they had established. In this work, we generalize the affine information theory to the setting of matrices. These inequalities on the space of matrices are induced by the interaction between with its Euclidean structure and equipped with a pseudo-norm.

Paper Structure

This paper contains 4 sections, 9 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of our main results
  4. Other related inequalities

Key Result

Theorem 1.1

Fix $m,n\in\mathbb{N}$ and $Q\in\mathcal{K}^{m}_o$. Let $p\geq 1$ and $\lambda \geq \frac{nm}{nm+p}$. Suppose $\mathfrak{X}$ and $\mathcal{Y}$ are independent random vectors on $M_{n,m}(\mathbb{R})$ and $\mathbb{R}^n$ respectively, each of which has a finite moment of order $p$. Suppose that either Here, $D_{n,p,\lambda}$ is a sharp constant given by eq_my_sharp below. There is equality if and on

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3: The $m$th-order $L^p$ Santaló inequality
  • Lemma 3.1
  • proof
  • proof : Proof of Theorem \ref{['t:BS_on_sphere']}
  • Proposition 3.2
  • Lemma 3.3
  • ...and 6 more