Entropy bounds for the absolute convex hull of tensors

Abstract

We derive entropy bounds for the absolute convex hull of vectors $X= (x_1 , \ldots , x_p)\in \mathbb{R}^{n \times p} $ in $\mathbb{R}^n$ and apply this to the case where $X$ is the $d$-fold tensor matrix $$X = \underbrace{Ψ\otimes \cdots \otimes Ψ}_{d \ {\rm times} }\in \mathbb{R}^{m^d \times r^d },$$ with a given $Ψ= ( ψ_1 , \ldots , ψ_r ) \in \mathbb{R}^{m \times r} $, normalized to that $ \| ψ_j \|_2 \le 1$ for all $j \in \{1 , \ldots , r\}$. For $ε>0$ we let ${\cal V} \subset \mathbb{R}^m$ be the linear space with smallest dimension $M ( ε, Ψ)$ such that $ \max_{1 \le j \le r } \min_{v \in {\cal V} } \| ψ_j - v \|_2 \le ε$. We call $M( ε, ψ)$ the $ε$-approximation of $Ψ$ and assume it is -- up to log terms -- polynomial in $ε$. We show that the entropy of the absolute convex hull of the $d$-fold tensor matrix $X$ is up to log-terms of the same order as the entropy for the case $d=1$. The results are generalized to absolute convex hulls of tensors of functions in $L_2 (μ)$ where $μ$ is Lebesgue measure on $[0,1]$. As an application we consider the space of functions on $[0,1]^d$ with bounded $q$-th order Vitali total variation for a given $q \in \mathbb{N}$. As a by-product, we construct an orthonormal, piecewise polynomial, wavelet dictionary for functions that are well-approximated by piecewise polynomials.

Theorems & Definitions (24)

• Example 1
• Example 2
• Definition 2.1
• Definition 3.1
• Definition 3.2
• Theorem 3.3
• Remark 3.1
• Remark 3.2
• Definition 4.1
• Definition 4.2