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 $$X = \underbrace{\Psi \otimes \cdots \otimes \Psi}_{d \ {\rm times} }\in \mathbb{R}^{m^d \times r^d },$$ with a given $\Psi = ( \psi_1 , \ldots , \psi_r ) \in \mathbb{R}^{m \times r} $, normalized to that $ \| \psi_j \|_2 \le 1$ for all $j \in \{1 , \ldots , r\}$. For $\epsilon >0$ we let ${\cal V} \subset \mathbb{R}^m$ be the linear space with smallest dimension $M ( \epsilon , \Psi)$ such that $ \max_{1 \le j \le r } \min_{v \in {\cal V} } \| \psi_j - v \|_2 \le \epsilon$. We call $M( \epsilon , \psi)$ the $\epsilon$-approximation of $\Psi$ and assume it is -- up to log terms -- polynomial in $\epsilon$. 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 $L_2 (\mu)$ where $\mu$ 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.

翻译：我们推导了向量 $X= (x_1 , \ldots , x_p)\in \mathbb{R}^{n \times p} $ 在 $\mathbb{R}^n$ 中绝对凸包的熵界，并将其应用于 $X$ 为 $d$ 重张量矩阵的情形：$$X = \underbrace{\Psi \otimes \cdots \otimes \Psi}_{d \ {\rm 次} }\in \mathbb{R}^{m^d \times r^d },$$ 其中 $\Psi = ( \psi_1 , \ldots , \psi_r ) \in \mathbb{R}^{m \times r} $ 给定，且对所有 $j \in \{1 , \ldots , r\}$ 满足归一化条件 $\| \psi_j \|_2 \le 1$。对 $\epsilon >0$，设 ${\cal V} \subset \mathbb{R}^m$ 为最小维数 $M ( \epsilon , \Psi)$ 的线性空间，使得 $ \max_{1 \le j \le r } \min_{v \in {\cal V} } \| \psi_j - v \|_2 \le \epsilon$。我们称 $M( \epsilon , \psi)$ 为 $\Psi$ 的 $\epsilon$-逼近，并假设其关于 $\epsilon$ 呈多项式阶（忽略对数项）。我们证明：$d$ 重张量矩阵 $X$ 绝对凸包的熵在忽略对数项意义下与 $d=1$ 情形具有相同阶数。该结果可推广至 $L_2 (\mu)$ 中函数张量的绝对凸包，其中 $\mu$ 为 $[0,1]$ 上的 Lebesgue 测度。作为应用，我们考虑给定 $q \in \mathbb{N}$ 下 $[0,1]^d$ 上具有有界 $q$ 阶 Vitali 全变差的函数空间。作为副产品，我们为可用分片多项式良好逼近的函数构造了一个正交、分片多项式小波字典。