Entropy bounds for the absolute convex hull of tensors
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913233617551360 |
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| author | van de Geer, Sara |
| author_facet | van de Geer, Sara |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08388 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Entropy bounds for the absolute convex hull of tensors van de Geer, Sara Statistics Theory Functional Analysis 41A10, 41A25, 94A12, 62G08 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. |
| title | Entropy bounds for the absolute convex hull of tensors |
| topic | Statistics Theory Functional Analysis 41A10, 41A25, 94A12, 62G08 |
| url | https://arxiv.org/abs/2402.08388 |