张量绝对凸包的熵界
统计理论
2024-02-14 v1 泛函分析
统计理论
摘要
我们推导了向量X = ( x 1 , … , x p ) ∈ R n × p X= (x_1 , \ldots , x_p)\in \mathbb{R}^{n \times p} X = ( x 1 , … , x p ) ∈ R n × p R n \mathbb{R}^n R n X X X d d d X = Ψ ⊗ ⋯ ⊗ Ψ ⏟ d t i m e s ∈ R m d × r d , X = \underbrace{\Psi \otimes \cdots \otimes \Psi}_{d \ {\rm times} }\in \mathbb{R}^{m^d \times r^d }, X = d times Ψ ⊗ ⋯ ⊗ Ψ ∈ R m d × r d , Ψ = ( ψ 1 , … , ψ r ) ∈ R m × r \Psi = ( \psi_1 , \ldots , \psi_r ) \in \mathbb{R}^{m \times r} Ψ = ( ψ 1 , … , ψ r ) ∈ R m × r j ∈ { 1 , … , r } j \in \{1 , \ldots , r\} j ∈ { 1 , … , r } ∥ ψ j ∥ 2 ≤ 1 \| \psi_j \|_2 \le 1 ∥ ψ j ∥ 2 ≤ 1 ϵ > 0 \epsilon >0 ϵ > 0 V ⊂ R m {\cal V} \subset \mathbb{R}^m V ⊂ R m M ( ϵ , Ψ ) M ( \epsilon , \Psi) M ( ϵ , Ψ ) max 1 ≤ j ≤ r min v ∈ V ∥ ψ j − v ∥ 2 ≤ ϵ \max_{1 \le j \le r } \min_{v \in {\cal V} } \| \psi_j - v \|_2 \le \epsilon max 1 ≤ j ≤ r min v ∈ V ∥ ψ j − v ∥ 2 ≤ ϵ M ( ϵ , ψ ) M( \epsilon , \psi) M ( ϵ , ψ ) Ψ \Psi Ψ ϵ \epsilon ϵ ϵ \epsilon ϵ d d d X X X d = 1 d=1 d = 1 L 2 ( μ ) L_2 (\mu) L 2 ( μ ) μ \mu μ [ 0 , 1 ] [0,1] [ 0 , 1 ] [ 0 , 1 ] d [0,1]^d [ 0 , 1 ] d q q q q ∈ N q \in \mathbb{N} q ∈ N
引用
@article{arxiv.2402.08388,
title = {Entropy bounds for the absolute convex hull of tensors},
author = {Sara van de Geer},
journal= {arXiv preprint arXiv:2402.08388},
year = {2024}
}