Entropy bounds for the absolute convex hull of tensors

We derive entropy bounds for the absolute convex hull of vectors X= (x₁ , … , xₚ)∈ Rn × p in Rⁿ and apply this to the case where X is the d-fold tensor matrix X = Ψ ⊗ ⋯ ⊗ Ψ_d \ times ∈ Rmᵈ × rᵈ, with a given Ψ = ( ψ₁ , … , ψᵣ ) ∈ Rm × r, normalized to that \| ψⱼ \|₂ ≤ 1 for all j ∈ \1 , … , r\. For ε >0 we let V ⊂ Rᵐ be the linear space with smallest dimension M ( ε , Ψ) such that max1 ≤ j ≤ r min_v ∈ V \| ψⱼ - v \|₂ ≤ 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 L₂ (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 ∈ N. As a by-product, we construct an orthonormal, piecewise polynomial, wavelet dictionary for functions that are well-approximated by piecewise polynomials.