On Equivalence of Anchored and ANOVA Spaces; Lower Bounds

TL;DR

This paper establishes lower bounds for the norms of embeddings between anchored and ANOVA function spaces, revealing polynomial or faster growth depending on weight types and p-norms, which informs the understanding of high-dimensional function approximation.

Contribution

It provides new lower bounds for embedding norms between anchored and ANOVA spaces, clarifying their behavior across different weight classes and p-norms.

Findings

Polynomial growth of norms for finite order and diameter weights when p>1

Super-polynomial growth for product order-dependent weights for any p

Norm behavior depends critically on weight type and p-norm

Abstract

We provide lower bounds for the norms of embeddings between $\boldsymbol{\gamma}$-weighted Anchored and ANOVA spaces of $s$-variate functions with mixed partial derivatives of order one bounded in $L_p$ norm ($p\in[1,\infty]$). In particular we show that the norms behave polynomially in $s$ for Finite Order Weights and Finite Diameter Weights if $p>1$, and increase faster than any polynomial in $s$ for Product Order-Dependent Weights and any $p$.