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Sharp Lower Bounds on the Manifold Widths of Sobolev and Besov Spaces

Jonathan W. Siegel

TL;DR

The paper addresses the sharp decay of manifold $n$-widths for the unit balls of Sobolev and Besov spaces in $L_p$, establishing limits for continuous non-linear approximation. It identifies a regime ($p < q$, with $q > 2$) where Bernstein widths fail to yield sharp bounds and introduces sphere embedding widths to capture Borsuk-Ulam obstructions, enabling sharp lower bounds. A key result is the exact value $\\delta_n(K_q^M)_p = (M-n)^{1/p - 1/q}$ for finite-dimensional $\\ell_q^M$-balls in $\\ell_p$ when $p \le q$, derived via sphere embeddings. Applying these insights, the authors prove $\\delta_n(\\mathcal{B}_{r,q}^s)_{L_p(\Omega)} \ge C n^{-s/d}$ (and the analogous Sobolev case) under the Sobolev embedding condition, thereby completing the sharp characterization of manifold widths for these spaces and highlighting that Bernstein widths can decay too fast in certain regimes.

Abstract

We consider the problem of determining the manifold $n$-widths of Sobolev and Besov spaces with error measured in the $L_p$-norm. The manifold widths control how efficiently these spaces can be approximated by general non-linear parametric methods with the restriction that the parameter selection and parameterization maps must be continuous. Existing upper and lower bounds only match when the Sobolev or Besov smoothness index $q$ satisfies $q\leq p$ or $1 \leq p \leq 2$. We close this gap and obtain sharp lower bounds for all $1 \leq p,q \leq \infty$ for which a compact embedding holds. A key part of our analysis is to determine the exact value of the manifold widths of finite dimensional $\ell^M_q$-balls in the $\ell_p$-norm when $p\leq q$. Although this result is not new, we provide a new proof and apply it to lower bounding the manifold widths of Sobolev and Besov spaces. Our results show that the Bernstein widths, which are typically used to lower bound the manifold widths, decay asymptotically faster than the manifold widths in many cases.

Sharp Lower Bounds on the Manifold Widths of Sobolev and Besov Spaces

TL;DR

The paper addresses the sharp decay of manifold -widths for the unit balls of Sobolev and Besov spaces in , establishing limits for continuous non-linear approximation. It identifies a regime (, with ) where Bernstein widths fail to yield sharp bounds and introduces sphere embedding widths to capture Borsuk-Ulam obstructions, enabling sharp lower bounds. A key result is the exact value for finite-dimensional -balls in when , derived via sphere embeddings. Applying these insights, the authors prove (and the analogous Sobolev case) under the Sobolev embedding condition, thereby completing the sharp characterization of manifold widths for these spaces and highlighting that Bernstein widths can decay too fast in certain regimes.

Abstract

We consider the problem of determining the manifold -widths of Sobolev and Besov spaces with error measured in the -norm. The manifold widths control how efficiently these spaces can be approximated by general non-linear parametric methods with the restriction that the parameter selection and parameterization maps must be continuous. Existing upper and lower bounds only match when the Sobolev or Besov smoothness index satisfies or . We close this gap and obtain sharp lower bounds for all for which a compact embedding holds. A key part of our analysis is to determine the exact value of the manifold widths of finite dimensional -balls in the -norm when . Although this result is not new, we provide a new proof and apply it to lower bounding the manifold widths of Sobolev and Besov spaces. Our results show that the Bernstein widths, which are typically used to lower bound the manifold widths, decay asymptotically faster than the manifold widths in many cases.
Paper Structure (4 sections, 5 theorems, 58 equations)

This paper contains 4 sections, 5 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Sharp Bounds on the Manifold Widths of $\ell_q^M$-balls
  3. Lower Bounds on Manifold Widths of Sobolev and Besov Spaces
  4. Acknowledgements

Key Result

Theorem 1

Let $1\leq n < M$ be integers, $0< p \leq q\leq \infty$ and let $K^M_q$ denote the $\ell_q^M$ unit ball as in ell-q-unit-ball-definition. Then the manifold widths $\delta_n$ satisfy

Theorems & Definitions (9)

  • Theorem 1: Main result in khodulev1989remark
  • Proposition 1
  • proof
  • Proposition 2
  • Lemma 1
  • proof
  • proof : Proof of Proposition \ref{['sphere-embedding-lq-lower-bounds-prop']}
  • Theorem 2
  • proof