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Constructive discretization and approximation in reproducing kernel Hilbert spaces

Abdellah Chkifa, Matthieu Dolbeault, David Krieg, Mario Ullrich

Abstract

We generalize the sparsification algorithm of Batson, Spielman and Srivastava, making one part of the result dimension-independent. In particular, we recover discretization inequalities in $L_2$- and sup-norms on general finite-dimensional subspaces, prove a suitable infinite-dimensional variant, and discuss the implications for the error of least-squares approximation based on samples. This gives a more constructive version of several recently established approximation bounds, some of which relied on the stronger and less constructive result of Marcus, Spielman and Srivastava. We also improve the constants and oversampling factors in these results.

Constructive discretization and approximation in reproducing kernel Hilbert spaces

Abstract

We generalize the sparsification algorithm of Batson, Spielman and Srivastava, making one part of the result dimension-independent. In particular, we recover discretization inequalities in - and sup-norms on general finite-dimensional subspaces, prove a suitable infinite-dimensional variant, and discuss the implications for the error of least-squares approximation based on samples. This gives a more constructive version of several recently established approximation bounds, some of which relied on the stronger and less constructive result of Marcus, Spielman and Srivastava. We also improve the constants and oversampling factors in these results.
Paper Structure (12 sections, 15 theorems, 127 equations, 1 figure, 1 algorithm)

This paper contains 12 sections, 15 theorems, 127 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Norm discretization
  3. Sampling Recovery
  4. Proofs of Theorem \ref{['thm:main']} and Proposition \ref{['coro:intro-equal']}
  5. Recap on linear algebra in infinite dimension
  6. Proof of Theorem \ref{['thm:main']}
  7. Proof of Proposition \ref{['coro:intro-equal']}
  8. Construction of points and weights
  9. Christoffel sampling
  10. Constructive results via truncation
  11. Fast computations
  12. Numerical results

Key Result

Proposition 1

Let $D$ be a finite set and let $\mu$ be the uniform distribution on $D$. Let $a=(a_1,\hdots,a_m)^\top$ be an orthonormal family in $L_2(D,\mu)$. Then, for any $n>m$, there exist points $x_1,\dots,x_n\in D$ and weights ${w_1,\dots,w_n>0}$ such that and

Figures (1)

  • Figure 1: Summary of the results of Sections \ref{['sec:intro']}, \ref{['sec:discretization']} and \ref{['sec:sampling']}, with arrows denoting implications. The first line contains linear algebra results, the second deals with norm discretization and the third with sampling recovery. The lower left block details the reproducing kernel Hilbert space (RKHS) setting, with the second column directly applying Thm. \ref{['thm:main']}, while the first column adds a control on the weights, and the third looks at sampling numbers. Lastly, the fourth column is focused on the $L_p$-setting, in which all weights are equal.

Theorems & Definitions (42)

  • Proposition 1: BSS, Theorem 3.1
  • Proposition 2: BDM14, Lemma 13
  • Theorem 3
  • Remark 4
  • Remark 5
  • Corollary 6
  • proof
  • Proposition 7
  • proof
  • Proposition 8
  • ...and 32 more