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On the approximation of vector-valued functions by volume sampling

Daniel Kressner, Tingting Ni, André Uschmajew

TL;DR

The paper addresses the problem of approximating a vector-valued function $f$ in $L^2(\Omega;\mathcal{H})$ by a $k$-dimensional subspace and, specifically, how sampling $f$ at $k$ points to generate a subspace affects the approximation error. Building on a Schmidt decomposition framework and a volume-sampling approach, the authors establish an existence result: there are $k$ samples such that the induced error is at most $\sqrt{k+1}$ times the best possible $d^{(2)}_k(f)$, matching the order of the analogous finite-dimensional column-subset results. The key idea is to use a density defined by the Gramian determinant $\det G^{(k)}$ over $\Omega^k$, compute the expected error under this density, and deduce the existence of favorable sample sets; the argument carefully handles the infinite-dimensional Bochner setting and the rank-deficient case. This provides a sharp, nontrivial improvement over prior $p=\infty$ bounds in the sampling-based subspace approximation context, with implications for reduced basis methods and related dimension-reduction techniques.

Abstract

Given a Hilbert space $\mathcal H$ and a finite measure space $Ω$, the approximation of a vector-valued function $f: Ω\to \mathcal H$ by a $k$-dimensional subspace $\mathcal U \subset \mathcal H$ plays an important role in dimension reduction techniques, such as reduced basis methods for solving parameter-dependent partial differential equations. For functions in the Lebesgue-Bochner space $L^2(Ω;\mathcal H)$, the best possible subspace approximation error $d_k^{(2)}$ is characterized by the singular values of $f$. However, for practical reasons, $\mathcal U$ is often restricted to be spanned by point samples of $f$. We show that this restriction only has a mild impact on the attainable error; there always exist $k$ samples such that the resulting error is not larger than $\sqrt{k+1} \cdot d_k^{(2)}$. Our work extends existing results by Binev at al. (SIAM J. Math. Anal., 43(3):1457-1472, 2011) on approximation in supremum norm and by Deshpande et al. (Theory Comput., 2:225-247, 2006) on column subset selection for matrices.

On the approximation of vector-valued functions by volume sampling

TL;DR

The paper addresses the problem of approximating a vector-valued function in by a -dimensional subspace and, specifically, how sampling at points to generate a subspace affects the approximation error. Building on a Schmidt decomposition framework and a volume-sampling approach, the authors establish an existence result: there are samples such that the induced error is at most times the best possible , matching the order of the analogous finite-dimensional column-subset results. The key idea is to use a density defined by the Gramian determinant over , compute the expected error under this density, and deduce the existence of favorable sample sets; the argument carefully handles the infinite-dimensional Bochner setting and the rank-deficient case. This provides a sharp, nontrivial improvement over prior bounds in the sampling-based subspace approximation context, with implications for reduced basis methods and related dimension-reduction techniques.

Abstract

Given a Hilbert space and a finite measure space , the approximation of a vector-valued function by a -dimensional subspace plays an important role in dimension reduction techniques, such as reduced basis methods for solving parameter-dependent partial differential equations. For functions in the Lebesgue-Bochner space , the best possible subspace approximation error is characterized by the singular values of . However, for practical reasons, is often restricted to be spanned by point samples of . We show that this restriction only has a mild impact on the attainable error; there always exist samples such that the resulting error is not larger than . Our work extends existing results by Binev at al. (SIAM J. Math. Anal., 43(3):1457-1472, 2011) on approximation in supremum norm and by Deshpande et al. (Theory Comput., 2:225-247, 2006) on column subset selection for matrices.
Paper Structure (4 sections, 5 theorems, 38 equations)

This paper contains 4 sections, 5 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Schmidt decomposition
  3. Expected volume
  4. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1

Let $f \in L^2(\Omega;\mathcal{H})$. Then there exists a measurable set $\hat{{\mathbf{\Omega}}} \subseteq \Omega \times \cdots \times \Omega = \Omega^k$ of positive product measure $\mu^{\otimes k}$ such that for $\mu^{\otimes k}$-almost every ${\mathbf{y}} = (y_1,\dots,y_k) \in \hat{{\mathbf{\Omeg where $\Pi_{{\mathbf{y}}} f$ denotes the $\mathcal{H}$-orthogonal projection of $f$ onto the span o

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2: Schmidt decomposition
  • proof
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof