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A new analysis of empirical interpolation methods and Chebyshev greedy algorithms

Yuwen Li

TL;DR

The paper develops entropy-number–based convergence bounds for generalized empirical interpolation methods (EIM) and the Chebyshev greedy algorithm (CGA), yielding sharper rates than traditional $d_n(K)$-width analyses. By treating EIM as a weak reduced-basis greedy method and leveraging metric entropy numbers $\varepsilon_n({\rm co}(K))$, the authors derive bounds that hold in Banach spaces and specialize to Hilbert spaces, with explicit dependence on Banach-space geometry through $\delta_{X,n}$ and Lebesgue-type constants. For the CGA, they establish entropy-based convergence estimates under a modulus-of-smoothness condition $\rho_X(t)\le C_X t^s$, and extend to interpolation spaces $X_\theta$, providing improved rates when $\varepsilon_n({\rm co}(K))$ decays rapidly. Numerical experiments on parametrized PDE-inspired dictionaries (e.g., ReLU$\_m$) corroborate the theoretical predictions, demonstrating faster convergence in regimes of fast entropy decay and highlighting practical implications for reduced-order modeling and sparse nonlinear approximation.

Abstract

We present new convergence estimates of generalized empirical interpolation methods in terms of the entropy numbers of the parametrized function class. Our analysis is transparent and leads to sharper convergence rates than the classical analysis via the Kolmogorov n-width. In addition, we also derive novel entropy-based convergence estimates of the Chebyshev greedy algorithm for sparse n-term nonlinear approximation of a target function. This also improves classical convergence analysis when corresponding entropy numbers decay fast enough.

A new analysis of empirical interpolation methods and Chebyshev greedy algorithms

TL;DR

The paper develops entropy-number–based convergence bounds for generalized empirical interpolation methods (EIM) and the Chebyshev greedy algorithm (CGA), yielding sharper rates than traditional -width analyses. By treating EIM as a weak reduced-basis greedy method and leveraging metric entropy numbers , the authors derive bounds that hold in Banach spaces and specialize to Hilbert spaces, with explicit dependence on Banach-space geometry through and Lebesgue-type constants. For the CGA, they establish entropy-based convergence estimates under a modulus-of-smoothness condition , and extend to interpolation spaces , providing improved rates when decays rapidly. Numerical experiments on parametrized PDE-inspired dictionaries (e.g., ReLU) corroborate the theoretical predictions, demonstrating faster convergence in regimes of fast entropy decay and highlighting practical implications for reduced-order modeling and sparse nonlinear approximation.

Abstract

We present new convergence estimates of generalized empirical interpolation methods in terms of the entropy numbers of the parametrized function class. Our analysis is transparent and leads to sharper convergence rates than the classical analysis via the Kolmogorov n-width. In addition, we also derive novel entropy-based convergence estimates of the Chebyshev greedy algorithm for sparse n-term nonlinear approximation of a target function. This also improves classical convergence analysis when corresponding entropy numbers decay fast enough.
Paper Structure (13 sections, 6 theorems, 84 equations, 3 figures, 4 algorithms)

This paper contains 13 sections, 6 theorems, 84 equations, 3 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Greedy Algorithms
  3. Reduced Basis Greedy Algorithm
  4. Chebyshev Greedy Algorithm
  5. Convergence of the EIM
  6. Error Bounds of EIMs in Banach Spaces
  7. Convergent Examples of the EIM
  8. Numerical Convergence of the EIM
  9. Convergence of the CGA
  10. Error Bounds of the weak CGA
  11. Convergent Examples of the CGA
  12. Numerical Convergence of the CGA
  13. Concluding Remarks

Key Result

Lemma 3.1

\newlabelmainlemma0 Let $K$ be a compact set in a Banach space $X$. Let $V_n=\pi^{\frac{n}{2}}/\Gamma(\frac{n}{2}+1)$ be the volume of an $\ell_2$ unit ball in $\mathbb{R}^n$. For any $v_1, \ldots, v_n\in K$ with $X_k={\rm span}\{v_1,\ldots,v_k\}$ and $X_0=\{0\}$, we have

Figures (3)

  • Figure 1: Errors of the EIM in $L_\infty[0,1]$ (left); errors of the EIM in $L_2[0,1]$ (right).
  • Figure 1: Convergence history of the CGA error.
  • Figure 2: Estimates of $\Lambda_n$ in $L_\infty[0,1]$.

Theorems & Definitions (9)

  • Lemma 3.1: Lemma 3.1 from LiSiegel2024
  • Theorem 3.2
  • Proof 1
  • Corollary 3.3
  • Proof 2
  • Lemma 3.4: Theorem 3 from Stern2015
  • Theorem 4.1
  • Proof 3
  • Corollary 4.2