Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings

Motonobu Kanagawa, Bharath Kumar Sriperumbudur, Kenji Fukumizu

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

This paper presents convergence analysis of kernel-based quadrature rules in misspecified settings, focusing on deterministic quadrature in Sobolev spaces. In particular, we deal with misspecified settings where a test integrand is less smooth than a Sobolev RKHS based on which a quadrature rule is constructed. We provide convergence guarantees based on two different assumptions on a quadrature rule: one on quadrature weights and the other on design points. More precisely, we show that convergence rates can be derived (i) if the sum of absolute weights remains constant (or does not increase quickly), or (ii) if the minimum distance between design points does not decrease very quickly. As a consequence of the latter result, we derive a rate of convergence for Bayesian quadrature in misspecified settings. We reveal a condition on design points to make Bayesian quadrature robust to misspecification, and show that, under this condition, it may adaptively achieve the optimal rate of convergence in the Sobolev space of a lesser order (i.e., of the unknown smoothness of a test integrand), under a slightly stronger regularity condition on the integrand.

Original languageEnglish (US)
JournalFoundations of Computational Mathematics
DOIs
StateAccepted/In press - Jan 1 2019

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Quadrature Rules
Convergence Analysis
Quadrature
Sobolev spaces
Integrand
kernel
Sobolev Spaces
Strong Regularity
Optimal Rate of Convergence
Misspecification
Minimum Distance
Regularity Conditions
Convergence Rate
Smoothness
Rate of Convergence
Unknown
Decrease
Design

All Science Journal Classification (ASJC) codes

  • Analysis
  • Computational Mathematics
  • Computational Theory and Mathematics
  • Applied Mathematics

Cite this

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abstract = "This paper presents convergence analysis of kernel-based quadrature rules in misspecified settings, focusing on deterministic quadrature in Sobolev spaces. In particular, we deal with misspecified settings where a test integrand is less smooth than a Sobolev RKHS based on which a quadrature rule is constructed. We provide convergence guarantees based on two different assumptions on a quadrature rule: one on quadrature weights and the other on design points. More precisely, we show that convergence rates can be derived (i) if the sum of absolute weights remains constant (or does not increase quickly), or (ii) if the minimum distance between design points does not decrease very quickly. As a consequence of the latter result, we derive a rate of convergence for Bayesian quadrature in misspecified settings. We reveal a condition on design points to make Bayesian quadrature robust to misspecification, and show that, under this condition, it may adaptively achieve the optimal rate of convergence in the Sobolev space of a lesser order (i.e., of the unknown smoothness of a test integrand), under a slightly stronger regularity condition on the integrand.",
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Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings. / Kanagawa, Motonobu; Sriperumbudur, Bharath Kumar; Fukumizu, Kenji.

In: Foundations of Computational Mathematics, 01.01.2019.

Research output: Contribution to journalArticle

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