Worst-case optimal approximation with increasingly flat Gaussian kernels

Karvonen, Toni; Särkkä, Simo

doi:10.1007/s10444-020-09767-1

Cited by 6 publications

(2 citation statements)

References 35 publications

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“…There are many appearances of the assumption (1) in the literature. Besides the detailed study of certain weighted Hilbert spaces of analytic functions [8,24,25,30,47], it appears naturally in the context of approximation with (increasingly flat) Gaussian kernels [12,17,26,41], or in tensor product approximations [14,16], or for certain smoothness spaces on complex spheres [7]. Moreover, it is a typical assumption for the construction of greedy bases [4,5,15].…”

Section: Expositionmentioning

confidence: 99%

Exponential tractability of L2-approximation with function values

et al. 2023

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We study the complexity of high-dimensional approximation in the L2-norm when different classes of information are available; we compare the power of function evaluations with the power of arbitrary continuous linear measurements. Here, we discuss the situation when the number of linear measurements required to achieve an error ε ∈ (0,1) in dimension $d\in \mathbb {N}$ d ∈ ℕ depends only poly-logarithmically on ε− 1. This corresponds to an exponential order of convergence of the approximation error, which often happens in applications. However, it does not mean that the high-dimensional approximation problem is easy, the main difficulty usually lies within the dependence on the dimension d. We determine to which extent the required amount of information changes if we allow only function evaluation instead of arbitrary linear information. It turns out that in this case we only lose very little, and we can even restrict to linear algorithms. In particular, several notions of tractability hold simultaneously for both types of available information.

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Section: Expositionmentioning

confidence: 99%

Exponential tractability of L2-approximation with function values

et al. 2023

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“…Instead of the abstract information-theoretic quantities, we present an orthogonal approach that minimises the worstcase error in a deterministic sense, inspired by the interpolation theory on RKHSs [24,25]. Nonetheless, we show that the worst-case error minimisation problem admits an information-theoretic analogue that is an instance of the Gaussian belief space planning [26][27][28].…”

Section: Related Workmentioning

confidence: 99%

Informative Planning for Worst-Case Error Minimisation in Sparse Gaussian Process Regression

Jennifer¹,

Lee²,

Yoo³

et al. 2022

Preprint

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We present a planning framework for minimising the deterministic worst-case error in sparse Gaussian process (GP) regression. We first derive a universal worst-case error bound for sparse GP regression with bounded noise using interpolation theory on reproducing kernel Hilbert spaces (RKHSs). By exploiting the conditional independence (CI) assumption central to sparse GP regression, we show that the worst-case error minimisation can be achieved by solving a posterior entropy minimisation problem. In turn, the posterior entropy minimisation problem is solved using a Gaussian belief space planning algorithm. We corroborate the proposed worst-case error bound in a simple 1D example, and test the planning framework in simulation for a 2D vehicle in a complex flow field. Our results demonstrate that the proposed posterior entropy minimisation approach is effective in minimising deterministic error, and outperforms the conventional measurement entropy maximisation formulation when the inducing points are fixed.

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