An analytic comparison of regularization methods for Gaussian Processes

Abstract : Gaussian Processes (GPs) are a popular approach to predict the output of a parameterized experiment. They have many applications in the field of Computer Experiments, in particular to perform sensitivity analysis, adaptive design of experiments and global optimization. Nearly all of the applications of GPs require the inversion of a covariance matrix that, in practice, is often ill-conditioned. Regularization methodologies are then employed with consequences on the GPs that need to be better understood. The two principal methods to deal with ill-conditioned covariance matrices are i) pseudoinverse and ii) adding a positive constant to the diagonal (the so-called nugget regularization). The first part of this paper provides an algebraic comparison of PI and nugget regularizations. Redundant points, responsible for covariance matrix singularity, are defined. It is proven that pseudoinverse regularization, contrarily to nugget regularization, averages the output values and makes the variance zero at redundant points. However, pseudoinverse and nugget regularizations become equivalent as the nugget value vanishes. A measure for data-model discrepancy is proposed which serves for choosing a regularization technique. In the second part of the paper, a distribution-wise GP is introduced that interpolates Gaussian distributions instead of data points. Distribution-wise GP can be seen as an improved regularization method for GPs.
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[Research Report] Ecole Nationale Supérieure des Mines de Saint-Etienne; LIMOS. 2017
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Contributeur : Le Riche Rodolphe <>
Soumis le : vendredi 5 mai 2017 - 02:25:58
Dernière modification le : mardi 23 octobre 2018 - 14:36:09
Document(s) archivé(s) le : dimanche 6 août 2017 - 12:35:35

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  • HAL Id : hal-01264192, version 4
  • ARXIV : 1602.00853

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Hossein Mohammadi, Rodolphe Le Riche, Nicolas Durrande, Eric Touboul, Xavier Bay. An analytic comparison of regularization methods for Gaussian Processes. [Research Report] Ecole Nationale Supérieure des Mines de Saint-Etienne; LIMOS. 2017. 〈hal-01264192v4〉

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