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Journal Articles Journal of Machine Learning Research Year : 2023

Kernel-Matrix Determinant Estimates from stopped Cholesky Decomposition

Abstract

Algorithms involving Gaussian processes or determinantal point processes typically require computing the determinant of a kernel matrix. Frequently, the latter is computed from the Cholesky decomposition, an algorithm of cubic complexity in the size of the matrix. We show that, under mild assumptions, it is possible to estimate the determinant from only a sub-matrix, with probabilistic guarantee on the relative error. We present an augmentation of the Cholesky decomposition that stops under certain conditions before processing the whole matrix. Experiments demonstrate that this can save a considerable amount of time while having an overhead of less than $5\%$ when not stopping early. More generally, we present a probabilistic stopping strategy for the approximation of a sum of known length where addends are revealed sequentially. We do not assume independence between addends, only that they are bounded from below and decrease in conditional expectation.

Dates and versions

hal-03359274 , version 1 (30-09-2021)

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Simon Bartels, Wouter Boomsma, Jes Frellsen, Damien Garreau. Kernel-Matrix Determinant Estimates from stopped Cholesky Decomposition. Journal of Machine Learning Research, 2023. ⟨hal-03359274⟩
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