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arXiv · 2012.02119

Robustly Learning Mixtures of $k$ Arbitrary Gaussians

Ainesh Bakshi·Ilias Diakonikolas·He Jia·Daniel M. Kane·Pravesh K. Kothari·Santosh S. Vempala
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Abstract

We give a polynomial-time algorithm for the problem of robustly estimating a mixture of $k$ arbitrary Gaussians in $\mathbb{R}^d$, for any fixed $k$, in the presence of a constant fraction of arbitrary corruptions. This resolves the main open problem in several previous works on algorithmic robust statistics, which addressed the special cases of robustly estimating (a) a single Gaussian, (b) a mixture of TV-distance separated Gaussians, and (c) a uniform mixture of two Gaussians. Our main tools are an efficient \emph{partial clustering} algorithm that relies on the sum-of-squares method, and a novel \emph{tensor decomposition} algorithm that allows errors in both Frobenius norm and low-rank terms.

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BibTeXRIS

Ainesh Bakshi, Ilias Diakonikolas, He Jia, Daniel M. Kane, Pravesh K. Kothari, Santosh S. Vempala. 2020-12-03. Robustly Learning Mixtures of $k$ Arbitrary Gaussians. https://arxiv.org/abs/2012.02119

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