A Spectral Algorithm for List-Decodable Covariance Estimation in Relative Frobenius Norm

• URL: http://arxiv.org/abs/2305.00966v1
• Date: Mon, 1 May 2023 17:54:35 GMT
• Title: A Spectral Algorithm for List-Decodable Covariance Estimation in Relative Frobenius Norm
• Authors: Ilias Diakonikolas, Daniel M. Kane, Jasper C. H. Lee, Ankit Pensia, Thanasis Pittas
• Abstract summary: We produce a list of hypotheses that are close to $Sigma$ in relative Frobenius norm. As a corollary, we obtain an efficient spectral algorithm for robust partial clustering of Gaussian mixture models. Our new method yields the first Sum-of-Squares-free algorithm for robustly learning GMMs.
• Score: 41.03423042792559
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the problem of list-decodable Gaussian covariance estimation. Given a multiset $T$ of $n$ points in $\mathbb R^d$ such that an unknown $\alpha<1/2$ fraction of points in $T$ are i.i.d. samples from an unknown Gaussian $\mathcal{N}(\mu, \Sigma)$, the goal is to output a list of $O(1/\alpha)$ hypotheses at least one of which is close to $\Sigma$ in relative Frobenius norm. Our main result is a $\mathrm{poly}(d,1/\alpha)$ sample and time algorithm for this task that guarantees relative Frobenius norm error of $\mathrm{poly}(1/\alpha)$. Importantly, our algorithm relies purely on spectral techniques. As a corollary, we obtain an efficient spectral algorithm for robust partial clustering of Gaussian mixture models (GMMs) -- a key ingredient in the recent work of [BDJ+22] on robustly learning arbitrary GMMs. Combined with the other components of [BDJ+22], our new method yields the first Sum-of-Squares-free algorithm for robustly learning GMMs. At the technical level, we develop a novel multi-filtering method for list-decodable covariance estimation that may be useful in other settings.

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