We give the first polynomial time algorithm for list-decodable covariance estimation. For any πΌ > 0, our algorithm takes input a sample π β β π of size π β©Ύ π poly(1/πΌ) obtained by adversarially corrupting an (1 β πΌ)π points in an i.i.d. sample π of size π from the Gaussian distribution with unknown mean π * and covariance Ξ£ * . In π poly(1/πΌ) time, it outputs a constant-size list of π = π (πΌ) = (1/πΌ) poly(1/πΌ) candidate parameters that, with high probability, contains a ( ΞΌ, Ξ£) such that the total variation distance ππ (N (π * , Ξ£ * ), N ( ΞΌ, Ξ£)) < 1βπ πΌ (1). This is a statistically strongest notion of distance and implies multiplicative spectral and relative Frobenius distance approximation with dimension independent error. Our algorithm works more generally for any distribution π· that possesses low-degree sum-of-squares certificates of two natural analytic properties: 1) anti-concentration of one-dimensional marginals and 2) hypercontractivity of degree 2 polynomials.Prior to our work, the only known results for estimating covariance in the list-decodable setting were for the special cases of list-decodable linear regression and subspace recovery [Karmalkar-Klivans-Kothari 2019. Even for these special cases, the known error guarantees are weak and in particular, the algorithms need super-polynomial time for any sub-constant (in dimension π) target error in natural norms. Our result, as a corollary, yields the first polynomial time exact algorithm for list-decodable linear regression and subspace recovery that, in particular, obtain 2 β poly(π) error in polynomial-time in the underlying dimension. CCS CONCEPTSβ’ Theory of computation β Design and analysis of algorithms; Complexity theory and logic.
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