List-decodable covariance estimation

Abstract

We give the first polynomial time algorithm for list-decodable covariance estimation. For any 𝛼 > 0, our algorithm takes input a sample 𝑌 ⊆ ℝ 𝑑 of size 𝑛 ⩾ 𝑑 poly(1/𝛼) obtained by adversarially corrupting an (1 − 𝛼)𝑛 points in an i.i.d. sample 𝑋 of size 𝑛 from the Gaussian distribution with unknown mean 𝜇 * and covariance Σ * . In 𝑛 poly(1/𝛼) time, it outputs a constant-size list of 𝑘 = 𝑘 (𝛼) = (1/𝛼) poly(1/𝛼) candidate parameters that, with high probability, contains a ( μ, Σ) such that the total variation distance 𝑇𝑉 (N (𝜇 * , Σ * ), N ( μ, Σ)) < 1−𝑂 𝛼 (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 𝐷 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 𝑑) 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 − poly(𝑑) error in polynomial-time in the underlying dimension. CCS CONCEPTS• Theory of computation → Design and analysis of algorithms; Complexity theory and logic.