List-Decodable Covariance Estimation

We give the first polynomial time algorithm for \emph{list-decodable covariance estimation}. For any $\alpha > 0$, our algorithm takes input a sample $Y \subseteq \mathbb{R}^d$ of size $n\geq d^{\mathsf{poly}(1/\alpha)}$ obtained by adversarially corrupting an $(1-\alpha)n$ points in an i.i.d. sample $X$ of size $n$ from the Gaussian distribution with unknown mean $\mu_*$ and covariance $\Sigma_*$. In $n^{\mathsf{poly}(1/\alpha)}$ time, it outputs a constant-size list of $k = k(\alpha)= (1/\alpha)^{\mathsf{poly}(1/\alpha)}$ candidate parameters that, with high probability, contains a $(\hat{\mu},\hat{\Sigma})$ such that the total variation distance $TV(\mathcal{N}(\mu_*,\Sigma_*),\mathcal{N}(\hat{\mu},\hat{\Sigma}))<1-O_{\alpha}(1)$. This is the statistically strongest notion of distance and implies multiplicative spectral and relative Frobenius distance approximation for parameters with dimension independent error. Our algorithm works more generally for $(1-\alpha)$-corruptions of 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 due to Karmarkar, Klivans, and Kothari (2019), Raghavendra and Yau (2019 and 2020) and Bakshi and Kothari (2020). These results need superpolynomial time for obtaining any subconstant error in the underlying dimension. Our result implies the first polynomial-time \emph{exact} algorithm for list-decodable linear regression and subspace recovery that allows, in particular, to obtain $2^{-\mathsf{poly}(d)}$ error in polynomial-time. Our result also implies an improved algorithm for clustering non-spherical mixtures.