Faster Algorithms for HighDimensional Robust Covariance Estimation
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Proceedings of the ThirtySecond Conference on Learning Theory, PMLR 99:727757, 2019.
Abstract
We study the problem of estimating the covariance matrix of a highdimensional distribution when a small constant fraction of the samples can be arbitrarily corrupted. Recent work gave the first polynomial time algorithms for this problem with nearoptimal error guarantees for several natural structured distributions. Our main contribution is to develop faster algorithms for this problem whose running time nearly matches that of computing the empirical covariance. Given $N = \tilde{\Omega}(d^2/\epsilon^2)$ samples from a $d$dimensional Gaussian distribution, an $\epsilon$fraction of which may be arbitrarily corrupted, our algorithm runs in time $\tilde{O}(d^{3.26})/\mathrm{poly}(\epsilon)$ and approximates the unknown covariance matrix to optimal error up to a logarithmic factor. Previous robust algorithms with comparable error guarantees all have runtimes $\tilde{\Omega}(d^{2 \omega})$ when $\epsilon = \Omega(1)$, where $\omega$ is the exponent of matrix multiplication. We also provide evidence that improving the running time of our algorithm may require new algorithmic techniques.
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