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Section: New Results

Affine Invariant Covariance Estimation for Heavy-Tailed Distributions

In this work we provide an estimator for the covariance matrix of a heavy-tailed multivariate distribution. We prove that the proposed estimator S^ admits an affine-invariant bound of the form

( 1 - ϵ ) S S ^ ( 1 + ϵ ) S

in high probability, where S is the unknown covariance matrix, and is the positive semidefinite order on symmetric matrices. The result only requires the existence of fourth-order moments, and allows for ϵ=O(k4dlog(d/δ)/n) where k4 is a measure of kurtosis of the distribution, d is the dimensionality of the space, n is the sample size, and 1δ is the desired confidence level. More generally, we can allow for regularization with level λ, then d gets replaced with the degrees of freedom number. Denoting cond(S) the condition number of S, the computational cost of the novel estimator is O(d2n+d3log(cond(S))), which is comparable to the cost of the sample covariance estimator in the statistically interesting regime nd. We consider applications of our estimator to eigenvalue estimation with relative error, and to ridge regression with heavy-tailed random design.