## 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 $\widehat{S}$ admits an *affine-invariant* bound of the form

in high probability, where S is the unknown covariance matrix, and $\le $ is the positive semidefinite order on symmetric matrices. The result only requires the existence of fourth-order moments, and allows for $\u03f5=O\left(\sqrt{{k}^{4}dlog(d/\delta )/n}\right)$ where ${k}^{4}$ is a measure of kurtosis of the distribution, d is the dimensionality of the space, n is the sample size, and $1-\delta $ is the desired confidence level. More generally, we can allow for regularization with level $\lambda $, then d gets replaced with the degrees of freedom number. Denoting $cond\left(S\right)$ the condition number of S, the computational cost of the novel estimator is $O({d}^{2}n+{d}^{3}log\left(cond\left(S\right)\right))$, which is comparable to the cost of the sample covariance estimator in the statistically interesting regime $n\ge d$. We consider applications of our estimator to eigenvalue estimation with relative error, and to ridge regression with heavy-tailed random design.