2019-06-25-cheng19a.md

abstract	section	title	layout	series	id	month	tex_title	firstpage	lastpage	page	order	cycles	bibtex_author	author	date	address	publisher	container-title	volume	genre	issued	pdf	extras
We study the problem of estimating the covariance matrix of a high-dimensional 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 near-optimal 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.	contributed	Faster Algorithms for High-Dimensional Robust Covariance Estimation	inproceedings	Proceedings of Machine Learning Research	cheng19a	0	Faster Algorithms for High-Dimensional Robust Covariance Estimation	727	757	727-757	727	false	Cheng, Yu and Diakonikolas, Ilias and Ge, Rong and Woodruff, David P.	given family Yu Cheng given family Ilias Diakonikolas given family Rong Ge given family David P. Woodruff	2019-06-25		PMLR	Proceedings of the Thirty-Second Conference on Learning Theory	99	inproceedings	date-parts 2019 6 25	http://proceedings.mlr.press/v99/cheng19a/cheng19a.pdf