On the sample complexity of robust PCA

Matthew Coudron, Gilad Lerman

Research output: Chapter in Book/Report/Conference proceedingConference contribution

6 Scopus citations

Abstract

We estimate the rate of convergence and sample complexity of a recent robust estimator for a generalized version of the inverse covariance matrix. This estimator is used in a convex algorithm for robust subspace recovery (i.e., robust PCA). Our model assumes a sub-Gaussian underlying distribution and an i.i.d. sample from it. Our main result shows with high probability that the norm of the difference between the generalized inverse covariance of the underlying distribution and its estimator from an i.i.d. sample of size N is of order O(N-0.5+∈) for arbitrarily small ∈ > 0 (affecting the probabilistic estimate); this rate of convergence is close to the one of direct covariance estimation, i.e., O(N-0.5). Our precise probabilistic estimate implies for some natural settings that the sample complexity of the generalized inverse covariance estimation when using the Frobenius norm is O(D2+δ) for arbitrarily small δ > 0 (whereas the sample complexity of direct covariance estimation with Frobenius norm is O(D2)). These results provide similar rates of convergence and sample complexity for the corresponding robust subspace recovery algorithm. To the best of our knowledge, this is the only work analyzing the sample complexity of any robust PCA algorithm.

Original languageEnglish (US)
Title of host publicationAdvances in Neural Information Processing Systems 25
Subtitle of host publication26th Annual Conference on Neural Information Processing Systems 2012, NIPS 2012
Pages3221-3229
Number of pages9
StatePublished - Dec 1 2012
Event26th Annual Conference on Neural Information Processing Systems 2012, NIPS 2012 - Lake Tahoe, NV, United States
Duration: Dec 3 2012Dec 6 2012

Publication series

NameAdvances in Neural Information Processing Systems
Volume4
ISSN (Print)1049-5258

Other

Other26th Annual Conference on Neural Information Processing Systems 2012, NIPS 2012
CountryUnited States
CityLake Tahoe, NV
Period12/3/1212/6/12

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  • Cite this

    Coudron, M., & Lerman, G. (2012). On the sample complexity of robust PCA. In Advances in Neural Information Processing Systems 25: 26th Annual Conference on Neural Information Processing Systems 2012, NIPS 2012 (pp. 3221-3229). (Advances in Neural Information Processing Systems; Vol. 4).