### 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(D^{2+δ}) for arbitrarily small δ > 0 (whereas the sample complexity of direct covariance estimation with Frobenius norm is O(D_{2})). 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 language | English (US) |
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Title of host publication | Advances in Neural Information Processing Systems 25 |

Subtitle of host publication | 26th Annual Conference on Neural Information Processing Systems 2012, NIPS 2012 |

Pages | 3221-3229 |

Number of pages | 9 |

State | Published - Dec 1 2012 |

Event | 26th Annual Conference on Neural Information Processing Systems 2012, NIPS 2012 - Lake Tahoe, NV, United States Duration: Dec 3 2012 → Dec 6 2012 |

### Publication series

Name | Advances in Neural Information Processing Systems |
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Volume | 4 |

ISSN (Print) | 1049-5258 |

### Other

Other | 26th Annual Conference on Neural Information Processing Systems 2012, NIPS 2012 |
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Country | United States |

City | Lake Tahoe, NV |

Period | 12/3/12 → 12/6/12 |

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## Cite this

*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).