Constrained Cramér-Rao Bound on Robust Principal Component Analysis

We investigate the behavior of the mean-square error (MSE) of low-rank and sparse matrix decomposition, in particular the special case of the robust principal component analysis (RPCA), and its generalization matrix completion and correction (MCC). We derive a constrained Cramér-Rao bound (CRB) for...

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Vydáno v:	IEEE transactions on signal processing Ročník 59; číslo 10; s. 5070 - 5076
Hlavní autoři:	Gongguo Tang, Nehorai, Arye
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York, NY IEEE 01.10.2011 Institute of Electrical and Electronics Engineers The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Témata:	Accelerated proximal gradient algorithm Applied sciences Approximation constrained Cramér-Rao bound Constraints Cramer-Rao bounds Design engineering Detection, estimation, filtering, equalization, prediction Estimators Exact sciences and technology Information, signal and communications theory matrix completion and correction Matrix decomposition Maximum likelihood estimation mean-square error Modeling Optimization Principal component analysis robust principal component analysis Robustness Signal and communications theory Signal to noise ratio Signal, noise Sparse matrices Telecommunications and information theory Performance evaluation robust principal component analysis Matrix decomposition maximum likelihood estimation Mean square error Statistical method Unbiased estimation Sparse matrix Signal processing Maximum likelihood Accelerated proximal gradient algorithm Gradient method constrained Cramér―Rao bound matrix completion and correction Principal component analysis Cramer Rao inequality Signal to noise ratio mean-square error
ISSN:	1053-587X, 1941-0476
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Shrnutí:	We investigate the behavior of the mean-square error (MSE) of low-rank and sparse matrix decomposition, in particular the special case of the robust principal component analysis (RPCA), and its generalization matrix completion and correction (MCC). We derive a constrained Cramér-Rao bound (CRB) for any locally unbiased estimator of the low-rank matrix and of the sparse matrix. We analyze the typical behavior of the constrained CRB for MCC where a subset of entries of the underlying matrix are randomly observed, some of which are grossly corrupted. We obtain approximated constrained CRBs by using a concentration of measure argument. We design an alternating minimization procedure to compute the maximum-likelihood estimator (MLE) for the low-rank matrix and the sparse matrix, assuming knowledge of the rank and the sparsity level. For relatively small rank and sparsity level, we demonstrate numerically that the performance of the MLE approaches the constrained CRB when the signal-to-noise-ratio is high. We discuss the implications of these bounds and compare them with the empirical performance of the accelerated proximal gradient algorithm as well as other existing bounds in the literature.
Bibliografie:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 ObjectType-Article-2 ObjectType-Feature-1 content type line 23
ISSN:	1053-587X 1941-0476
DOI:	10.1109/TSP.2011.2161984