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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Bibliographic Details
Published in:IEEE transactions on signal processing Vol. 59; no. 10; pp. 5070 - 5076
Main Authors: Gongguo Tang, Nehorai, Arye
Format: Journal Article
Language:English
Published: New York, NY IEEE 01.10.2011
Institute of Electrical and Electronics Engineers
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
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
Online Access:Get full text
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Summary: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.
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ISSN:1053-587X
1941-0476
DOI:10.1109/TSP.2011.2161984