Maximum likelihood estimation under partial sparsity constraints

We consider the problem of estimating two deterministic vectors in a linear Gaussian model where one of the unknown vectors is subject to a sparsity constraint. We derive the maximum likelihood estimator for this problem and develop the Projected Orthogonal Matching Pursuit (POMP) algorithm for its practical implementation. The corresponding constrained Cramér-Rao bound (CCRB) on the mean-square-error is developed under the sparsity constraint. We then show that estimation in linear dynamical systems with a sparse control can be formulated as a special case of this problem.

[1]  Richard G. Baraniuk,et al.  Exact signal recovery from sparsely corrupted measurements through the Pursuit of Justice , 2009, 2009 Conference Record of the Forty-Third Asilomar Conference on Signals, Systems and Computers.

[2]  Georgios B. Giannakis,et al.  Doubly Robust Smoothing of Dynamical Processes via Outlier Sparsity Constraints , 2011, IEEE Transactions on Signal Processing.

[3]  Stéphane Mallat,et al.  Matching pursuits with time-frequency dictionaries , 1993, IEEE Trans. Signal Process..

[4]  Yonina C. Eldar,et al.  Introduction to Compressed Sensing , 2022 .

[5]  Yi Ma,et al.  Robust principal component analysis? , 2009, JACM.

[6]  Yonina C. Eldar,et al.  The Cramér-Rao Bound for Estimating a Sparse Parameter Vector , 2010, IEEE Transactions on Signal Processing.

[7]  Joseph Tabrikian,et al.  Performance bounds for constrained parameter estimation , 2012, 2012 IEEE 7th Sensor Array and Multichannel Signal Processing Workshop (SAM).

[8]  Helmut Bölcskei,et al.  Recovery of Sparsely Corrupted Signals , 2011, IEEE Transactions on Information Theory.

[9]  Yonina C. Eldar,et al.  Structured Compressed Sensing: From Theory to Applications , 2011, IEEE Transactions on Signal Processing.

[10]  Alfred O. Hero,et al.  Lower bounds for parametric estimation with constraints , 1990, IEEE Trans. Inf. Theory.