# American Institute of Mathematical Sciences

August  2012, 6(3): 447-464. doi: 10.3934/ipi.2012.6.447

## Optimal estimation of $\ell_1$-regularization prior from a regularized empirical Bayesian risk standpoint

 1 Shenzhen Key Lab of Visual Computing and Visual Analytics, Shenzhen Institute of Advanced Technology, Shenzhen, Guangdong, 518055, China 2 Department of Mathematics and Earth and Ocean Science, The University of British Columbia, Vancouver, BC, V6T 1Z2, Canada 3 Business Analytics and Mathematical Sciences, IBM T J Watson Research Center, Yorktown Heights, NY, 10598, United States

Received  March 2011 Revised  April 2012 Published  September 2012

We address the problem of prior matrix estimation for the solution of $\ell_1$-regularized ill-posed inverse problems. From a Bayesian viewpoint, we show that such a matrix can be regarded as an influence matrix in a multivariate $\ell_1$-Laplace density function. Assuming a training set is given, the prior matrix design problem is cast as a maximum likelihood term with an additional sparsity-inducing term. This formulation results in an unconstrained yet nonconvex optimization problem. Memory requirements as well as computation of the nonlinear, nonsmooth sub-gradient equations are prohibitive for large-scale problems. Thus, we introduce an iterative algorithm to design efficient priors for such large problems. We further demonstrate that the solutions of ill-posed inverse problems by incorporation of $\ell_1$-regularization using the learned prior matrix perform generally better than commonly used regularization techniques where the prior matrix is chosen a-priori.
Citation: Hui Huang, Eldad Haber, Lior Horesh. Optimal estimation of $\ell_1$-regularization prior from a regularized empirical Bayesian risk standpoint. Inverse Problems & Imaging, 2012, 6 (3) : 447-464. doi: 10.3934/ipi.2012.6.447
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