Convex approaches to model wavelet coefficients

Statistical dependencies among wavelet coefficients are commonly represented by graphical models such as hidden Markov trees. However, in linear inverse problems such as deconvolution, tomography, and compressed sensing, the presence of a sensing or observation matrix leads to reconstruction problems that are intractable or non-convex optimizations. Past work has dealt with this issue by resorting to greedy or suboptimal iterative reconstruction methods. We propose new modeling approaches based on group sparsity of the wavelet coefficients that leads to convex optimizations that can be solved efficiently. We show that the methods we develop perform significantly better in deconvolution and compressed sensing applications, while being as computationally efficient as standard coefficient-wise approaches such as Lasso. (Joint work with Prof. Robert Nowak and Prof. Stephen Wright )