Accelerating Sparse Recovery by Reducing Chatter

Compressive Sensing has driven a resurgence of sparse recovery algorithms with l_1-norm minimization. While these minimizations are relatively well understood for small underdetermined, possibly inconsistent systems, their behavior for large over-determined and inconsistent systems has received much less attention. Specifically, we focus on large systems where computational restrictions call for algorithms that use randomized subsets of rows that are touched a limited number of times. In that regime, l_1-norm minimization algorithms exhibit unwanted fluctuations near the desired solution, and the Linear Bregman iterations are no exception. We explain this observed lack of performance in terms of chatter, a well-known phenomena observed in non-smooth dynamical systems, where intermediate solutions wander between different states stifling convergence. By identifying chatter as the culprit, we modify the Bregman iterations with chatter reducing adaptive element-wise step lengths in combination with potential support detection via threshold crossing. We demonstrate the performance of our algorithm on carefully selected stylized examples and a realistic seismic imaging problem involving millions of unknowns and matrix-free matrix-vector products that involve expensive wave-equation solves.