Dynamic ℓ1 Reconstruction

Sparse signal recovery often involves solving an ℓ1-regularized optimization problem. Most of the existing algorithms focus on the static settings, where the goal is to recover a fixed signal from a fixed system of equations. This talk will have two parts. In the first, we present a collection of homotopy-based algorithms that dynamically update the solution of the underlying ℓ1 problem as the system changes. The sparse Kalman filter solves an ℓ1-regularized Kalman filter equation for a time-varying signal that follows a linear dynamical system. Our proposed algorithm sequentially updates the solution as the new measurements are added and the old measurements are removed from the system. In the second part of the talk, we will discuss a continuous time "algorithm" (i.e. a set of coupled nonlinear differential equations) for solving a class of sparsity regularized least-square problems. We characterize the convergence properties of this neural-net type system, with a special emphasis on the case when the final solution is indeed sparse. This is joint work with M. Salman Asif, Aurele Balavoine, and Chris Rozell