Petros Boufounos

Title: L1 minimization without amplitude information

Abstract: In many practical situations, we need to reconstruct sparse signals from measurements that eliminate amplitude information of the acquired signal. This talk discusses two specific acquisition scenarios. In the first scenario the sparse signal is acquired by recording the timing of its zero crossings. Signal reconstruction from zero crossings is well know to be unstable. The sparsity prior and L1 regularization stabilizes the reconstruction and allows us to recover the signal. In the second scenario we only acquire the signs of linear measurements of a sparse signal. This introduces significant reconstruction ambiguity which the sparsity prior allows us to resolve. 

In both cases, the measurements eliminate amplitude information on the signal, resulting to a fundamental ambiguity. To resolve this ambiguity we impose the constraint that the signal lies on the unit L2 sphere. This constraint significantly reduces the search space and improves reconstruction results. We demonstrate a variation of the Fixed Point Continuation algorithm that runs on the unit sphere and, despite the nonconvexity of the problems, produces very good results.