DMS Applied Mathematics Seminar

Speaker: Chrysoula Tsogka, University of California, Merced

Title: Fast Signal Recovery from Quadratic Measurements

Abstract: We present a novel approach for recovering a sparse signal from quadratic measurements corresponding to a rank-one tensorization of the data vector. Such quadratic measurements, referred to as interferometric or cross-correlated data, naturally arise in many fields such as remote sensing, spectroscopy, holography and seismology. Compared to the sparse signal recovery problem that uses linear measurements, the unknown in this case is a matrix, \(X=\rho \rho^*\), formed by the cross correlations of \(\rho \in C^K\). This creates a bottleneck for the inversion since the number of unknowns grows quadratically in \(K\). The main idea of the proposed approach is to reduce the dimensionality of the problem by recovering only the diagonal elements of the unknown matrix, \(| \rho_i|^2\), \(i=1,\ldots,K\). The contribution of the off-diagonal terms \(\rho_i \rho_j^*\) for \(i \neq j\) to the data is treated as noise and is absorbed using the Noise Collector approach introduced in [Moscoso et al, The noise collector for sparse recovery in high dimensions, PNAS 117 (2020)]. With this strategy, we recover the unknown by solving a convex linear problem whose cost is similar to the one that uses linear measurements. The proposed approach provides exact support recovery when the data is not too noisy, and there are no false positives for any level of noise.