Fast Signal Recovery from Quadratic Measurements

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 formed by the cross correlations of the sought signal. This creates a bottleneck for the inversion since the number of unknowns grows quadratically with the dimension of the signal. The main idea of the proposed approach is to reduce the dimensionality of the problem by recovering only the diagonal of the unknown matrix, whose dimension grows linearly with the size of the signal, and use an efficient Noise Collector to absorb the cross-correlated data that come from the off-diagonal elements of this matrix. These elements do not carry extra information about the support of the signal, but significantly contribute to these data. With this strategy, we recover the unknown matrix by solving a convex linear problem whose cost is similar to the one that uses linear measurements. Our theory shows that the proposed approach provides exact support recovery when the data is not too noisy, and that there are no false positives for any level of noise. It also demonstrates that the level of sparsity that can be recovered scales almost linearly with the number of data. The numerical experiments presented in the paper corroborate these findings.