The Geometry of Off-the-Grid Compressed Sensing

Compressed sensing (CS) ensures the recovery of sparse vectors from a number randomized measurements proportional to their sparsity. The initial theory considers discretized domains, and randomness makes physical positions grid nodes irrelevant. Most imaging devices, however, operate over some continuous domain, it sense consider Dirac masses with arbitrary positions. In this article, we such setup analyze performance BLASSO algorithm, which is extension celebrated LASSO $$\ell ^1$$ regularization method. This approach appealing numerical perspective because avoids discretize domain interest. Previous works considered translation-invariant measurements, as Fourier coefficients, in clear that discrete should be extended by imposing minimum distance separation constraint (often called “Rayleigh limit”) between Diracs. These prior works, rule out many domains operators interest, are not translation invariant. includes, for instance, Laplace positive reals Gaussian mixture models mean-covariance space. Our theoretical advances crucially rely on introduction canonical metric associated measurement operator, so-called Fisher geodesic distance. case one recovers Euclidean metric, but can cope (possibly non-translation invariant) domains. Furthermore, naturally invariant under joint reparameterization both operator locations. second main contribution shows if spikes larger than Rayleigh constant, then stable way stream Diracs, provided (up log factors) We measure stability using an optimal transport constructed top result factor) sharp does require any assumption amplitudes underlying measure. proof technique relies infinite-dimensional golfing scheme operates space measures general