Sketched Over-Parametrized Projected Gradient Descent for Sparse Spike Estimation

We consider the problem of recovering off-the-grid spikes from linear measurements in the context of Single Molecule Localization Microscopy (SMLM). State of the art model-based methods such as Over-Parametrized Continuous Orthogonal Matching Pursuit (OP-COMP) with Projected Gradient Descent (PGD) have been shown to successfully recover those signals. The computational cost of these methods scales linearly with the number of measurements. When this number of measurements is large with respect to the dimensionality of the signal, we propose to reduce it with a so-called sketching operator. Based on recent results on compressive sensing in the space of measures, we approximate the ideal sketching operator (benefiting from theoretical recovery guarantees), in the context of SMLM. This sketching method coupled to OP-COMP with PGD shows significant improvements in calculation time in realistic synthetic microscopy experiments.