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Support Stability of Spike Deconvolution via Total Variation Minimization

Spike deconvolution is an inverse problem aiming at recovering point sources from their convolution with a point spread function (PSF). The stability of this problem in the presence of noise is long known to be closely related to the separation between those sources. It is therefore essential to characterize the resolution limit above which the point sources can be stably recovered from a given estimator, without spurious or missing sources from the estimate. In this paper, we establish the resolution limit above which the Beurling-LASSO estimator can stably recover two point sources, and show that the limit depends only on the PSF. Our result highlights the impact of PSF on the resolution limit in the noisy setting, which was not evident in previous studies of the noiseless setting. We further confirm our findings by comparing the theoretical limit with the empirical performance of the Beurling-LASSO estimator.