Super-resolution with Binary Priors: Theory and Algorithms.

The problem of super-resolution is concerned with the reconstruction of temporally/spatially localized events (or spikes) from samples of their convolution with a low-pass filter. Distinct from prior works which exploit sparsity in appropriate domains in order to solve the resulting ill-posed problem, this paper explores the role of binary priors in super-resolution, where the spike (or source) amplitudes are assumed to be binary-valued. Our study is inspired by the problem of neural spike deconvolution, but also applies to other applications such as symbol detection in hybrid millimeter wave communication systems. This paper makes several theoretical and algorithmic contributions to enable binary super-resolution with very few measurements. Our results show that binary constraints offer much stronger identifiability guarantees than sparsity, allowing us to operate in "extreme compression" regimes, where the number of measurements can be significantly smaller than the sparsity level of the spikes. To ensure exact recovery in this "extreme compression" regime, it becomes necessary to design algorithms that exactly enforce binary constraints without relaxation. In order to overcome the ensuing computational challenges, we consider a first order auto-regressive filter (which appears in neural spike deconvolution), and exploit its special structure. This results in a novel formulation of the super-resolution binary spike recovery in terms of binary search in one dimension. We perform numerical experiments that validate our theory and also show the benefits of binary constraints in neural spike deconvolution from real calcium imaging datasets.