Recovery of continuous quantities from discrete and binary data with applications to neural data

We consider three problems, motivated by questions in computational neuroscience, related to recovering continuous quantities from binary or discrete data or measurements in the context of sparse structure. First, we show that it is possible to recover the norms of sparse vectors given one-bit compressive measurements, and provide associated guarantees. Second, we present a novel algorithm for spike-sorting in neural data, which involves recovering continuous times and amplitudes of events using discrete bases. This method, Continuous Orthogonal Matching Pursuit, builds on algorithms used in compressive sensing. It exploits the sparsity of the signal and proceeds greedily, achieving gains in speed and accuracy over previous methods. Lastly, we present a Bayesian method making use of hierarchical priors for entropy rate estimation from binary sequences.