Paper:
Inferring sparse representations of continuous signals with continuous orthogonal matching pursuit
Many signals, such as spike trains recorded in multi-channel electrophysiological recordings, may be represented as the sparse sum of translated and scaled copies of waveforms whose timing and amplitudes are of interest. From the aggregate signal, one may seek to estimate the identities, amplitudes, and translations of the waveforms that compose the signal. Here we present a fast method for recovering these identities, amplitudes, and translations. The method involves greedily selecting component waveforms and then refining estimates of their amplitudes and translations, moving iteratively between these steps in a process analogous to the well-known Orthogonal Matching Pursuit (OMP) algorithm [11]. Our approach for modeling translations borrows from Continuous Basis Pursuit (CBP) [4], which we extend in several ways: by selecting a subspace that optimally captures translated copies of the waveforms, replacing the convex optimization problem with a greedy approach, and moving to the Fourier domain to more precisely estimate time shifts. We test the resulting method, which we call Continuous Orthogonal Matching Pursuit (COMP), on simulated and neural data, where it shows gains over CBP in both speed and accuracy.

Review:
"Inferring sparse representations of continuous signals with continuous orthogonal matching pursuit" is a nice, compact, well-written paper on a method that should be of relevance for anyone dealing with spike sorting. The authors build up on work of Ekanadham and all (2011a,b) by considering an alternative basis for decompositions, and a different optimisation method. They provide comparison with previous results and report significant improvement in computation time, and evidence for increased accuracy of detection and estimation of waveform parameters.One question about the comparison - in order to be fair to the older method, the authors pruned waveforms with amplitude <.3. Was that value optimized for (was it equal to the lower bound on a in COMP case)? Could CBP score better, if that parameter was further increased? How was lambda of the L1-penalty chosen?Also - is it possible to plot hits and misses in separate plots? Or are these not telling more about the nature of typical errors of various methods?Are all "hit errors" of COMP calculated from the "Fourier" method?Figure 1 Might be a bit confusing to the reader - having observed the original waveform, I expected to see the reconstructed waveform. It took me a while to realise the black shapes in panels 2-4 were not the reconstructions. I would suggest to at least expand the legend.The average errors provided in the legend seem to be much lower than values plotted in the figure.    Since the paper is so well written, I include suggestion to double-check lines:41 (spurious "the")85 (spurious "here")153 (theta missing from the argument of sinus)299/300 please, correct all indices (j' and n are swapped, (i,j) came out of nowhere)311 and 315 - indices swapped411 eps=1 - do you mean 1 Delta?