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Recent research has shown that performance in signal processing tasks canoften be significantly improved by using signal models based on sparserepresentations, where a signal is approximated using a small number ofelements from a fixed dictionary. Unfortunately, inference in this modelinvolves solving non-smooth optimization problems that are computationallyexpensive. While significant efforts have focused on developing digitalalgorithms specifically for this problem, these algorithms are inappropriatefor many applications because of the time and power requirements necessary tosolve large optimization problems. Based on recent work in computationalneuroscience, we explore the potential advantages of continuous time dynamicalsystems for solving sparse approximation problems if they were implemented inanalog VLSI. Specifically, in the simulated task of recovering synthetic andMRI data acquired via compressive sensing techniques, we show that thesesystems can potentially perform recovery at time scales of 10-20{\mu}s,supporting datarates of 50-100 kHz (orders of magnitude faster that digitalalgorithms). Furthermore, we show analytically that a wide range of sparseapproximation problems can be solved in the same basic architecture, includingapproximate $\ell^p$ norms, modified $\ell^1$ norms, re-weighted $\ell^1$ and$\ell^2$, the block $\ell^1$ norm and classic Tikhonov regularization.

Analog Sparse Approximation with Applications to Compressed Sensing