Compressed Sensing with 1D Total Variation: Breaking Sample Complexity Barriers via Non-Uniform Recovery (iTWIST'20)
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This paper investigates total variation minimization in one spatial dimension for the recovery of gradient-sparse signals from undersampled Gaussian measurements. Recently established bounds for the required sampling rate state that uniform recovery of all s-gradient-sparse signals in ℝ^n is only possible with m ≳√(s n)·PolyLog(n) measurements. Such a condition is especially prohibitive for high-dimensional problems, where s is much smaller than n. However, previous empirical findings seem to indicate that the latter sampling rate does not reflect the typical behavior of total variation minimization. Indeed, this work provides a rigorous analysis that breaks the √(s n)-bottleneck for a large class of natural signals. The main result shows that non-uniform recovery succeeds with high probability for m ≳ s ·PolyLog(n) measurements if the jump discontinuities of the signal vector are sufficiently well separated. In particular, this guarantee allows for signals arising from a discretization of piecewise constant functions defined on an interval. The present paper serves as a short summary of the main results in our recent work [arxiv:2001.09952].
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