On the Impossibility of Uniform Sparse Reconstruction using Greedy Methods

It has previously shown that a trigonometric polynomial having at most M nonvanishing coecients can be recovered from N = O(M log(D)) random samples by the greedy methods thresholding and orthogonal matching pursuit with high probability. In this note we show that these results cannot be made uniform in the sense that a single (random) sampling set cannot guarantee recovery of all such M-sparse trigonometric polynomials simultaneously with high probability using the two greedy methods.

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