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The problem central to sparse recovery and compressive sensing is that of stable sparse recovery: we want a distribution of matrices A in R^{m\times n} such that, for any x \in R^n and with probability at least 2/3 over A, there is an algorithm to recover x* from Ax with ||x* - x||_p <= C min_{k-sparse x'} ||x - x'||_p for some constant C > 1 and norm p.
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