Abstract
In compressed sensing a sparse vector is approximately retrieved from an under-determined equation system Ax = b. Exact retrieval would mean solving a large combinatorial problem which is well known to be NP-hard. For b of the form Ax 0 + ϵ, where x 0 is the ground truth and ϵ is noise, the ‘oracle solution’ is the one you get if you a priori know the support of x 0, and is the best solution one could hope for. We provide a non-convex functional whose global minimum is the oracle solution, with the property that any other local minimizer necessarily has high cardinality. We provide estimates of the type with constants C that are significantly lower than for competing methods or theorems, and our theory relies on soft assumptions on the matrix A, in comparison with standard results in the field. The framework also allows to incorporate a priori information on the cardinality of the sought vector. In this case we show that despite being non-convex, our cost functional has no spurious local minima and the global minima is again the oracle solution, thereby providing the first method which is guaranteed to find this point for reasonable levels of noise, without resorting to combinatorial methods.
Highlights
In particular we show that the global minimizer with the minimax concave penalty (MCP)-penalty (i.e. Q2(card)) is the oracle solution; it follows that the MCP is unbiased, not merely nearly unbiased as claimed in [52]
In the present article we give simple conditions which imply that the global minima of (9) for f(x) = μ card(x) is the oracle solution, and that any local minima necessarily has a high cardinality unless it is the global minima
Classical results from compressed sensing literature usually require that the numbers δk are small, something which we have found is hard to fulfill in practice
Summary
The result holds given certain assumptions on the matrix A, related to the restricted isometry property (RIP) of A, which in a separate publication (theorem 1.5, [15]) was shown to hold with ‘overwhelming probability’. These results give the impression that the theory is more or less complete and that improvements only can be marginal. A fairly well-known non-convex alternative is the minimax concave penalty (MCP) by Zhang, which was coined nearly unbiased since the results in [52] imply that the method does find the oracle solution with probability tending to one under the assumptions of that paper. The ‘oracle solution’ is sort of the holy grail of compressed sensing, and aside from Zhang’s work and this publication, there seems to be no reliable methods (with proofs) of how to find it
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