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Approximate Sparsity Pattern Recovery: Information-Theoretic Lower Bounds

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2 Author(s)
Reeves, G. ; Dept. of Electr. Eng. & Comput. Sci., Univ. of California, Berkeley, Berkeley, CA, USA ; Gastpar, M.C.

Recovery of the sparsity pattern (or support) of an unknown sparse vector from a small number of noisy linear measurements is an important problem in compressed sensing. In this paper, the high-dimensional setting is considered. It is shown that if the measurement rate and per-sample signal-to-noise ratio (SNR) are finite constants independent of the length of the vector, then the optimal sparsity pattern estimate will have a constant fraction of errors. Lower bounds on the measurement rate needed to attain a desired fraction of errors are given in terms of the SNR and various key parameters of the unknown vector. The tightness of the bounds in a scaling sense, as a function of the SNR and the fraction of errors, is established by comparison with existing achievable bounds. Near optimality is shown for a wide variety of practically motivated signal models.

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Information Theory, IEEE Transactions on  (Volume:59 ,  Issue: 6 )