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Recovery of the sparsity pattern (or support) of an unknown sparse vectorfrom a limited number of noisy linear measurements is an important problem incompressed sensing. In the high-dimensional setting, it is known that recoverywith a vanishing fraction of errors is impossible if the measurement rate andthe per-sample signal-to-noise ratio (SNR) are finite constants, independent ofthe vector length. In this paper, it is shown that recovery with an arbitrarilysmall but constant fraction of errors is, however, possible, and that in somecases computationally simple estimators are near-optimal. Bounds on themeasurement rate needed to attain a desired fraction of errors are given interms of the SNR and various key parameters of the unknown vector for severaldifferent recovery algorithms. The tightness of the bounds, in a scaling sense,as a function of the SNR and the fraction of errors, is established bycomparison with existing information-theoretic necessary bounds. Nearoptimality is shown for a wide variety of practically motivated signal models.

The Sampling Rate-Distortion Tradeoff for Sparsity Pattern Recovery in Compressed Sensing