Abstract
Signal recovery from unitarily invariant measurements is investigated in this paper. A message-passing algorithm is formulated on the basis of expectation propagation (EP). A rigorous analysis is presented for the dynamics of the algorithm in the large system limit, where both input and output dimensions tend to infinity while the compression rate is kept constant. The main result is the justification of state evolution (SE) equations conjectured by Ma and Ping. This result implies that the EP-based algorithm achieves the Bayes-optimal performance that was originally derived via a non-rigorous tool in statistical physics and proved partially in a recent paper, when the compression rate is larger than a threshold. The proof is based on an extension of a conventional conditioning technique for the standard Gaussian matrix to the case of the Haar matrix.
Highlights
A breakthrough for signal recovery is to construct messagepassing (MP) that is Bayes-optimal in the large system limit, Manuscript received April 5, 2017; revised May 20, 2019; accepted August 8, 2019
When the compression rate is larger than the so-called belief propagation (BP) threshold [6], the BP-based algorithm was numerically shown to achieve the Bayes-optimal performance in the large system limit, which was originally conjectured by Tanaka [7] via the replica method—a non-rigorous tool in statistical physics, and proved in [8], [9] for i.i.d. zero-mean Gaussian measurements
It is recognized that approximate message-passing (AMP) fails to converge when the i.i.d. zero-mean assumption of measurement matrices is broken [15], unless damping [16] is employed. As solutions to this convergence issue, since Opper and Winther’s pioneering work [17, Appendix D], as well as [18], several algorithms have been proposed on the basis of expectation propagation (EP) [19], expectation consistent (EC) approximations [17], [20], [21], S-transform [22], vector AMP [23], or turbo principle [24]–[27]
Summary
When the compression rate is larger than the so-called BP threshold [6], the BP-based algorithm was numerically shown to achieve the Bayes-optimal performance in the large system limit, which was originally conjectured by Tanaka [7] via the replica method—a non-rigorous tool in statistical physics, and proved in [8], [9] for i.i.d. zero-mean Gaussian measurements. It is recognized that AMP fails to converge when the i.i.d. zero-mean assumption of measurement matrices is broken [15], unless damping [16] is employed As solutions to this convergence issue, since Opper and Winther’s pioneering work [17, Appendix D], as well as [18], several algorithms have been proposed on the basis of expectation propagation (EP) [19], expectation consistent (EC) approximations [17], [20], [21], S-transform [22], vector AMP [23], or turbo principle [24]–[27]. The purpose of this paper is to present a rigorous proof for the conjecture
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