Rigorous dynamics of expectation-propagation-based signal recovery from unitarily invariant measurements

Abstract

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 EPbased algorithm achieves the Bayes-optimal performance derived by Takeda et al. in 2006 via a non-rigorous tool in statistical physics, 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.

On the Convergence of Orthogonal/Vector AMP: Long-Memory Message-Passing Strategy

“…In the initial iteration t = 0, the initial values x B→A,0 = 0 and v B→A,0,0 = E[ x 2 ]/N are used to compute (10) and (11). The sufficient statistic ( 10) is equivalent to optimized LM damping of all preceding messages {x B→A,t ′ } t t ′ =0 in [18], which was obtained as a solution to an optimization problem based on state evolution results.…”

“…In this paper, these MP algorithms are referred to as OAMP. When the sensing matrix is right-orthogonally invariant, OAMP was proved to be Bayes-optimal in the large system limit [6], [11].…”

“…Strictly speaking, the Bayes-optimality of OAMP needs an implicit assumption under which state evolution recursions for OAMP converge to a fixed point (FP) after an infinite number of iterations [6], [11]. Thus, this assumption needs to be confirmed for individual problems [12]- [14].…”

A Family of Counterexamples to the Central Limit Theorem Based on Binary Linear Codes

“…Suppose that H is a parity-check matrix of an [n, k, d] binary linear code, and consider the sequence X defined in (1). Then, the CLT fails for all d ≤ n.…”

“…The CLT is useful in the field of information theory, communications, and signal processing. For example, it provides a foundation for the additive white Gaussian noise (AWGN) channel in information theory, and was utilized to prove the asymptotic convergence property of message-passing algorithms in communications or compressed sensing [1].…”