On the Convergence of Orthogonal/Vector AMP: Long-Memory Message-Passing Strategy
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Orthogonal/vector approximate message-passing (AMP) is a powerful message-passing (MP) algorithm for signal reconstruction in compressed sensing. This paper proves the convergence of Bayes-optimal orthogonal/vector AMP in the large system limit. The proof strategy is based on a novel long-memory (LM) MP approach: A first step is a construction of LM-MP that is guaranteed to converge in principle. A second step is a large-system analysis of LM-MP via an existing framework of state evolution. A third step is to prove the convergence of Bayes-optimal LM-MP via a new statistical interpretation of existing LM damping. The last is an exact reduction of Bayes-optimal LM-MP to Bayes-optimal orthogonal/vector AMP. Numerical simulations are presented to show the verification of state evolution results for damped orthogonal/vector AMP and a challenge in LM-MP for finite-sized systems.
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