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Approximate message passing (AMP) is a family of iterative algorithms thatgeneralize matrix power iteration. AMP algorithms are known to optimally solvemany average-case optimization problems. In this paper, we show that a largeclass of AMP algorithms can be simulated in polynomial time by \emph{localstatistics hierarchy} semidefinite programs (SDPs), even when an unknownprincipal minor of measure $1/\mathrm{polylog}(\mathrm{dimension})$ isadversarially corrupted. Ours are the first robust guarantees for many of theseproblems. Further, our results offer an interesting counterpoint to stronglower bounds against less constrained SDP relaxations for average-casemax-cut-gain (a.k.a. "optimizing the Sherrington-Kirkpatrick Hamiltonian") andother problems.

Semidefinite programs simulate approximate message passing robustly