Approximate message passing (AMP) is a family of iterative algorithms that generalize matrix power iteration. AMP algorithms are known to optimally solve many average-case optimization problems. In this paper, we show that a large class of AMP algorithms can be simulated in polynomial time by local statistics hierarchy semidefinite programs (SDPs), even when an unknown principal minor of measure 1/polylog(dimension) is adversarially corrupted. Ours are the first robust guarantees for many of these problems. Further, our results offer an interesting counterpoint to strong lower bounds against less constrained SDP relaxations for average-case max-cut-gain (a.k.a. “optimizing the Sherrington-Kirkpatrick Hamiltonian”) and other problems.

We give the first polynomial time algorithm for list-decodable covariance estimation. For any α > 0, our algorithm takes input a sample Y ⊆ d of size n≥ dpoly(1/α) obtained by adversarially corrupting an (1−α)n points in an i.i.d. sample X of size n from the Gaussian distribution with unknown mean µ* and covariance Σ*. In npoly(1/α) time, it outputs a constant-size list of k = k(α)= (1/α)poly(1/α) candidate parameters that, with high probability, contains a (µ,Σ) such that the total variation distance TV(N(µ*,Σ*),N(µ,Σ))<1−Oα(1). This is a statistically strongest notion of distance and implies multiplicative spectral and relative Frobenius distance approximation with dimension independent error. Our algorithm works more generally for any distribution D that possesses low-degree sum-of-squares certificates of two natural analytic properties: 1) anti-concentration of one-dimensional marginals and 2) hypercontractivity of degree 2 polynomials.