Easy, robust approximate message passing for planted spike models

We present a simple and efficient algorithm for robust approximate message passing (AMP) in the spiked matrix setting. In particular, let \varepsilon be a sufficiently small constant, and suppose that X \in \mathbb R^{n \times n} is a Gaussian matrix with a planted rank-1 spike, and E \in \mathbb R^{n \times n} is an adversarially chosen matrix supported on an \varepsilon n \times \varepsilon n principal minor. Let v_{AMP}(X) be the output of an AMP iteration on the uncorrupted matrix X. We give a procedure that, given access only to the corrupted matrix Y = X + E, computes a vector v_{ALG}(Y) which is \tilde{O}(\sqrt{\varepsilon})-close to v_{AMP}(X), for any of a class of AMP iterations which includes sparse Principal Component Analysis (PCA), non-negative PCA, and \mathbb Z_2 synchronization. Our algorithm consists of a spectral pre-processing step combined with a robust spectral initialization procedure; given these inputs, we prove that (perhaps surprisingly) AMP is robust out-of-the-box.