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 ∈ mathbb Rn × n is a Gaussian matrix with a planted rank-1 spike, and E ∈ mathbb Rn × n is an adversarially chosen matrix supported on an varepsilon n × varepsilon n principal minor. Let vAMP(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 vALG(Y) which is tildeO(√varepsilon)-close to vAMP(X), for any of a class of AMP iterations which includes sparse Principal Component Analysis (PCA), non-negative PCA, and mathbb Z2 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.