Principal Component Analysis

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Period ending 2026-09-21

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May 30, 2026 · cs.DSJ/K move · Enter open · S save

Easy, robust approximate message passing for planted spike models

Misha Ivkov, Tselil Schramm

*Stanford University. · †Stanford University.

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_{\mathrm{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_{\mathrm{ALG}}(Y)

which is

\tilde{O}(\sqrt{\varepsilon})

-close to

v_{\mathrm{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.

Principal Component Analysis Message Passing Neural Networks Binary Spikes

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