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We study symmetric spiked matrix models with respect to a general class ofnoise distributions. Given a rank-1 deformation of a random noise matrix, whoseentries are independently distributed with zero mean and unit variance, thegoal is to estimate the rank-1 part. For the case of Gaussian noise, the topeigenvector of the given matrix is a widely-studied estimator known to achieveoptimal statistical guarantees, e.g., in the sense of the celebrated BBP phasetransition. However, this estimator can fail completely for heavy-tailed noise.In this work, we exhibit an estimator that works for heavy-tailed noise up tothe BBP threshold that is optimal even for Gaussian noise. We give anon-asymptotic analysis of our estimator which relies only on the variance ofeach entry remaining constant as the size of the matrix grows: higher momentsmay grow arbitrarily fast or even fail to exist. Previously, it was only knownhow to achieve these guarantees if higher-order moments of the noises arebounded by a constant independent of the size of the matrix. Our estimator canbe evaluated in polynomial time by counting self-avoiding walks via a color-coding technique. Moreover, we extend our estimator to spiked tensor modelsand establish analogous results.

Estimating Rank-One Spikes from Heavy-Tailed Noise via Self-Avoiding Walks