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How do statistical dependencies in measurement noise influencehigh-dimensional inference? To answer this, we study the paradigmatic spikedmatrix model of principal components analysis (PCA), where a rank-one matrix iscorrupted by additive noise. We go beyond the usual independence assumption onthe noise entries, by drawing the noise from a low-order polynomial orthogonalmatrix ensemble. The resulting noise correlations make the setting relevant forapplications but analytically challenging. We provide the firstcharacterization of the Bayes-optimal limits of inference in this model. If thespike is rotation-invariant, we show that standard spectral PCA is optimal.However, for more general priors, both PCA and the existing approximate messagepassing algorithm (AMP) fall short of achieving the information-theoreticlimits, which we compute using the replica method from statistical mechanics.We thus propose a novel AMP, inspired by the theory of AdaptiveThouless-Anderson-Palmer equations, which saturates the theoretical limit. ThisAMP comes with a rigorous state evolution analysis tracking its performance.Although we focus on specific noise distributions, our methodology can begeneralized to a wide class of trace matrix ensembles at the cost of moreinvolved expressions. Finally, despite the seemingly strong assumption ofrotation-invariant noise, our theory empirically predicts algorithmicperformance on real data, pointing at remarkable universality properties.

Bayes-optimal limits in structured PCA, and how to reach them