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We consider the problem of parameter estimation from observations given by ageneralized linear model. Spectral methods are a simple yet effective approachfor estimation: they estimate the parameter via the principal eigenvector of amatrix obtained by suitably preprocessing the observations. Despite their wideuse, a rigorous performance characterization of spectral estimators, as well asa principled way to preprocess the data, is available only for unstructured(i.e., i.i.d. Gaussian and Haar) designs. In contrast, real-world designmatrices are highly structured and exhibit non-trivial correlations. To addressthis problem, we consider correlated Gaussian designs which capture theanisotropic nature of the measurements via a feature covariance matrix$\Sigma$. Our main result is a precise asymptotic characterization of theperformance of spectral estimators in this setting. This then allows toidentify the optimal preprocessing that minimizes the number of samples neededto meaningfully estimate the parameter. Remarkably, such an optimal spectralestimator depends on $\Sigma$ only through its normalized trace, which can beconsistently estimated from the data. Numerical results demonstrate theadvantage of our principled approach over previous heuristic methods.  Existing analyses of spectral estimators crucially rely on the rotationalinvariance of the design matrix. This key assumption does not hold forcorrelated Gaussian designs. To circumvent this difficulty, we develop a novelstrategy based on designing and analyzing an approximate message passingalgorithm whose fixed point coincides with the desired spectral estimator. Ourmethodology is general, and opens the way to the precise characterization ofspiked matrices and of the corresponding spectral methods in a variety ofsettings.

Spectral Estimators for Structured Generalized Linear Models via Approximate Message Passing