For multi-index models, the eigenvalues, eigenvector overlaps, and optimal preprocessing of spectral estimators are characterized exactly in the proportional asymptotics.
The merged-staircase property: a necessary and nearly sufficient condition for sgd learning of sparse functions on two-layer neural networks
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Spectral Estimators for Multi-Index Models: Precise Asymptotics and Optimal Weak Recovery
For multi-index models, the eigenvalues, eigenvector overlaps, and optimal preprocessing of spectral estimators are characterized exactly in the proportional asymptotics.