Hugo Latourelle-Vigeant

We study the problem of learning multi-index models (MIMs), where the label depends on the input x ∈\mathbbR^d only through an unknown s-dimensional projection W_*^⊤ x ∈\mathbbR^s. Exploiting the equivariance of this problem under the orthogonal group O_d, we obtain a sharp harmonic-analytic characterization of the learning complexity for MIMs with spherically symmetric inputs—which refines and generalizes previous Gaussian-specific analyses. Specifically, we derive statistical and computational complexity lower bounds within the Statistical Query (SQ) and Low-Degree Polynomial (LDP) frameworks. These bounds decompose naturally across spherical harmonic subspaces. Guided by this decomposition, we construct a family of spectral algorithms based on harmonic tensor unfolding that sequentially recover the latent directions and (nearly) achieve these SQ and LDP lower bounds. Depending on the choice of harmonic degree sequence, these estimators can realize a broad range of trade-offs between sample and runtime complexity. From a technical standpoint, our results build on the semisimple decomposition of the O_d-action on L^2(S^d-1) and the intertwining isomorphism between spherical harmonics and traceless symmetric tensors.

This paper develops some theory of the Dyson equation for correlated linearizations and uses it to solve a problem on asymptotic deterministic equivalent for the test error in random features regression. The theory developed for the correlated Dyson equation includes existence-uniqueness, spectral support bounds and stability properties. This theory is new for constructing deterministic equivalents for pseudo-resolvents of a class of linearizations with correlated entries. In the application, this theory is used to give a deterministic equivalent of the test error in random features ridge regression, in a proportional scaling regime, wherein we have conditioned on both training and test datasets.