Optimal Spectral Transitions in High-Dimensional Multi-Index Models
Abstract
We consider the problem of how many samples from a Gaussian multi-index model are required to weakly reconstruct the relevant index subspace. Despite its increasing popularity as a testbed for investigating the computational complexity of neural networks, results beyond the single-index setting remain elusive. In this work, we introduce spectral algorithms based on the linearization of a message passing scheme tailored to this problem. Our main contribution is to show that the proposed methods achieve the optimal reconstruction threshold. Leveraging a high-dimensional characterization of the algorithms, we show that above the critical threshold the leading eigenvector correlates with the relevant index subspace, a phenomenon reminiscent of the Baik-Ben Arous-Peche (BBP) transition in spiked models arising in random matrix theory. Supported by numerical experiments and a rigorous theoretical framework, our work bridges critical gaps in the computational limits of weak learnability in multi-index model.
Cite
@article{arxiv.2502.02545,
title = {Optimal Spectral Transitions in High-Dimensional Multi-Index Models},
author = {Leonardo Defilippis and Yatin Dandi and Pierre Mergny and Florent Krzakala and Bruno Loureiro},
journal= {arXiv preprint arXiv:2502.02545},
year = {2025}
}