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arXiv · 2502.02545

Optimal Spectral Transitions in High-Dimensional Multi-Index Models

Leonardo Defilippis·Yatin Dandi·Pierre Mergny·Florent Krzakala·Bruno Loureiro
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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.

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Leonardo Defilippis, Yatin Dandi, Pierre Mergny, Florent Krzakala, Bruno Loureiro. 2025-02-04. Optimal Spectral Transitions in High-Dimensional Multi-Index Models. https://arxiv.org/abs/2502.02545

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