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Spectral Estimators for Multi-Index Models: Precise Asymptotics and Optimal Weak Recovery

3 February 2025
Filip Kovačević
Yihan Zhang
Marco Mondelli
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Abstract

Multi-index models provide a popular framework to investigate the learnability of functions with low-dimensional structure and, also due to their connections with neural networks, they have been object of recent intensive study. In this paper, we focus on recovering the subspace spanned by the signals via spectral estimators -- a family of methods that are routinely used in practice, often as a warm-start for iterative algorithms. Our main technical contribution is a precise asymptotic characterization of the performance of spectral methods, when sample size and input dimension grow proportionally and the dimension ppp of the space to recover is fixed. Specifically, we locate the top-ppp eigenvalues of the spectral matrix and establish the overlaps between the corresponding eigenvectors (which give the spectral estimators) and a basis of the signal subspace. Our analysis unveils a phase transition phenomenon in which, as the sample complexity grows, eigenvalues escape from the bulk of the spectrum and, when that happens, eigenvectors recover directions of the desired subspace. The precise characterization we put forward enables the optimization of the data preprocessing, thus allowing to identify the spectral estimator that requires the minimal sample size for weak recovery.

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@article{kovačević2025_2502.01583,
  title={ Spectral Estimators for Multi-Index Models: Precise Asymptotics and Optimal Weak Recovery },
  author={ Filip Kovačević and Yihan Zhang and Marco Mondelli },
  journal={arXiv preprint arXiv:2502.01583},
  year={ 2025 }
}
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