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arXiv · 2506.05500

The Generative Leap: Sharp Sample Complexity for Efficiently Learning Gaussian Multi-Index Models

Alex Damian·Jason D. Lee·Joan Bruna
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Abstract

In this work we consider generic Gaussian Multi-index models, in which the labels only depend on the (Gaussian) $d$-dimensional inputs through their projection onto a low-dimensional $r = O_d(1)$ subspace, and we study efficient agnostic estimation procedures for this hidden subspace. We introduce the \emph{generative leap} exponent $k^\star$, a natural extension of the generative exponent from [Damian et al.'24] to the multi-index setting. We first show that a sample complexity of $n=\Theta(d^{1 \vee \k/2})$ is necessary in the class of algorithms captured by the Low-Degree-Polynomial framework. We then establish that this sample complexity is also sufficient, by giving an agnostic sequential estimation procedure (that is, requiring no prior knowledge of the multi-index model) based on a spectral U-statistic over appropriate Hermite tensors. We further compute the generative leap exponent for several examples including piecewise linear functions (deep ReLU networks with bias), and general deep neural networks (with $r$-dimensional first hidden layer).

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Alex Damian, Jason D. Lee, Joan Bruna. 2025-06-05. The Generative Leap: Sharp Sample Complexity for Efficiently Learning Gaussian Multi-Index Models. https://arxiv.org/abs/2506.05500

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