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.

We study the high-dimensional asymptotics of empirical risk minimization (ERM) in over-parametrized two-layer neural networks with quadratic activations trained on synthetic data. We derive sharp asymptotics for both training and test errors by mapping the ℓ2-regularized learning problem to a convex matrix sensing task with nuclear norm penalization. This reveals that capacity control in such networks emerges from a low-rank structure in the learned feature maps. Our results characterize the global minima of the loss and yield precise generalization thresholds, showing how the width of the target function governs learnability. This analysis bridges and extends ideas from spin-glass methods, matrix factorization, and convex optimization and emphasizes the deep link between low-rank matrix sensing and learning in quadratic neural networks.

This appendix presents the proof of the main technical result, Theorem 1. Throughout the whole proof, we assume that the set of conditions from Sec. 2 is verified. A.1 Required background In this Section, we give an overview of the main concepts and tools on approximate message passing algorithms which will be required for the proof. We start with some definitions that commonly appear in the approximate message-passing literature, see e.g. The main regularity class of functions we will use is that of pseudo-Lipschitz functions, which roughly amounts to functions with polynomially bounded first derivatives. We include the required scaling w.r.t. the dimensions in the definition for convenience. Since K will be kept finite, it can be absorbed in any of the constants. For example, the function f: Rn R,x7 1nkxk22 is pseudo-Lipshitz of order 2. Moreau envelopes and Bregman proximal operators -- In our proof, we will also frequently use the notions of Moreau envelopes and proximal operators, see e.g.

Signal recovery under generative neural network priors has emerged as a promising direction in statistical inference and computational imaging. Theoretical analysis of reconstruction algorithms under generative priors is, however, challenging. For generative priors with fully connected layers and Gaussian i.i.d.

We study a spiked Wigner problem with an inhomogeneous noise profile.

We rigorously prove the aforementioned formula in two particular cases.

We study the high-dimensional asymptotics of empirical risk minimization (ERM) in over-parametrized two-layer neural networks with quadratic activations trained on synthetic data. We derive sharp asymptotics for both training and test errors by mapping the $\ell_2$-regularized learning problem to a convex matrix sensing task with nuclear norm penalization. This reveals that capacity control in such networks emerges from a low-rank structure in the learned feature maps. Our results characterize the global minima of the loss and yield precise generalization thresholds, showing how the width of the target function governs learnability. This analysis bridges and extends ideas from spin-glass methods, matrix factorization, and convex optimization and emphasizes the deep link between low-rank matrix sensing and learning in quadratic neural networks.