Newest papers

Depth Separation for Neural Networks under Gaussian Measure

Cynthia Bortolotto, Francisco Ganacim, Roberto I. Oliveira · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.28800 · cited by 0 works (OpenAlex) · open-access PDF

Abstract

We revisit the problem of depth separation for neural networks, as studied by Eldan and Shamir, Daniely, and others. Specifically, we consider the natural setting of $L^2$ approximation of radial functions under the standard Gaussian measure in $d$ dimensions. We show that there is a radial function in $L^2$ under the standard Gaussian measure that can be approximated to a fixed sufficiently small $L^2$ error by a depth-3 ReLU neural network of polynomial size in $d$ but can only be approximated by depth-2 networks with subexponentially bounded weights if their size is at least exponential in $d$. For that, we prove a general result showing that if a radial function has substantial mass in the high frequencies of its Hermite expansion, then it cannot be well approximated by a depth-2 neural network with small width and bounded weights. All theorems, lemmas, propositions, and corollaries are verified in Lean~4.

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Topics: Bounded function, Measure (data warehouse), Mathematics, Artificial neural network, Gaussian, Hermite polynomials

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