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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

Korean

Eldan과 Shamir, Daniely, 및 다른 사람들에 의해 연구된 것처럼, 우리는 신경 네트워크에 대한 깊이 분리 문제를 재검토합니다. 구체적으로, 우리는 $L^2$의 표준 Gaussian 측정에 따라 방사선 기능의 자연 설정을 고려합니다. $L^2$의 표준 Gaussian 측정에 따라 $L^2$의 방사선 기능이 있습니다. 우리는 표준 Gaussian 측정에 따라 $L^2$의 방사선 기능이 $D$의 표준 Gaussian 측정에 따라 존재한다는 것을 보여줍니다. 이를 위해, 우리는 방사선 기능이 Hermite 확장의 높은 주파수에서 상당한 질량을 가지고 있다면, 그것은 작은 경계와 작은 경계를 가진 Ne-2 depural 네트워크에 의해 잘 접근 할 수 없다는

Machine translation (Meta M2M100); the English original follows.

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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