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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

Japanese

エルダンとシャミール、ダニエリー、およびその他が研究したように、神経ネットワークの深度分離の問題を再検討します。具体的には、我々は、標準のガウシアの測定値の下で、標準的なガウシアの測定値の下で、ラジオ機能の自然な設定L^2$を考慮します。我々は、標準的なガウシアの測定値の下で、L^2$のラジオ機能があることを示しますが、その大きさが少なくとも数値的にL^2$である場合に、下指数に限定された重さを持つ深度2ネットワークによってのみ近似することができることを示します。それのために、我々は、ラジオ機能がそのヘルミット拡張の高い周波数で大量の質量を持っている場合

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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