Sobolev Approximation by Fixed-Size Neural Networks with Arbitrary Accuracy

• URL: http://arxiv.org/abs/2606.16975v1
• Date: Mon, 15 Jun 2026 17:12:47 GMT
• Title: Sobolev Approximation by Fixed-Size Neural Networks with Arbitrary Accuracy
• Abstract summary: We prove that any function in $W2,infty((a,b)d)$ can be approximated with arbitrary accuracy by a fixed-size $mathrmDUAF_infty$-activated network.<n>For every $1leq sleq n$, fixed-size $widetildemathrmDUAF_n$-activated networks still approximate any $fin Ws,infty((a,b)d)$ with arbitrary accuracy in the $Ws-1,
• Score: 11.112131888798174
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this work, we investigate new activation functions for achieving arbitrary-accuracy Sobolev approximation by fixed-size neural networks. We first show that any function in $W^{2,\infty}((a,b)^d)$ can be approximated with arbitrary accuracy, measured in the $W^{1,\infty}$-norm, by a fixed-size neural network using the Elementary Universal Activation Function ($\mathrm{EUAF}$). To extend this result to $W^{s,\infty}((a,b)^d)$ for $s\in\mathbb{N}$, we introduce a smooth activation $\mathrm{DUAF}_{\infty}$ from the family of Differentiable Universal Activation Functions ($\mathrm{DUAF}_n$). We prove that any function in $W^{s,\infty}((a,b)^d)$ can be approximated with arbitrary accuracy in the $W^{s-1,\infty}$-norm by a fixed-size $\mathrm{DUAF}_{\infty}$-activated network. We further construct sigmoidal variants $\widetilde{\mathrm{DUAF}}_n$ and show that, for every $1\leq s\leq n$, fixed-size $\widetilde{\mathrm{DUAF}}_n$-activated networks still approximate any $f\in W^{s,\infty}((a,b)^d)$ with arbitrary accuracy in the $W^{s-1,\infty}$-norm. In all these results, the width and depth bounds are computed explicitly, and the proposed activations are elementary.

Related papers

• On Explicit Super-Expressive Approximation for Neural Networks [14.904203072235434]
  We investigate the fixed-architecture neural network approximation with explicit parameter bounds and elementary activations.<n>While prior work demonstrated super-expressive approximation using fixed-size networks, they lack quantitative and non-asymptotic characterizations of parameter magnitude with respect to the approximation error.
  arXiv  Detail & Related papers  (2026-07-07T20:21:57Z)
• On the optimal approximation of Sobolev and Besov functions using deep ReLU neural networks [2.4112990554464235]
  We show that the rate $mathcalO((WL)-2s/d)$ indeed holds under the Sobolev embedding condition.<n>Key tool in our proof is a novel encoding of sparse vectors by using deep ReLU neural networks with varied width and depth.
  arXiv  Detail & Related papers  (2024-09-02T02:26:01Z)
• A Unified Framework for Uniform Signal Recovery in Nonlinear Generative Compressed Sensing [68.80803866919123]
  Under nonlinear measurements, most prior results are non-uniform, i.e., they hold with high probability for a fixed $mathbfx*$ rather than for all $mathbfx*$ simultaneously. Our framework accommodates GCS with 1-bit/uniformly quantized observations and single index models as canonical examples. We also develop a concentration inequality that produces tighter bounds for product processes whose index sets have low metric entropy.
  arXiv  Detail & Related papers  (2023-09-25T17:54:19Z)
• Distribution-Independent Regression for Generalized Linear Models with Oblivious Corruptions [49.69852011882769]
  We show the first algorithms for the problem of regression for generalized linear models (GLMs) in the presence of additive oblivious noise. We present an algorithm that tackles newthis problem in its most general distribution-independent setting. This is the first newalgorithmic result for GLM regression newwith oblivious noise which can handle more than half the samples being arbitrarily corrupted.
  arXiv  Detail & Related papers  (2023-09-20T21:41:59Z)
• Minimum width for universal approximation using ReLU networks on compact domain [8.839687029212673]
  We show that the minimum width for $Lp$ approximation of $Lp$ functions is exactly $maxd_x,d_y,2$ if an activation function is ReLU-Like (e.g., ReLU, GELU, Softplus) Compared to the known result for ReLU networks, $w_min=maxd_x+1,d_y$ when the domain is $smashmathbb Rd_x$, our result first shows that approximation on a compact domain requires smaller width than on
  arXiv  Detail & Related papers  (2023-09-19T08:04:48Z)
• Fast $(1+\varepsilon)$-Approximation Algorithms for Binary Matrix Factorization [54.29685789885059]
  We introduce efficient $(1+varepsilon)$-approximation algorithms for the binary matrix factorization (BMF) problem. The goal is to approximate $mathbfA$ as a product of low-rank factors. Our techniques generalize to other common variants of the BMF problem.
  arXiv  Detail & Related papers  (2023-06-02T18:55:27Z)
• Near-Linear Time and Fixed-Parameter Tractable Algorithms for Tensor Decompositions [51.19236668224547]
  We study low rank approximation of tensors, focusing on the tensor train and Tucker decompositions. For tensor train decomposition, we give a bicriteria $(1 + eps)$-approximation algorithm with a small bicriteria rank and $O(q cdot nnz(A))$ running time. In addition, we extend our algorithm to tensor networks with arbitrary graphs.
  arXiv  Detail & Related papers  (2022-07-15T11:55:09Z)
• Learning a Single Neuron with Adversarial Label Noise via Gradient Descent [50.659479930171585]
  We study a function of the form $mathbfxmapstosigma(mathbfwcdotmathbfx)$ for monotone activations. The goal of the learner is to output a hypothesis vector $mathbfw$ that $F(mathbbw)=C, epsilon$ with high probability.
  arXiv  Detail & Related papers  (2022-06-17T17:55:43Z)
• Active Sampling for Linear Regression Beyond the $\ell_2$ Norm [70.49273459706546]
  We study active sampling algorithms for linear regression, which aim to query only a small number of entries of a target vector. We show that this dependence on $d$ is optimal, up to logarithmic factors. We also provide the first total sensitivity upper bound $O(dmax1,p/2log2 n)$ for loss functions with at most degree $p$ growth.
  arXiv  Detail & Related papers  (2021-11-09T00:20:01Z)
• Neural networks with superexpressive activations and integer weights [91.3755431537592]
  An example of an activation function $sigma$ is given such that networks with activations $sigma, lfloorcdotrfloor$, integer weights and a fixed architecture is given. The range of integer weights required for $varepsilon$-approximation of H"older continuous functions is derived.
  arXiv  Detail & Related papers  (2021-05-20T17:29:08Z)
• Neural Network Approximation: Three Hidden Layers Are Enough [4.468952886990851]
  A three-hidden-layer neural network with super approximation power is introduced. Network is built with the floor function ($lfloor xrfloor$), the exponential function ($2x$), the step function ($1_xgeq 0$), or their compositions as the activation function in each neuron.
  arXiv  Detail & Related papers  (2020-10-25T18:30:57Z)
• Deep Network with Approximation Error Being Reciprocal of Width to Power of Square Root of Depth [4.468952886990851]
  A new network with super approximation power is introduced. This network is built with Floor ($lfloor xrfloor$) or ReLU ($max0,x$) activation function in each neuron.
  arXiv  Detail & Related papers  (2020-06-22T13:27:33Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.