Sharp Lower Bounds for Linearized ReLU^k Approximation on the Sphere

• URL: http://arxiv.org/abs/2510.04060v2
• Date: Mon, 03 Nov 2025 12:49:13 GMT
• Title: Sharp Lower Bounds for Linearized ReLU^k Approximation on the Sphere
• Authors: Tong Mao, Jinchao Xu,
• Abstract summary: We prove a saturation theorem for linearized shallow ReLU$k$ neural networks on the unit sphere $mathbb Sd$.<n>Our results place linearized neural-network approximation firmly within the classical saturation framework.
• Score: 3.8969433233965205
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We prove a saturation theorem for linearized shallow ReLU$^k$ neural networks on the unit sphere $\mathbb S^d$. For any antipodally quasi-uniform set of centers, if the target function has smoothness $r>\tfrac{d+2k+1}{2}$, then the best $\mathcal{L}^2(\mathbb S^d)$ approximation cannot converge faster than order $n^{-\frac{d+2k+1}{2d}}$. This lower bound matches existing upper bounds, thereby establishing the exact saturation order $\tfrac{d+2k+1}{2d}$ for such networks. Our results place linearized neural-network approximation firmly within the classical saturation framework and show that, although ReLU$^k$ networks outperform finite elements under equal degrees $k$, this advantage is intrinsically limited.

Related papers

• Sublinear Time Quantum Sensitivity Sampling [57.356528942341534]
  We present a unified framework for quantum sensitivity sampling, extending the advantages of quantum computing to a broad class of classical approximation problems.<n>Our framework provides a streamlined approach for constructing coresets and offers significant runtime improvements in applications such as clustering, regression, and low-rank approximation.
  arXiv  Detail & Related papers  (2025-09-20T20:18:49Z)
• Enhancing Convergence of Decentralized Gradient Tracking under the KL Property [10.925931212031692]
  We establish convergence of the same type for the notorious decentralized gradient-tracking-based algorithm SONATA.<n>This matches the convergence behavior of centralized-gradient algorithms except when.<n>thetain (1/2,1)$, sub rate is certified; and.<n>textbf(iii)$ when $thetain (1/2,1)$, sub rate is certified; and.<n>textbf(iii)$ when $thetain (1/2,1)$, sub rate is certified.
  arXiv  Detail & Related papers  (2024-12-12T18:44:36Z)
• Approximation Rates for Shallow ReLU$^k$ Neural Networks on Sobolev Spaces via the Radon Transform [4.096453902709292]
  We consider the problem of how efficiently shallow neural networks with the ReLU$k$ activation function can approximate functions from Sobolev spaces.<n>We provide a simple proof of nearly optimal approximation rates in a variety of cases, including when $qleq p$, $pgeq 2$, and $s leq k + (d+1)/2$.
  arXiv  Detail & Related papers  (2024-08-20T16:43:45Z)
• Neural network learns low-dimensional polynomials with SGD near the information-theoretic limit [75.4661041626338]
  We study the problem of gradient descent learning of a single-index target function $f_*(boldsymbolx) = textstylesigma_*left(langleboldsymbolx,boldsymbolthetarangleright)$<n>We prove that a two-layer neural network optimized by an SGD-based algorithm learns $f_*$ with a complexity that is not governed by information exponents.
  arXiv  Detail & Related papers  (2024-06-03T17:56:58Z)
• Learning with Norm Constrained, Over-parameterized, Two-layer Neural Networks [54.177130905659155]
  Recent studies show that a reproducing kernel Hilbert space (RKHS) is not a suitable space to model functions by neural networks. In this paper, we study a suitable function space for over- parameterized two-layer neural networks with bounded norms.
  arXiv  Detail & Related papers  (2024-04-29T15:04:07Z)
• 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)
• Optimal Approximation of Zonoids and Uniform Approximation by Shallow Neural Networks [2.7195102129095003]
  We study the following two related problems.<n>The first is to determine what error an arbitrary zonoid in $mathbbRd+1$ can be approximated in the Hausdorff distance by a sum of $n$ line segments.<n>The second is to determine optimal approximation rates in the uniform norm for shallow ReLU$k$ neural networks on their variation spaces.
  arXiv  Detail & Related papers  (2023-07-28T03:43:17Z)
• Shallow neural network representation of polynomials [91.3755431537592]
  We show that $d$-variables of degreeR$ can be represented on $[0,1]d$ as shallow neural networks of width $d+1+sum_r=2Rbinomr+d-1d-1d-1[binomr+d-1d-1d-1[binomr+d-1d-1d-1[binomr+d-1d-1d-1d-1[binomr+d-1d-1d-1d-1
  arXiv  Detail & Related papers  (2022-08-17T08:14:52Z)
• Approximation of Smoothness Classes by Deep Rectifier Networks [0.0]
  We show that alertdeep networks with a fixed activation function attain optimal or near to optimal approximation rates for functions in the Besov space. Using theory, this implies that the entire range of smoothness classes at or above the critical line is (near to) optimally approximated by deep ReLU/RePU networks.
  arXiv  Detail & Related papers  (2020-07-30T17:58:05Z)
• Learning Over-Parametrized Two-Layer ReLU Neural Networks beyond NTK [58.5766737343951]
  We consider the dynamic of descent for learning a two-layer neural network. We show that an over-parametrized two-layer neural network can provably learn with gradient loss at most ground with Tangent samples.
  arXiv  Detail & Related papers  (2020-07-09T07:09:28Z)
• Complexity of Finding Stationary Points of Nonsmooth Nonconvex Functions [84.49087114959872]
  We provide the first non-asymptotic analysis for finding stationary points of nonsmooth, nonsmooth functions. In particular, we study Hadamard semi-differentiable functions, perhaps the largest class of nonsmooth functions.
  arXiv  Detail & Related papers  (2020-02-10T23:23:04Z)
• Curse of Dimensionality on Randomized Smoothing for Certifiable Robustness [151.67113334248464]
  We show that extending the smoothing technique to defend against other attack models can be challenging. We present experimental results on CIFAR to validate our theory.
  arXiv  Detail & Related papers  (2020-02-08T22:02:14Z)
• A Corrective View of Neural Networks: Representation, Memorization and Learning [26.87238691716307]
  We develop a corrective mechanism for neural network approximation. We show that two-layer neural networks in the random features regime (RF) can memorize arbitrary labels. We also consider three-layer neural networks and show that the corrective mechanism yields faster representation rates for smooth radial functions.
  arXiv  Detail & Related papers  (2020-02-01T20:51:09Z)

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.