mlsanityData to 7 Oct 2026

Spherical Analysis of Learning Nonlinear Functionals

arXiv 2410.01047 ↗First posted 1 Oct 2024cs.LG math.FA stat.MLcited by 3

Not on the map. Why?

Abstract (arXiv)

In recent years, there has been growing interest in the field of functional neural networks. They have been proposed and studied with the aim of approximating continuous functionals defined on sets of functions on Euclidean domains. In this paper, we consider functionals defined on sets of functions on spheres. The approximation ability of deep ReLU neural networks is investigated by novel spherical analysis using an encoder-decoder framework. An encoder comes up first to accommodate the infinite-dimensional nature of the domain of functionals. It utilizes spherical harmonics to help us extract the latent finite-dimensional information of functions, which in turn facilitates in the next step of approximation analysis using fully connected neural networks. Moreover, real-world objects are frequently sampled discretely and are often corrupted by noise. Therefore, encoders with discrete input and those with discrete and random noise input are constructed, respectively. The approximation rates with different encoder structures are provided therein.

Key facts. Cited by 3 papers in the map (data through 7 Oct 2026). Builds on 19 of its 36 references that are in the map. First posted on arXiv 1 Oct 2024.

Builds on

19 of 36 references.

References · 19

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7 more
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Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 1; 2025: 0; 2026: 2.

3 papers.

  1. 2024DeepONet for Solving Nonlinear Partial Differential Equations with Physics-Informed Trainingcited by 1
  2. 2026Sparse-Aware Neural Networks for Nonlinear Functionals: Mitigating the Exponential Dependence on Dimensioncited by 0
  3. 2026Optimal Neural Network Approximation via Empirical Least Squares with Deterministic Samplescited by 0