用球面分析方法研究神经网络对球面上函数泛函的逼近能力。
Spherical Analysis of Learning Nonlinear Functionals
- 基于编码器-解码器框架，利用球谐函数提取函数低维特征
- 在离散采样与含噪输入下均给出逼近速率，证明网络有效
- 适用于需要处理球面数据的科学计算与几何学习场景
近年来，功能神经网络受到越来越多关注，旨在逼近定义在欧氏域函数集合上的连续泛函。本文研究定义在球面函数集合上的泛函，通过新颖的球面分析方法，结合编码器-解码器框架，考察深度ReLU神经网络的逼近能力。编码器首先应对泛函定义域的无穷维特性，利用球谐函数提取函数的潜在有限维信息，从而为后续全连接网络的逼近分析提供基础。此外，现实世界对象常以离散方式采样且受噪声污染，因此分别构建了处理离散输入和含随机噪声离散输入的编码器。文中给出了不同编码器结构下的逼近速率，验证了方法的有效性。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。