利用稀疏性提升函数学习的效率与稳定性，缓解维度灾难。
Sparse-Aware Neural Networks for Nonlinear Functionals: Mitigating the Exponential Dependence on Dimension
- 用卷积网络提取样本中的稀疏特征，结合全连接网络逼近非线性泛函。
- 在快速衰减频率和混合光滑性空间中，逼近精度提升且所需样本更少。
- 理论揭示稀疏性可缓解函数学习中的维度诅咒，适合高维数据建模研究者。
深度神经网络已成为学习定义在无限维函数空间上的算子的强大工具。然而，现有理论常面临维度困境与可解释性不足的问题。本文研究稀疏性在函数学习中的作用，提出一种新框架：利用卷积架构从有限样本中提取稀疏特征，并结合深层全连接网络有效逼近非线性泛函。通过通用离散化方法，证明稀疏近似器可从离散采样中实现稳定恢复。无论确定性还是随机采样方案均适用于该分析。结果表明，在具有快速频率衰减和混合光滑性的函数空间中，逼近率得到改进，所需样本量显著减少。这些发现为稀疏性如何缓解函数学习中的维度诅咒提供了新的理论见解。
原文摘要 · Abstract (English)
Deep neural networks have emerged as powerful tools for learning operators defined over infinite-dimensional function spaces. However, existing theories frequently encounter difficulties related to dimensionality and limited interpretability. This work investigates how sparsity can help address these challenges in functional learning, a central ingredient in operator learning. We propose a framework that employs convolutional architectures to extract sparse features from a finite number of samples, together with deep fully connected networks to effectively approximate nonlinear functionals. Using universal discretization methods, we show that sparse approximators enable stable recovery from discrete samples. In addition, both the deterministic and the random sampling schemes are sufficient for our analysis. These findings lead to improved approximation rates and reduced sample sizes in various function spaces, including those with fast frequency decay and mixed smoothness. They also provide new theoretical insights into how sparsity can alleviate the curse of dimensionality in functional learning.
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