用多面体分解提升神经网络逼近效率，尤其在函数奇异点附近表现更优。
Neural Network Approximation: A View from Polytope Decomposition
- 基于多面体划分输入空间，结合核多项式方法构建逼近模型。
- 在连续函数逼近中达到更优精度，尤其在函数奇异点处性能显著提升。
- 适用于需要高精度逼近的场景，如科学计算与复杂函数建模。
通用逼近理论为验证神经网络表达能力提供了基础框架，使其可在实际应用中被合理使用。然而，现有理论大多通过均匀划分输入空间为微小超立方体来构建，未考虑目标函数的局部规律性。本文从多面体分解视角研究ReLU网络的通用逼近能力，提出一种显式的核多项式方法，其特征不仅包括改进的Totik-Ditzian型连续性模，还包含多面体子域分解。随后，在每个子域内分别构造ReLU网络以逼近核多项式。实验表明，多面体分解使逼近在多数情况下更高效灵活，尤其在目标函数奇异点附近优势明显。最后，该方法扩展至解析函数，实现了更高的逼近速率。
原文摘要 · Abstract (English)
Universal approximation theory offers a foundational framework to verify neural network expressiveness, enabling principled utilization in real-world applications. However, most existing theoretical constructions are established by uniformly dividing the input space into tiny hypercubes without considering the local regularity of the target function. In this work, we investigate the universal approximation capabilities of ReLU networks from a view of polytope decomposition, which offers a more realistic and task-oriented approach compared to current methods. To achieve this, we develop an explicit kernel polynomial method to derive an universal approximation of continuous functions, which is characterized not only by the refined Totik-Ditzian-type modulus of continuity, but also by polytopical domain decomposition. Then, a ReLU network is constructed to approximate the kernel polynomial in each subdomain separately. Furthermore, we find that polytope decomposition makes our approximation more efficient and flexible than existing methods in many cases, especially near singular points of the objective function. Lastly, we extend our approach to analytic functions to reach a higher approximation rate.
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。