Neural Network Approximation: A View from Polytope Decomposition
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910000922755072 |
|---|---|
| author | Li, ZeYu Zhang, ShiJun Zeng, TieYong Fan, FengLei |
| author_facet | Li, ZeYu Zhang, ShiJun Zeng, TieYong Fan, FengLei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_18264 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Neural Network Approximation: A View from Polytope Decomposition Li, ZeYu Zhang, ShiJun Zeng, TieYong Fan, FengLei Machine Learning Artificial Intelligence 68Q25, 68Q32, 68T99 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. |
| title | Neural Network Approximation: A View from Polytope Decomposition |
| topic | Machine Learning Artificial Intelligence 68Q25, 68Q32, 68T99 |
| url | https://arxiv.org/abs/2601.18264 |