# Neural Network Approximation: A View from Polytope Decomposition

arXiv: https://arxiv.org/abs/2601.18264 · HTML: https://mlsanity.com/paper/2601.18264/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 42 references that are in the map.
- First posted on arXiv 26 Jan 2026.

## Builds on (most cited first, 18 of 18)

- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [The Expressive Power of Neural Networks: A View from the Width](https://mlsanity.com/paper/1709.02540/) (2017)
- [Universal Approximation with Deep Narrow Networks](https://mlsanity.com/paper/1905.08539/) (2019)
- [Deep Network Approximation for Smooth Functions](https://mlsanity.com/paper/2001.03040/) (2020)
- [Approximating Continuous Functions by ReLU Nets of Minimal Width](https://mlsanity.com/paper/1710.11278/) (2017)
- [Deep Network Approximation Characterized by Number of Neurons](https://mlsanity.com/paper/1906.05497/) (2019)
- [Universal Function Approximation by Deep Neural Nets with Bounded Width and ReLU Activations](https://mlsanity.com/paper/1708.02691/) (2017)
- [Optimal Approximation Rate of ReLU Networks in terms of Width and Depth](https://mlsanity.com/paper/2103.00502/) (2021)
- [Error bounds for approximations with deep ReLU neural networks in Ws,p norms](https://mlsanity.com/paper/1902.07896/) (2019)
- [Minimum Width for Universal Approximation](https://mlsanity.com/paper/2006.08859/) (2020)
- [Representation Benefits of Deep Feedforward Networks](https://mlsanity.com/paper/1509.08101/) (2015)
- [Why Deep Neural Networks for Function Approximation?](https://mlsanity.com/paper/1610.04161/) (2016)
- [ResNet with one-neuron hidden layers is a Universal Approximator](https://mlsanity.com/paper/1806.10909/) (2018)
- [Nonparametric Regression on Low-Dimensional Manifolds using Deep ReLU Networks : Function Approximation and Statistical Recovery](https://mlsanity.com/paper/1908.01842/) (2019)
- [Neural Network Approximation: Three Hidden Layers Are Enough](https://mlsanity.com/paper/2010.14075/) (2020)
- [On a Sparse Shortcut Topology of Artificial Neural Networks](https://mlsanity.com/paper/1811.09003/) (2018)
- [Neural Network Approximation of Refinable Functions](https://mlsanity.com/paper/2107.13191/) (2021)
- [Deep Network Approximation in Terms of Intrinsic Parameters](https://mlsanity.com/paper/2111.07964/) (2021)

## Abstract (arXiv)

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.