# Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ2

arXiv: https://arxiv.org/abs/2503.13001 · HTML: https://mlsanity.com/paper/2503.13001/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 24 references that are in the map.
- First posted on arXiv 17 Mar 2025.

## Builds on (most cited first, 12 of 12)

- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [Understanding Deep Neural Networks with Rectified Linear Units](https://mlsanity.com/paper/1611.01491/) (2016)
- [Bounding and Counting Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1711.02114/) (2017)
- [Universal Function Approximation by Deep Neural Nets with Bounded Width and ReLU Activations](https://mlsanity.com/paper/1708.02691/) (2017)
- [Towards Lower Bounds on the Depth of ReLU Neural Networks](https://mlsanity.com/paper/2105.14835/) (2021)
- [Lower Bounds on the Depth of Integral ReLU Neural Networks via Lattice Polytopes](https://mlsanity.com/paper/2302.12553/) (2023)
- [Lower bounds over Boolean inputs for deep neural networks with ReLU gates](https://mlsanity.com/paper/1711.03073/) (2017)
- [A Framework for the construction of upper bounds on the number of affine linear regions of ReLU feed-forward neural networks](https://mlsanity.com/paper/1806.01918/) (2018)
- [ReLU Deep Neural Networks from the Hierarchical Basis Perspective](https://mlsanity.com/paper/2105.04156/) (2021)
- [Decomposition Polyhedra of Piecewise Linear Functions](https://mlsanity.com/paper/2410.04907/) (2024)
- [Representing Piecewise Linear Functions by Functions with Small Arity](https://mlsanity.com/paper/2305.16933/) (2023)
- [Bounds on the Number of Pieces in Continuous Piecewise Affine Functions](https://mlsanity.com/paper/2503.09525/) (2025)

## Built on by (most cited first, 1 of 1)

- [The Computational Complexity of Counting Linear Regions in ReLU Neural Networks](https://mlsanity.com/paper/2505.16716/) (2025)

## Abstract (arXiv)

It is shown that any continuous piecewise affine (CPA) function ℝ2→ℝ with p pieces can be represented by a ReLU neural network with two hidden layers and O(p) neurons. Unlike prior work, which focused on convex pieces, this analysis considers CPA functions with connected but potentially non-convex pieces.