# Representing Piecewise-Linear Functions by Functions with Minimal Arity

arXiv: https://arxiv.org/abs/2406.02421 · HTML: https://mlsanity.com/paper/2406.02421/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 23 references that are in the map.
- First posted on arXiv 4 Jun 2024.

## Builds on (most cited first, 8 of 8)

- [Understanding Deep Neural Networks with Rectified Linear Units](https://mlsanity.com/paper/1611.01491/) (2016)
- [Towards Lower Bounds on the Depth of ReLU Neural Networks](https://mlsanity.com/paper/2105.14835/) (2021)
- [Sharp bounds for the number of regions of maxout networks and vertices of Minkowski sums](https://mlsanity.com/paper/2104.08135/) (2021)
- [Lower Bounds on the Depth of Integral ReLU Neural Networks via Lattice Polytopes](https://mlsanity.com/paper/2302.12553/) (2023)
- [Improved Bounds on Neural Complexity for Representing Piecewise Linear Functions](https://mlsanity.com/paper/2210.07236/) (2022)
- [The Real Tropical Geometry of Neural Networks](https://mlsanity.com/paper/2403.11871/) (2024)
- [Lower bounds over Boolean inputs for deep neural networks with ReLU gates](https://mlsanity.com/paper/1711.03073/) (2017)
- [Representing Piecewise Linear Functions by Functions with Small Arity](https://mlsanity.com/paper/2305.16933/) (2023)

## Built on by (most cited first, 2 of 2)

- [Decomposition Polyhedra of Piecewise Linear Functions](https://mlsanity.com/paper/2410.04907/) (2024)
- [Positivity sets of hinge functions](https://mlsanity.com/paper/2503.13512/) (2025)

## Abstract (arXiv)

Any continuous piecewise-linear function Fcolon ℝn→ ℝ can be represented as a linear combination of max functions of at most n+1 affine-linear functions. In our previous paper [``Representing piecewise linear functions by functions with small arity'', AAECC, 2023], we showed that this upper bound of n+1 arguments is tight. In the present paper, we extend this result by establishing a correspondence between the function F and the minimal number of arguments that are needed in any such decomposition. We show that the tessellation of the input space ℝn induced by the function F has a direct connection to the number of arguments in the max functions.