# Bounds on the Number of Pieces in Continuous Piecewise Affine Functions

arXiv: https://arxiv.org/abs/2503.09525 · HTML: https://mlsanity.com/paper/2503.09525/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Mar 2025.

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

- [The Computational Complexity of Counting Linear Regions in ReLU Neural Networks](https://mlsanity.com/paper/2505.16716/) (2025)
- [Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ2](https://mlsanity.com/paper/2503.13001/) (2025)

## Abstract (arXiv)

The complexity of continuous piecewise affine (CPA) functions can be measured by the number of pieces p or the number of distinct affine functions n. For CPA functions on ℝd, this paper shows an upper bound of p=O(nd+1) and constructs a family of functions achieving a lower bound of p=Omega(n^d+1-fracc√log2(n)).