mlsanityData to 14 Oct 2026

Representing Piecewise-Linear Functions by Functions with Minimal Arity

arXiv 2406.02421 ↗First posted 4 Jun 2024cs.DM cs.LG cs.SCcited by 2

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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.

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

8 of 23 references.

References · 8

  1. 2016Understanding Deep Neural Networks with Rectified Linear Unitscited by 257
  2. 2021Towards Lower Bounds on the Depth of ReLU Neural Networkscited by 56
  3. 2021Sharp bounds for the number of regions of maxout networks and vertices of Minkowski sumscited by 37
  4. 2023Lower Bounds on the Depth of Integral ReLU Neural Networks via Lattice Polytopescited by 36
  5. 2022Improved Bounds on Neural Complexity for Representing Piecewise Linear Functionscited by 26
  6. 2024The Real Tropical Geometry of Neural Networkscited by 24
  7. 2017Lower bounds over Boolean inputs for deep neural networks with ReLU gatescited by 21
  8. 2023Representing Piecewise Linear Functions by Functions with Small Aritycited by 7

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Cited by year

Citing papers per year: 2024: 1; 2025: 1.

2 papers.

  1. 2024Decomposition Polyhedra of Piecewise Linear Functionscited by 9
  2. 2025Positivity sets of hinge functionscited by 0