# Piecewise Linear Regression via a Difference of Convex Functions

arXiv: https://arxiv.org/abs/2007.02422 · HTML: https://mlsanity.com/paper/2007.02422/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Jul 2020.

## Built on by (most cited first, 14 of 14)

- [The Real Tropical Geometry of Neural Networks](https://mlsanity.com/paper/2403.11871/) (2024)
- [Reliably-stabilizing piecewise-affine neural network controllers](https://mlsanity.com/paper/2111.07183/) (2021)
- [On Solution Functions of Optimization: Universal Approximation and Covering Number Bounds](https://mlsanity.com/paper/2212.01314/) (2022)
- [Data-Driven Subgroup Identification for Linear Regression](https://mlsanity.com/paper/2305.00195/) (2023)
- [Faster Algorithms for Learning Convex Functions](https://mlsanity.com/paper/2111.01348/) (2021)
- [Data-driven Piecewise Affine Decision Rules for Stochastic Programming with Covariate Information](https://mlsanity.com/paper/2304.13646/) (2023)
- [Interactive Generalized Additive Model and Its Applications in Electric Load Forecasting](https://mlsanity.com/paper/2310.15662/) (2023)
- [An Efficient Data Analysis Method for Big Data using Multiple-Model Linear Regression](https://mlsanity.com/paper/2308.12691/) (2023)
- [Identifiable Convex-Concave Regression via Sub-gradient Regularised Least Squares](https://mlsanity.com/paper/2506.18078/) (2025)
- [Cellular Learning: Scattered Data Regression in High Dimensions via Voronoi Cells](https://mlsanity.com/paper/2510.03810/) (2025)
- [Near-optimal Delta-convex Estimation of Lipschitz Functions](https://mlsanity.com/paper/2511.15615/) (2025)
- [Hidden Monotonicity: Explaining Deep Neural Networks via their DC Decomposition](https://mlsanity.com/paper/2601.07700/) (2026)
- [Locally Near Optimal Piecewise Linear Regression in High Dimensions via Difference of Max-Affine Functions](https://mlsanity.com/paper/2605.06959/) (2026)
- [Patchwork: A compact representation for 3D polygonal shapes](https://mlsanity.com/paper/2605.16266/) (2026)

## Abstract (arXiv)

We present a new piecewise linear regression methodology that utilizes fitting a difference of convex functions (DC functions) to the data. These are functions f that may be represented as the difference φ1 - φ2 for a choice of convex functions φ1, φ2. The method proceeds by estimating piecewise-liner convex functions, in a manner similar to max-affine regression, whose difference approximates the data. The choice of the function is regularised by a new seminorm over the class of DC functions that controls the ℓ_∞ Lipschitz constant of the estimate. The resulting methodology can be efficiently implemented via Quadratic programming even in high dimensions, and is shown to have close to minimax statistical risk. We empirically validate the method, showing it to be practically implementable, and to have comparable performance to existing regression/classification methods on real-world datasets.