# Piecewise linear regression and classification

arXiv: https://arxiv.org/abs/2103.06189 · HTML: https://mlsanity.com/paper/2103.06189/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 53 references that are in the map.
- First posted on arXiv 10 Mar 2021.

## Builds on (most cited first, 9 of 9)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Minimizing Finite Sums with the Stochastic Average Gradient](https://mlsanity.com/paper/1309.2388/) (2013)
- [PMLB: A Large Benchmark Suite for Machine Learning Evaluation and Comparison](https://mlsanity.com/paper/1703.00512/) (2017)
- [Nonlinear System Identification: A User-Oriented Roadmap](https://mlsanity.com/paper/1902.00683/) (2019)
- [Global optimization via inverse distance weighting and radial basis functions](https://mlsanity.com/paper/1906.06498/) (2019)
- [Parallel Selective Algorithms for Big Data Optimization](https://mlsanity.com/paper/1402.5521/) (2014)
- [Active preference learning based on radial basis functions](https://mlsanity.com/paper/1909.13049/) (2019)
- [A Convex Feasibility Approach to Anytime Model Predictive Control](https://mlsanity.com/paper/1502.07974/) (2015)
- [PIANO: A Fast Parallel Iterative Algorithm for Multinomial and Sparse Multinomial Logistic Regression](https://mlsanity.com/paper/2002.09133/) (2020)

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

- [Piecewise Linear Neural Networks and Deep Learning](https://mlsanity.com/paper/2206.09149/) (2022)
- [Learning Linear Complementarity Systems](https://mlsanity.com/paper/2112.13284/) (2021)

## Abstract (arXiv)

This paper proposes a method for solving multivariate regression and classification problems using piecewise linear predictors over a polyhedral partition of the feature space. The resulting algorithm that we call PARC (Piecewise Affine Regression and Classification) alternates between (i) solving ridge regression problems for numeric targets, softmax regression problems for categorical targets, and either softmax regression or cluster centroid computation for piecewise linear separation, and (ii) assigning the training points to different clusters on the basis of a criterion that balances prediction accuracy and piecewise-linear separability. We prove that PARC is a block-coordinate descent algorithm that optimizes a suitably constructed objective function, and that it converges in a finite number of steps to a local minimum of that function. The accuracy of the algorithm is extensively tested numerically on synthetic and real-world datasets, showing that the approach provides an extension of linear regression/classification that is particularly useful when the obtained predictor is used as part of an optimization model. A Python implementation of the algorithm described in this paper is available at http://cse.lab.imtlucca.it/~bemporad/parc .