# Piecewise Polynomial Regression of Tame Functions via Integer Programming

arXiv: https://arxiv.org/abs/2311.13544 · HTML: https://mlsanity.com/paper/2311.13544/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 57 references that are in the map.
- First posted on arXiv 22 Nov 2023.

## Builds on (most cited first, 17 of 17)

- [Complexity of Linear Regions in Deep Networks](https://mlsanity.com/paper/1901.09021/) (2019)
- [Bounding and Counting Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1711.02114/) (2017)
- [Stochastic subgradient method converges on tame functions](https://mlsanity.com/paper/1804.07795/) (2018)
- [Conservative set valued fields, automatic differentiation, stochastic gradient method and deep learning](https://mlsanity.com/paper/1909.10300/) (2019)
- [A mathematical model for automatic differentiation in machine learning](https://mlsanity.com/paper/2006.02080/) (2020)
- [Optimal Sparse Regression Trees](https://mlsanity.com/paper/2211.14980/) (2022)
- [Active manifolds, stratifications, and convergence to local minima in nonsmooth optimization](https://mlsanity.com/paper/2108.11832/) (2021)
- [Adaptive Estimation of Multivariate Piecewise Polynomials and Bounded Variation Functions by Optimal Decision Trees](https://mlsanity.com/paper/1911.11562/) (2019)
- [Progress in Mathematical Programming Solvers from 2001 to 2020](https://mlsanity.com/paper/2206.09787/) (2022)
- [ReLU Neural Networks, Polyhedral Decompositions, and Persistent Homolog](https://mlsanity.com/paper/2306.17418/) (2023)
- [Nonsmoothness in Machine Learning: specific structure, proximal identification, and applications](https://mlsanity.com/paper/2010.00848/) (2020)
- [Stochastic Subgradient Descent Escapes Active Strict Saddles on Weakly Convex Functions](https://mlsanity.com/paper/2108.02072/) (2021)
- [Nonconvex piecewise linear functions: Advanced formulations and simple modeling tools](https://mlsanity.com/paper/1708.00050/) (2017)
- [Global convergence of the gradient method for functions definable in o-minimal structures](https://mlsanity.com/paper/2303.03534/) (2023)
- [Taming Binarized Neural Networks and Mixed-Integer Programs](https://mlsanity.com/paper/2310.04469/) (2023)
- [Lattice partition recovery with dyadic CART](https://mlsanity.com/paper/2105.13504/) (2021)
- [Polynomial Matrix Inequalities within Tame Geometry](https://mlsanity.com/paper/2206.03941/) (2022)

## Built on by (most cited first, 1 of 1)

- [Deep Learning as the Disciplined Construction of Tame Objects](https://mlsanity.com/paper/2509.18025/) (2025)

## Abstract (arXiv)

Tame functions are a class of nonsmooth, nonconvex functions, which feature in a wide range of applications: functions encountered in the training of deep neural networks with all common activations, value functions of mixed-integer programs, or wave functions of small molecules. We consider approximating tame functions with piecewise polynomial functions. We bound the quality of approximation of a tame function by a piecewise polynomial function with a given number of segments on any full-dimensional cube. We also present the first mixed-integer programming formulation of piecewise polynomial regression. Together, these can be used to estimate tame functions. We demonstrate promising computational results.