# Delaunay-Triangulation-Based Learning with Hessian Total-Variation Regularization

arXiv: https://arxiv.org/abs/2208.07787 · HTML: https://mlsanity.com/paper/2208.07787/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 53 references that are in the map.
- First posted on arXiv 16 Aug 2022.

## Builds on (most cited first, 16 of 16)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [Understanding Deep Neural Networks with Rectified Linear Units](https://mlsanity.com/paper/1611.01491/) (2016)
- [The Power of Depth for Feedforward Neural Networks](https://mlsanity.com/paper/1512.03965/) (2015)
- [A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case](https://mlsanity.com/paper/1910.01635/) (2019)
- [Three Mechanisms of Weight Decay Regularization](https://mlsanity.com/paper/1810.12281/) (2018)
- [Minimum Width for Universal Approximation](https://mlsanity.com/paper/2006.08859/) (2020)
- [On Interpretability of Artificial Neural Networks: A Survey](https://mlsanity.com/paper/2001.02522/) (2020)
- [What Kinds of Functions do Deep Neural Networks Learn? Insights from Variational Spline Theory](https://mlsanity.com/paper/2105.03361/) (2021)
- [A unifying representer theorem for inverse problems and machine learning](https://mlsanity.com/paper/1903.00687/) (2019)
- [Ridges, Neural Networks, and the Radon Transform](https://mlsanity.com/paper/2203.02543/) (2022)
- [Sparsest Piecewise-Linear Regression of One-Dimensional Data](https://mlsanity.com/paper/2003.10112/) (2020)
- [Measuring Complexity of Learning Schemes Using Hessian-Schatten Total Variation](https://mlsanity.com/paper/2112.06209/) (2021)
- [The Problem of Projecting the Origin of Euclidean Space onto the Convex Polyhedron](https://mlsanity.com/paper/1605.05351/) (2016)

## Abstract (arXiv)

Regression is one of the core problems tackled in supervised learning. Rectified linear unit (ReLU) neural networks generate continuous and piecewise-linear (CPWL) mappings and are the state-of-the-art approach for solving regression problems. In this paper, we propose an alternative method that leverages the expressivity of CPWL functions. In contrast to deep neural networks, our CPWL parameterization guarantees stability and is interpretable. Our approach relies on the partitioning of the domain of the CPWL function by a Delaunay triangulation. The function values at the vertices of the triangulation are our learnable parameters and identify the CPWL function uniquely. Formulating the learning scheme as a variational problem, we use the Hessian total variation (HTV) as regularizer to favor CPWL functions with few affine pieces. In this way, we control the complexity of our model through a single hyperparameter. By developing a computational framework to compute the HTV of any CPWL function parameterized by a triangulation, we discretize the learning problem as the generalized least absolute shrinkage and selection operator (LASSO). Our experiments validate the usage of our method in low-dimensional scenarios.