  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.
