# Nonparametric Functional Approximation with Delaunay Triangulation

## TL;DR

The paper introduces a differentiable nonparametric method called Delaunay triangulation learner (DTL) that partitions feature space into simplices for functional approximation, combining geometric optimality with linear modeling.

## Contribution

It presents a novel DTL algorithm that leverages Delaunay triangulation for nonparametric function approximation, with theoretical analysis and empirical comparison.

## Key findings

- DTL effectively partitions feature space into simplices.
- Theoretical properties of DTL are rigorously analyzed.
- DTL outperforms some existing learners in numerical studies.

## Abstract

We propose a differentiable nonparametric algorithm, the Delaunay triangulation learner (DTL), to solve the functional approximation problem on the basis of a $p$-dimensional feature space. By conducting the Delaunay triangulation algorithm on the data points, the DTL partitions the feature space into a series of $p$-dimensional simplices in a geometrically optimal way, and fits a linear model within each simplex. We study its theoretical properties by exploring the geometric properties of the Delaunay triangulation, and compare its performance with other statistical learners in numerical studies.