Fused Density Estimation: Theory and Methods

TL;DR

This paper introduces a nonparametric density estimation method for geometric networks using fused density estimators with total variation regularization, supported by theoretical convergence guarantees and practical optimization strategies.

Contribution

It develops a tractable quadratic programming approach for fused density estimation on networks, with proven minimax convergence rates and applicability to univariate cases.

Findings

Achieves minimax convergence rate over univariate densities.

Transforms the variational problem into a finite-dimensional quadratic program.

Demonstrates the method's effectiveness on univariate and network data.

Abstract

In this paper we introduce a method for nonparametric density estimation on geometric networks. We define fused density estimators as solutions to a total variation regularized maximum-likelihood density estimation problem. We provide theoretical support for fused density estimation by proving that the squared Hellinger rate of convergence for the estimator achieves the minimax bound over univariate densities of log-bounded variation. We reduce the original variational formulation in order to transform it into a tractable, finite-dimensional quadratic program. Because random variables on geometric networks are simple generalizations of the univariate case, this method also provides a useful tool for univariate density estimation. Lastly, we apply this method and assess its performance on examples in the univariate and geometric network setting. We compare the performance of different optimization techniques to solve the problem, and use these results to inform recommendations for the computation of fused density estimators.