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
