Fused density estimation: theory and methods

We introduce a method for non‐parametric 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 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 various optimization techniques to solve the problem and use these results to inform recommendations for the computation of fused density estimators.