Fused Density Estimation: Theory and Methods

Robert Bassett, James Sharpnack

Abstract: In this paper we introduce a method for nonparametric density estimation on infras-tructure networks. We definefused density estimatorsas solutions to a total variationregularized maximum-likelihood density estimation problem. We provide theoreticalsupport for fused density estimation by proving that the squared Hellinger rate ofconvergence for the estimator achieves the minimax bound over univariate densitiesof log-bounded variation. We reduce the original variational formulation in order totransform it into a tractable, finite-dimensional quadratic program. Because randomvariables the networks we consider generalizations of the univariate case, this methodalso provides a useful tool for univariate density estimation. Lastly, we apply thismethod and assess its performance on examples in the univariate and infrastructurenetwork settings. We compare the performance of different optimization techniques tosolve the problem, and use these results to inform recommendations for the computa-tion of fused density estimators.