Abstract

Abstract. L1 penalties have proven to be an attractive regularization device for nonparametric regression, image reconstruction, and model selection. For function estimation, L1 penalties, interpreted as roughness of the candidate function measured by their total variation, are known to be capable of capturing sharp changes in the target function while still maintaining a general smoothing objective. We explore the use of penalties based on total variation of the estimated density, its square root, and its logarithm – and their derivatives – in the context of univariate and bivariate density estimation, and compare the results to some other density estimation methods including L2 penalized likelihood methods. Our objective is to develop a unified approach to total variation penalized density estimation offering methods that are: capable of identifying qualitative features like sharp peaks, extendible to higher dimensions, and computationally tractable. Modern interior point methods for solving convex optimization problems play a critical role in achieving the final objective, as do piecewise linear finite element methods that facilitate the use of sparse linear algebra. 1.

Citations