Abstract

Numerous applications in statistics, signal pro-cessing, and machine learning regularize us-ing Total Variation (TV) penalties. We study anisotropic (`1-based) TV and also a related `2-norm variant. We consider for both vari-ants associated (1D) proximity operators, which lead to challenging optimization problems. We solve these problems by developing Newton-type methods that outperform the state-of-the-art al-gorithms. More importantly, our 1D-TV al-gorithms serve as building blocks for solving the harder task of computing 2- (and higher)-dimensional TV proximity. We illustrate the computational benefits of our methods by apply-ing them to several applications: (i) image de-noising; (ii) image deconvolution (by plugging in our TV solvers into publicly available software); and (iii) four variants of fused-lasso. The results show large speedups—and to support our claims, we provide software accompanying this paper. 1.