We consider the problem of minimizing the continuous valued total variation subject to different unary terms on trees and propose fast direct algorithms based on dynamic programming to solve these problems. We treat both the convex and the non-convex case and derive worst case complexities that are equal or better than existing methods. We show applications to total variation based 2D image processing and computer vision problems based on a Lagrangian decomposition approach. The resulting algorithms are very efficient, offer a high degree of parallelism and come along with memory requirements which are only in the order of the number of image pixels.
The paper considers the problem of nonparametric regression with emphasis on controlling the number of local extremes. Two methods, the run method and the taut-string multiresolution method, are introduced and analyzed on standard test beds. It is shown that the number and locations of local extreme values are consistently estimated. Rates of convergence are proved for both methods. The run method converges slowly but can withstand blocks as well as a high proportion of isolated outliers. The rate of convergence of the taut-string multiresolution method is almost optimal. The method is extremely sensitive and can detect very low power peaks. Section 1 contains an introduction with special reference to the number of local extreme values. The run method is described in Section 2 and the taut-string-multiresolution method in Section 3. Low power peaks are considered in Section 4. Section 5 contains a comparison with other methods and Section 6 a short conclusion. The proofs are given in Section 7 and the taut-string algorithm is described in the Appendix. A very fast noniterative algorithm is proposed for denoising or smoothing one-dimensional discrete signals, by solving the total variation regularized least-squares problem or the related fused lasso problem. A C code implementation is available on the web page of the author. Suppose that we observe independent, identically distributed random pairs (X1; Y1), (X2; Y2), . . . , (Xn; Yn). Our goal is to estimate regression functions such as the conditional mean or nquantile of Y given X, where 0 0 is some tuning parameter. This framework is extended further in order to include binary or Poisson regression, and to include local variation penalties. The latter are needed in order to construct estimators adapting to inhomogenous smoothness of f . For the general framework we develop noniterative algorithms for the solution of the minimization problems which are closely related to the taut string algorithm (cf. Davies and Kovac 2001). We propose a dynamic programming algorithm for the one-dimensional Fused Lasso Signal Approximator (FLSA). The proposed algorithm has a linear running time in the worst case. A similar approach is developed for the task of least squares segmentation, and simulations indicate substantial performance improvement over existing algorithms. Examples of R and C implementations are provided in the online Supplementary materials, posted on the journal web site.