We study recovery of piecewise-constant signals on graphs by the estimator minimizing an @math -edge-penalized objective. Although exact minimization of this objective may be computationally intractable, we show that the same statistical risk guarantees are achieved by the @math -expansion algorithm which computes an approximate minimizer in polynomial time. We establish that for graphs with small average vertex degree, these guarantees are minimax rate-optimal over classes of edge-sparse signals. For spatially inhomogeneous graphs, we propose minimization of an edge-weighted objective where each edge is weighted by its effective resistance or another measure of its contribution to the graph's connectivity. We establish minimax optimality of the resulting estimators over corresponding edge-weighted sparsity classes. We show theoretically that these risk guarantees are not always achieved by the estimator minimizing the @math total-variation relaxation, and empirically that the @math -based estimates are more accurate in high signal-to-noise settings.
For changepoint problems where @math is the linear chain, ) may be exactly minimized by dynamic programming in quadratic time @cite_26 @cite_50 @cite_55 . Pruning ideas may reduce runtime to be near-linear in practice @cite_56 . Correct changepoint recovery and distributional properties of @math minimizing ) were studied asymptotically in @cite_36 @cite_9 when the number of true changepoints is fixed. Non-asymptotic risk bounds similar to ) and ) were established for estimators minimizing similar objectives in @cite_51 @cite_21 ; we discuss this further below. Extension to the recovery of piecewise-constant functions over a continuous interval was considered in @cite_41 .
@cite_26 @cite_36 @cite_41 @cite_55 @cite_9 @cite_21 @cite_56 @cite_50 @cite_51
2060955827 2094089565 2097880821 2167121755 1973897753 1968512439
An estimator of the number of change-points in an independent normal sequence is proposed via Schwarz' criterion. Weak consistency of this estimator is established. We study the asymptotics for jump-penalized least squares regression aiming at approximating a regression function by piecewise constant functions. Besides conventional consistency and convergence rates of the estimates in L 2 ([0,1)) our results cover other metrics like Skorokhod metric on the space of cadlag functions and uniform metrics on C([0, 1]). We will show that these estimators are in an adaptive sense rate optimal over certain classes of "approximation spaces." Special cases are the class of functions of bounded variation (piecewise) Holder continuous functions of order 0 < α ≤ 1 and the class of step functions with a finite but arbitrary number of jumps. In the latter setting, we will also deduce the rates known from change-point analysis for detecting the jumps. Finally, the issue of fully automatic selection of the smoothing parameter is addressed. Many signal processing problems can be solved by maximizing the fitness of a segmented model over all possible partitions of the data interval. This letter describes a simple but powerful algorithm that searches the exponentially large space of partitions of N data points in time O(N sup 2 ). The algorithm is guaranteed to find the exact global optimum, automatically determines the model order (the number of segments), has a convenient real-time mode, can be extended to higher dimensional data spaces, and solves a surprising variety of problems in signal detection and characterization, density estimation, cluster analysis, and classification. This paper is mainly devoted to a precise analysis of what kind of penalties should be used in order to perform model selection via the minimiza- tion of a penalized least-squares type criterion within some general Gaussian framework including the classical ones. As compared to our previous paper on this topic (Birge and Massart in J. Eur. Math. Soc. 3, 203-268 (2001)), more elaborate forms of the penalties are given which are shown to be, in some sense, optimal. We indeed provide more precise upper bounds for the risk of the penalized estimators and lower bounds for the penalty terms, showing that the use of smaller penalties may lead to disastrous results. These lower bounds may also be used to design a practical strategy that allows to estimate the penalty from the data when the amount of noise is unknown. We provide an illustra- tion of the method for the problem of estimating a piecewise constant signal in Gaussian noise when neither the number, nor the location of the change points are known. We discuss the interplay between local M -smoothers, Bayes smoothers and some nonlinear filters for edge-preserving signal reconstruction. We prove that all smoothers in question are nonlinear filters in a precise sense and characterize their fixed points. Then a Potts model is adopted for segmentation. For 1-d signals, an exact algorithm for the computation of maximum posterior modes is derived and applied to a phantom and to 1-d fMRI-data. This paper deals with the problem of detecting change-points in the mean of a signal corrupted by an additive Gaussian noise. The number of changes and their position are unknown. From a nonasymptotic point of view, we propose to estimate them with a method based on a penalized least-squares criterion. We choose the penalty function such that the resulting estimator minimizes the quadratic risk according to the results of Birge and Massart. This penalty depends on unknown constants and we propose a calibration to obtain an automatic method. The performance of the method is assessed through simulation experiments. An application to real data is shown.