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.
Performance bounds for criteria for model selection are developed using recent theory for sieves. The model selection criteria are based on an empirical loss or contrast function with an added penalty term motivated by empirical process theory and roughly proportional to the number of parameters needed to describe the model divided by the number of observations. Most of our examples involve density or regression estimation settings and we focus on the problem of estimating the unknown density or regression function. We show that the quadratic risk of the minimum penalized empirical contrast estimator is bounded by an index of the accuracy of the sieve. This accuracy index quantifies the trade-off among the candidate models between the approximation error and parameter dimension relative to sample size. 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. 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.