Article
Minimax adaptive dimension reduction for regression
In this paper, we address the problem of regression estimation in the context of a -dimensional predictor when is large. We propose a general model in which the regression function is a composite function. Our model consists in a nonlinear extension of the usual sufficient dimension reduction setting. The strategy followed for estimating the regression function is based on the estimation of a new parameter, called the reduced dimension. We adopt a minimax point of view and provide both lower and upper bounds for the optimal rates of convergence for the estimation of the regression function in the context of our model. We prove that our estimate adapts, in the minimax sense, to the unknown value of the reduced dimension and achieves therefore fast rates of convergence when .
This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.
Images can be represented as vectors in a high-dimensional Image space with components specifying light intensities at image pixels. To avoid the ‘curse of dimensionality’, the original high-dimensional image data are transformed into their lower-dimensional features preserving certain subject-driven data properties. These properties can include ‘information-preserving’ when using the constructed low-dimensional features instead of original high-dimensional vectors, as well preserving the distances and angles between the original high-dimensional image vectors. Under the commonly used Manifold assumption that the high-dimensional image data lie on or near a certain unknown low-dimensional Image manifold embedded in an ambient high-dimensional ‘observation’ space, a constructing of the lower-dimensional features consists in constructing an Embedding mapping from the Image manifold to Feature space, which, in turn, determines a low-dimensional parameterization of the Image manifold. We propose a new geometrically motivated Embedding method which constructs a low-dimensional parameterization of the Image manifold and provides the information-preserving property as well as the locally isometric and conformal properties. © (2015) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only.
Based on n randomly drawn vectors in a Hilbert space, we study the k-means clustering scheme. Here, clustering is performed by computing the Voronoi partition associated with centers that minimize an empirical criterion, called distorsion. The performance of the method is evaluated by comparing the theoretical distorsion of empirical optimal centers to the theoretical optimal distorsion. Our first result states that, provided that the underlying distribution satisfies an exponential moment condition, an upper bound for the above performance criterion isO(1/n√). Then, motivated by a broad range of applications, we focus on the case where the data are real-valued random fields. Assuming that they share a Hölder property in quadratic mean, we construct a numerically simple k-means algorithm based on a discretized version of the data. With a judicious choice of the discretization, we prove that the performance of this algorithm matches the performance of the classical algorithm.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traffic is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the final node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a finite-dimensional system of differential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of differential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.
A form for an unbiased estimate of the coefficient of determination of a linear regression model is obtained. It is calculated by using a sample from a multivariate normal distribution. This estimate is proposed as an alternative criterion for a choice of regression factors.