. In this paper classical matrix perturbation theory is approached from a probabilistic point of view. The perturbed quantity is approximated by a first-order perturbation expansion, in which the perturbation is assumed to be random. This permits the computation of statistics estimating the variation in the perturbed quantity. Up to the higher-order terms that are ignored in the expansion, these statistics tend to be more realistic than perturbation bounds obtained in terms of norms. The technique is applied to a number of problems in matrix perturbation theory, including least squares and the eigenvalue problem. Key words. perturbation theory, random matrix, linear system, least squares, eigenvalue, eigenvector, invariant subspace, singular value AMS(MOS) subject classifications. 15A06, 15A12, 15A18, 15A52, 15A60 1. Introduction. Let A be a matrix and let F be a matrix valued function of A. Two principal problems of matrix perturbation theory are the following. Given a matrix E, pr...

We consider policy evaluation algorithms within the context of infinite-horizon dynamic programming problems with discounted cost. We focus on discrete-time dynamic systems with a large number of states, and we discuss two methods, which use simulation, temporal differences, and linear cost function approximation. The first method is a new gradient-like algorithm involving least-squares subproblems and a diminishing stepsize, which is based on the #-policy iteration method of Bertsekas and Ioffe. The second method is the LSTD(#) algorithm recently proposed by Boyan, which for # =0coincides with the linear least-squares temporal-difference algorithm of Bradtke and Barto. At present, there is only a convergence result by Bradtke and Barto for the LSTD(0) algorithm. Here, we strengthen this result by showing the convergence of LSTD(#), with probability 1, for every # [0, 1].

A developmental trajectory describes the course of a behavior over age or time. A group-based method for identifying distinctive groups of individual trajectories within the population and for profiling the characteristics of group members is demonstrated. Such clusters might include groups of "increasers. " "decreasers," and "no changers. " Suitably defined probability distributions are used to handle 3 data types—count, binary, and psychometric scale data. Four capabilities are demonstrated: (a) the capability to identify rather than assume distinctive groups of trajectories, (b) the capability to estimate the proportion of the population following each such trajectory group, (c) the capability to relate group membership probability to individual characteristics and circumstances, and (d) the capability to use the group membership probabilities for various other purposes such as creating profiles of group members. Over the past decade, major advances have been made in methodology for analyzing individual-level developmental trajectories. The two main branches of methodology are hierarchical modeling (Bryk &

In the past decades linear scale-space theory was derived on the basis of various axiomatics. In this paper we revisit these axioms and show that they merely coincide with the following physical principles, namely that the image domain is a Galilean space, that the total energy exchange between a region and its surrounding is preserved under linear filtering and that the physical observables should be invariant under the group of similarity transformations. These observables are elements of the similarity jet spanned by natural coordinates and differential energies read out by a vision system. Furthermore, linear scale-space theory is extended to spatio-temporal images on bounded and curved domains. Our theory permits a delay-operation at the present moment which is in agreement with the motion detection model of Reichardt. In this respect our theory deviates from that of Koenderink which requires additional syntactical operators to realise such a delay-operation. Finally, the semi-d...

In this paper we establish a mathematical framework in which we develop measures for determining the contribution of individual features to the performance of a classifier. Corresponding to these measures, we design metrics that allow estimation of the importance of features for a specific multi-layer perceptron neural network. It is shown that all measures constitute lower bounds for the correctness that can be obtained when the feature under study is excluded and the classifier rebuilt. We also present a method for pruning input nodes from the network such that most of the knowledge encoded in its weights is retained. The proposed metrics and the pruning method are validated with a number of experiments with artificial classification tasks. The experiments indicate that the metric called replaceability results in the tightest error bounds. Both this metric and the metric called expected influence result in good rankings of the features. (c) 1998 Elsevier Science Ltd. All rights reserved.

We present a natural and realistic knowledge acquisition and processing scenario. In the first phase a domain expert identifies deduction rules that he thinks are good indicators of whether a specific target concept is likely to occur. In a second knowledge acquisition phase, a learning algorithm automatically adjusts, corrects and optimizes the deterministic rule hypothesis given by the domain expert by selecting an appropriate subset of the rule hypothesis and by attaching uncertainties to them. Then, in the running phase of the knowledge base we can arbitrarily combine the learned uncertainties of the rules with uncertain factual information. Formally, we introduce the natural class of disjunctive probabilistic concepts and prove that this class is efficiently distribution-free learnable. The distribution-free learning model of probabilistic concepts was introduced by Kearns and Schapire and generalizes Valiant's probably approximately correct learning model. We show how to simulate...

This is a draft of a manuscript of a postgraduate course taught at the Hong Kong University of Science and Technology in Spring 94. The course gives an introduction into the young and fascinating field of knowledge discovery in databases. The manuscript is suited for beginners who can leave out the more advanced sections, as well as people who would like to do research in this area. This manuscript is partly incomplete. Especially, the last section discussing approaches to learn knowledge involving time is missing. Contents 1 Introduction 2 1.1 Course Outline : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 1.2 Basic Notions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 1.3 A Case Study : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 1.4 Outlook : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 LIST OF FIGURES 2 2 Rule Languages 15 2.1 Proposit...

Current drawn by a static CMOS VLSI integrated circuit during quiescent periods is extremely small and is normally of the order of nanoamperes. However, it is remarkably susceptible to a number of failure modes. Many faults present in such ICs cause the quiescent power-supply cur- rent (IDDQ) to increase by several orders of magnitude. Some of these faults may not manifest as logical faults, and would not be detected by traditional IC test techniques.

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This paper develops new algorithms belonging to the class of context modelling methods, with direct application to lossless coding of gray level images. The prediction stage and the context modelling stage are performed using nonlinear techniques rooted in the field of order statistics nonlinear filtering, which proved competitive in image restoration applications. The new nonlinear predictors introduced here can be easily rephrased as adaptive nonlinear filtering tools, useful in image restoration applications. We propose a new variant of the Context algorithm, where the prediction, modelling of errors and coding are realized using a Finite State Machine modeller, (which reduces the complexity of tree modellers, by lumping together similar nodes). The new Context algorithm is shown to have coding performance better than the best available algorithms, as illustrated in the experimental section. 1. INTRODUCTION The application of the sound theory of universal modelling and coding to ...