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54
Stochastic Perturbation Theory
, 1988
"... . In this paper classical matrix perturbation theory is approached from a probabilistic point of view. The perturbed quantity is approximated by a firstorder perturbation expansion, in which the perturbation is assumed to be random. This permits the computation of statistics estimating the variatio ..."
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Cited by 617 (31 self)
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. In this paper classical matrix perturbation theory is approached from a probabilistic point of view. The perturbed quantity is approximated by a firstorder 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 higherorder 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...
Least Squares Policy Evaluation Algorithms With Linear Function Approximation
 Theory and Applications
, 2002
"... We consider policy evaluation algorithms within the context of infinitehorizon dynamic programming problems with discounted cost. We focus on discretetime dynamic systems with a large number of states, and we discuss two methods, which use simulation, temporal differences, and linear cost function ..."
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Cited by 65 (9 self)
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We consider policy evaluation algorithms within the context of infinitehorizon dynamic programming problems with discounted cost. We focus on discretetime 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 gradientlike algorithm involving leastsquares 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 leastsquares temporaldifference 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].
Analyzing Developmental Trajectories: A Semiparametric, GroupBased Approach
 Psychological Methods
, 1999
"... A developmental trajectory describes the course of a behavior over age or time. A groupbased 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 " ..."
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Cited by 56 (1 self)
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A developmental trajectory describes the course of a behavior over age or time. A groupbased 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 individuallevel developmental trajectories. The two main branches of methodology are hierarchical modeling (Bryk &
WorstCase Properties of the Uniform Distribution and Randomized Algorithms for Robustness Analysis
, 1996
"... Motivated by the current limitations of the existing algorithms for robustness analysis and design, in this paper we take a different direction which follows the socalled probabilistic approach. That is, we aim to estimate the probability that a control system with uncertain parameters q restricted ..."
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Cited by 12 (0 self)
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Motivated by the current limitations of the existing algorithms for robustness analysis and design, in this paper we take a different direction which follows the socalled probabilistic approach. That is, we aim to estimate the probability that a control system with uncertain parameters q restricted to a box Q attains a given level of performance fl. Since this probability depends on the underlying distribution, we address the following question: What is a "reasonable" distribution so that the estimated probability makes sense? To answer this question, we define two worstcase criteria and prove that the uniform distribution is optimal in both cases. In the second part of the paper, we turn our attention to a subsequent problem. That is, we estimate the sizes of both the socalled "good" and "bad" sets via sampling. Roughly speaking, the good set contains the parameters q 2 Q with performance level better than or equal to fl while the bad set is the set of parameters q 2 Q with perform...
A theory of interactive parallel processing: new capacity measures and predictions for a response time inequality series
, 2004
"... The authors present a theory of stochastic interactive parallel processing with special emphasis on channel interactions and their relation to system capacity. The approach is based both on linear systems theory augmented with stochastic elements and decisional operators and on a metatheory of paral ..."
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Cited by 11 (4 self)
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The authors present a theory of stochastic interactive parallel processing with special emphasis on channel interactions and their relation to system capacity. The approach is based both on linear systems theory augmented with stochastic elements and decisional operators and on a metatheory of parallel channels ’ dependencies that incorporates standard independent and coactive parallel models as special cases. The metatheory is applied to OR and AND experimental paradigms, and the authors establish new theorems relating response time performance in these designs to earlier and novel issues. One notable outcome is the remarkable processing efficiency associated with linear parallelchannel systems that include mutually positive interactions. The results may offer insight into perceptual and cognitive configural–holistic processing systems.
Assessing the Importance of Features for MultiLayer Perceptrons
, 1998
"... 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 multilay ..."
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Cited by 8 (3 self)
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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 multilayer 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.
On the Learning of Rule Uncertainties and their Integration into Probabilistic Knowledge Bases
, 1993
"... 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 au ..."
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Cited by 8 (4 self)
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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 distributionfree learnable. The distributionfree 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...
Knowledge Discovery in Databases
, 1994
"... 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 ..."
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Cited by 8 (4 self)
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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...
Linear ScaleSpace Theory from Physical Principles
 IN JOURNAL OF MATHEMATICAL IMAGING AND VISION
, 1995
"... In the past decades linear scalespace 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 re ..."
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Cited by 7 (5 self)
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In the past decades linear scalespace 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 scalespace theory is extended to spatiotemporal images on bounded and curved domains. Our theory permits a delayoperation 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 delayoperation. Finally, the semid...