Showing papers on "Generalization published in 2005"

The Multiple-Sets Split Feasibility Problem and Its Applications for Inverse Problems

Abstract: The multiple-sets split feasibility problem requires finding a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space. It can be a model for many inverse problems where constraints are imposed on the solutions in the domain of a linear operator as well as in the operator's range. It generalizes the convex feasibility problem as well as the two-sets split feasibility problem. We propose a projection algorithm that minimizes a proximity function that measures the distance of a point from all sets. The formulation, as well as the algorithm, generalize earlier work on the split feasibility problem. We offer also a generalization to proximity functions with Bregman distances. Application of the method to the inverse problem of intensity-modulated radiation therapy treatment planning is studied in a separate companion paper and is here only described briefly.

Generalization in Qualitative Research

Abstract: Earlier treatments of moderatum generalization (e.g. Williams, 2000a) explicitly addressed interpretivist sociology. This article extends that earlier argument by examining some of its implications for a wider range of qualitative research methods. It first adopts an empirical approach, providing concrete illustrations from the most recent volume of Sociology of what sociologists actually do when describing the meaning of their findings. In the light of this, we reconsider the significance of moderatum generalization for research practice and the status of sociological knowledge, in particular making the case that research design should plan for anticipated generalizations, and that generalization should be more explicitly formulated within a context of supporting evidence.

AdWords and generalized on-line matching

Abstract: How does a search engine company decide what ads to display with each query so as to maximize its revenue? This turns out to be a generalization of the online bipartite matching problem. We introduce the notion of a tradeoff revealing LP and use it to derive two optimal algorithms achieving competitive ratios of 1-1/e for this problem.

A 2-Component Generalization of the Camassa-Holm Equation and Its Solutions

Abstract: An explicit reciprocal transformation between a 2-component generalization of the Camassa-Holm equation, called the 2-CH system, and the first negative flow of the AKNS hierarchy is established, this transformation enables one to obtain solutions of the 2-CH system from those of the first negative flow of the AKNS hierarchy. Interesting examples of peakon and multi-kink solutions of the 2-CH system are presented.

A theory of concepts and their combinations I: The structure of the sets of contexts and properties.

Abstract: Purpose – To elaborate a theory for modeling concepts that incorporates how a context influences the typicality of a single exemplar and the applicability of a single property of a concept. To investigate the structure of the sets of contexts and properties.Design/methodology/approach – The effect of context on the typicality of an exemplar and the applicability of a property is accounted for by introducing the notion of “state of a concept”, and making use of the state‐context‐property formalism (SCOP), a generalization of the quantum formalism, whose basic notions are states, contexts and properties.Findings – The paper proves that the set of context and the set of properties of a concept is a complete orthocomplemented lattice, i.e. a set with a partial order relation, such that for each subset there exists a greatest lower bound and a least upper bound, and such that for each element there exists an orthocomplement. This structure describes the “and”, “or”, and “not”, respectively for contexts and pro...

K-SVD and its non-negative variant for dictionary design

Abstract: In recent years there is a growing interest in the study of sparse representation for signals. Using an overcomplete dictionary that contains prototype signal-atoms, signals are described as sparse linear combinations of these atoms. Recent activity in this field concentrated mainly on the study of pursuit algorithms that decompose signals with respect to a given dictionary. Designing dictionaries to better fit the above model can be done by either selecting pre-specified transforms, or by adapting the dictionary to a set of training signals. Both these techniques have been considered in recent years, however this topic is largely still open. In this paper we address the latter problem of designing dictionaries, and introduce the K-SVD algorithm for this task. We show how this algorithm could be interpreted as a generalization of the K-Means clustering process, and demonstrate its behavior in both synthetic tests and in applications on real data. Finally, we turn to describe its generalization to nonnegative matrix factorization problem that suits signals generated under an additive model with positive atoms. We present a simple and yet efficient variation of the K-SVD that handles such extraction of non-negative dictionaries.

Nonlinear expectations and nonlinear markov chains

Abstract: This paper deals with nonlinear expectations. The author obtains a nonlinear generalization of the well-known Kolmogorov's consistent theorem and then use it to construct filtration-consistent nonlinear expectations via nonlinear Markov chains. Compared to the author's previous results, i.e., the theory of g-expectations introduced via BSDE on a probability space, the present framework is not based on a given probability measure. Many fully nonlinear and singular situations are covered. The induced topology is a natural generalization of Lp-norms and L∞-norm in linear situations. The author also obtains the existence and uniqueness result of BSDE under this new framework and develops a nonlinear type of von Neumann-Morgenstern representation theorem to utilities and present dynamic risk measures.

Generalization and Justification: The Challenge of Introducing Algebraic Reasoning through Patterning Activities.

Abstract: The expectation that students be introduced to algebraic ideas at earlier grade levels places an increased burden on the classroom teacher to help students construct and justify generalizations. This study provides insight into the reasoning of 25 sixth-grade students as they approached patterning tasks in which they were required to develop and justify generalizations while using computer spreadsheets as an instructional tool. The students demonstrated both the potential and pitfalls of such activities. During whole-class discussions, students were generally able to provide appropriate generalizations and justify using generic examples. Students who used geometric schemes were more successful in providing general arguments and valid justifications. However, during small-group discussions, the students rarely justified their generalizations, with some students focusing more on particular values than on general relations. It is recommended that the various student strategies and justifications be brought t...

Supervised Neural Gas with General Similarity Measure

Abstract: Prototype based classification offers intuitive and sparse models with excellent generalization ability. However, these models usually crucially depend on the underlying Euclidian metric; moreover, online variants likely suffer from the problem of local optima. We here propose a generalization of learning vector quantization with three additional features: (I) it directly integrates neighborhood cooperation, hence is less affected by local optima; (II) the method can be combined with any differentiable similarity measure whereby metric parameters such as relevance factors of the input dimensions can automatically be adapted according to the given data; (III) it obeys a gradient dynamics hence shows very robust behavior, and the chosen objective is related to margin optimization.

Optimization approaches for generalization and data abstraction

Abstract: The availability of methods for abstracting and generalizing spatial data is vital for understanding and communicating spatial information. Spatial analysis using maps at different scales is a good example of this. Such methods are needed not only for analogue spatial data sets but even more so for digital data. In order to automate the process of generating different levels of detail of a spatial data set, generalization operations are used. The paper first gives an overview on current approaches for the automation of generalization and data abstraction, and then presents solutions for three generalization problems based on optimization techniques. Least‐Squares Adjustment is used for displacement and shape simplification (here, building groundplans), and Self‐Organizing Maps, a Neural Network technique, is applied for typification, i.e. a density preserving reduction of objects. The methods are validated with several examples and evaluated according to their advantages and disadvantages. Finally, a scen...

Mean value in invexity analysis

Abstract: In this paper, a generalization of the mean value theorem is considered in the case of functions defined on an invex set with respect to η (which is not necessarily connected).

Quantum search algorithms

Abstract: We review some of quantum algorithms for search problems: Grover's search algorithm, its generalization to amplitude amplification, the applications of amplitude amplification to various problems and the recent quantum algorithms based on quantum walks.

Quadric-based simplification in any dimension

Abstract: We present a novel generalization of the quadric error metric used in surface simplification that can be used for simplifying simplicial complexes of any type embedded in Euclidean spaces of any dimension. We demonstrate that our generalized simplification system can produce high quality approximations of plane and space curves, triangulated surfaces, tetrahedralized volume data, and simplicial complexes of mixed type. Our method is both efficient and easy to implement. It is capable of processing complexes of arbitrary topology, including nonmanifolds, and can preserve intricate boundaries.

Improving generalization of artificial neural networks in rainfall-runoff modelling

Abstract: Artificial neural networks (ANNs) are general-purpose techniques that can be used for nonlinear data-driven rainfall–runoff modelling. The key issue to construct a good model by means of ANNs is to understand their structural features and the problems related to their construction. Indeed, the quantity and quality of data, the type of noise and the mathematical properties of the algorithm for estimating the usual large number of parameters (weights) are crucial for the generalization performances of ANNs. However, it is well known that ANNs may suffer from poor generalization properties due to the high number of parameters and non-Gaussian data noise. Therefore, in the first part of this paper, the features and problems of ANNs are discussed. Eight Avoiding Overfitting Techniques are then presented, considering that these are methods for improving the generalization of ANNs. For this reason, they have been tested on two case studies—rainfall–runoff data from two drainage basins in the south of It...

A model-theoretic counterpart of loop formulas

Abstract: In an important recent paper, Lin and Zhao introduced the concept of a loop formula, and showed that the answer sets for a logic program are exactly the models of Clark's completion of the program that satisfy the loop formulas. Just as supported sets are a model-theoretic account of completion, "externally supported" sets, defined in this paper, are a model-theoretic counterpart of loop formulas. This reformulation of loop formulas shows that they are related to assumption sets (Sacca and Zaniolo) and to unfounded sets (Van Gelder, Ross and Schlipf; Leone, Rullo and Scarcello), invented many years earlier. Other contributions of this paper includes a simplification of the definition of a loop, extending it to programs with classical negation and infinite programs, and a generalization of the definition of a loop formula.

A General Diagnostic Model Applied to Language Testing Data. Research Report. ETS RR-05-16.

Abstract: Probabilistic models with one or more latent variables are designed to report on a corresponding number of skills or cognitive attributes. Multidimensional skill profiles offer additional information beyond what a single test score can provide, if the reported skills can be identified and distinguished reliably. Many recent approaches to skill profile models are limited to dichotomous data and have made use of computationally intensive estimation methods such as Markov chain Monte Carlo, since standard maximum likelihood (ML) estimation techniques were deemed infeasible. This paper presents a general diagnostic model (GDM) that can be estimated with standard ML techniques and applies to polytomous response variables as well as to skills with two or more proficiency levels. The paper uses one member of a larger class of diagnostic models, a compensatory diagnostic model for dichotomous and partial credit data. Many well-known models, such as univariate and multivariate versions of the Rasch model and the two-parameter logistic item response theory model, the generalized partial credit model, as well as a variety of skill profile models, are special cases of this GDM. In addition to an introduction to this model, the paper presents a parameter recovery study using simulated data and an application to real data from the field test for TOEFL Internet-based testing.

Continuous Generalization for Visualization on Small Mobile Devices

Abstract: Visualizing spatial information on small mobile displays is a big chance and challenge at the same time. In order to address the tradeoff between huge spatial data sets and small storage capacities and visualization screens, we propose to visualize only the information on the screen which adequately fits the current resolution. To realize this, we automatically decompose the generalization of an object into a sequence of elementary steps. From this, one can later easily obtain any desired generalization level by applying the appropriate subpart of the sequence. The method method does not only lead to smooth transitions between different object representations but also is useful for incremental transmission of maps through limited bandwidth channels.

Stability results in learning theory

Abstract: The problem of proving generalization bounds for the performance of learning algorithms can be formulated as a problem of bounding the bias and variance of estimators of the expected error. We show how various stability assumptions can be employed for this purpose. We provide a necessary and sufficient stability condition for bounding the bias and variance for the Empirical Risk Minimization algorithm, and various sufficient conditions for bounding bias and variance of estimators for general algorithms. We discuss settings in which it is possible to obtain exponential bounds, and we prove an extension of the bounded-difference inequality for "almost always" stable algorithms.

Stability and generalization of bipartite ranking algorithms

Abstract: The problem of ranking, in which the goal is to learn a real-valued ranking function that induces a ranking or ordering over an instance space, has recently gained attention in machine learning. We study generalization properties of ranking algorithms, in a particular setting of the ranking problem known as the bipartite ranking problem, using the notion of algorithmic stability.In particular, we derive generalization bounds for bipartite ranking algorithms that have good stability properties. We show that kernel-based ranking algorithms that perform regularization in a reproducing kernel Hilbert space have such stability properties, and therefore our bounds can be applied to these algorithms; this is in contrast with previous generalization bounds for ranking, which are based on uniform convergence and in many cases cannot be applied to these algorithms. A comparison of the bounds we obtain with corresponding bounds for classification algorithms yields some interesting insights into the difference in generalization behaviour between ranking and classification.

Generalization of a vision-based computational model of mind-reading

Abstract: This paper describes a vision-based computational model of mind-reading that infers complex mental states from head and facial expressions in real-time. The generalization ability of the system is evaluated on videos that were posed by lay people in a relatively uncontrolled recording environment for six mental states—agreeing, concentrating, disagreeing, interested, thinking and unsure. The results show that the system’s accuracy is comparable to that of humans on the same corpus.

Bloat Control and Generalization Pressure Using the Minimum Description Length Principle for a Pittsburgh Approach Learning Classifier System.

Abstract: Bloat control and generalization pressure are very important issues in the design of Pittsburgh Approach Learning Classifier Systems (LCS), in order to achieve simple and accurate solutions in a reasonable time. In this paper we propose a method to achieve these objectives based on the Minimum Description Length (MDL) principle. This principle is a metric which combines in a smart way the accuracy and the complexity of a theory (rule set, instance set, etc.). An extensive comparison with our previous generalization pressure method across several domains and using two knowledge representations has been done. The test show that the MDL based size control method is a good and robust choice.

Rough membership and bayesian confirmation measures for parameterized rough sets

Abstract: A generalization of the original idea of rough sets and variable precision rough sets is introduced. This generalization is based on the concept of absolute and relative rough membership. Similarly to variable precision rough set model, the generalization called parameterized rough set model, is aimed at modeling data relationships expressed in terms of frequency distribution rather than in terms of a full inclusion relation used in the classical rough set approach. However, differently from variable precision rough set model, one or more parameters modeling the degree to which the condition attribute values confirm the decision attribute value, are considered. The properties of this extended model are investigated and compared to the classical rough set model and the variable precision rough set model.

Ontology-driven map generalization

Abstract: Different users of geospatial information have different requirements of that information. Matching information to users' requirements demands an understanding of the ontological aspects of geospatial data. In this paper, we present an ontology-driven map generalization algorithm, called DMin, that can be tailored to particular users and users' tasks. The level of detail in a generated map is automatically adapted by DMin according to the semantics of the features represented. The DMin algorithm is based on a weighting function that has two components: (1) a geometric component that differs from previous approaches to map generalization in that no fixed threshold values are needed to parameterize the generalization process and (2) a semantic component that considers the relevance of map features to the user. The flexibility of DMin is demonstrated using the example of a transportation network.

Summation formulae for coefficients of L-functions

Abstract: With applications in mind we establish a summation formula for the coefficients of a general Dirichlet series satisfying a suitable functional equation. Among a number of consequences we derive a generalization of an elegant divisor sum bound due to F. V. Atkinson.

Topological and fractal properties of real numbers which are not normal

Abstract: The set L of essentially non-normal numbers of the unit interval (i.e., the set of real numbers having no asymptotic frequencies of all digits in their nonterminating s-adic expansion) is studied in details. It is proven that the set L is generic in the topological sense (it is of the second Baire category) as well as in the sense of fractal geometry (L is a superfractal set, i.e., the Hausdorff–Besicovitch dimension of the set L is equal 1). These results are substantial generalizations of the previous results of the two latter authors [M. Pratsiovytyi, G. Torbin, Ukrainian Math. J. 47 (7) (1995) 971–975]. The Q ∗ -representation of real numbers (which is a generalization of the s-adic expansion) is also studied. This representation is determined by the stochastic matrix Q ∗ . We prove the existence of such a Q ∗ -representation that almost all (in the sense of Lebesgue measure) real numbers have no asymptotic frequency of all digits. In the case where the matrix Q ∗ has additional asymptotic properties, the Hausdorff–Besicovitch dimension of the set of numbers with prescribed asymptotic properties of their digits is determined (this is a generalization of the Eggleston–Besicovitch theorem). The connections between the notions of “normality of numbers” respectively of “asymptotic frequencies” of their digits is also studied.

Tensor decomposition and approximation schemes for constraint satisfaction problems

Abstract: The only general class of MAX-rCSP problems for which Polynomial Time Approximation Schemes (PTAS) are known are the dense problems. In this paper, we give PTAS's for a much larger class of weighted MAX-rCSP problems which includes as special cases the dense problems and, for r = 2, all metric instances (where the weights satisfy the triangle inequality) and quasimetric instances; for r > 2, our class includes a generalization of metrics. Our algorithms are based on low-rank approximations with two novel features: (1) a method of approximating a tensor by the sum of a small number of "rank-1" tensors, akin to the traditional Singular Value Decomposition (this might be of independent interest) and (2) a simple way of scaling the weights. Besides MAX-rCSP problems, we also give PTAS's for problems with a constant number of global constraints such as maximum weighted graph bisection and some generalizations.

Taming the wild in impartial combinatorial games

Abstract: We introduce a misere quotient semigroup construction in impartial combinatorial game theory, and argue that it is the long-sought natural generalization of the normal-play Sprague-Grundy theory to misere play. Along the way, we illustrate how to use the theory to describe complete analyses of two wild taking and breaking games.