Showing papers on "Generalization published in 2005"
TL;DR: In this paper, a projection algorithm is proposed to minimize a proximity function that measures the distance of a point from all sets in the image space, which generalizes the convex feasibility problem as well as two-sets split feasibility problem.
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.
733 citations
TL;DR: The authors make the case that research design should plan for anticipated generalizations, and that generalization should be more explicitly formulated within a context of supporting evidence, in particular by using the most recent volume of Sociology.
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.
477 citations
23 Oct 2005
TL;DR: The notion of a tradeoff revealing LP is introduced and used to derive two optimal algorithms achieving competitive ratios of 1-1/e for this problem of online bipartite 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.
396 citations
TL;DR: In this paper, a continual effort on generalization of the Trotter formula to higher-order exponential product formulas is reviewed, particularly because it conserves important symmetries of the system dynamics.
Abstract: In the present article, we review a continual effort on generalization of the Trotter formula to higher-order exponential product formulas. The exponential product formula is a good and useful approximant, particularly because it conserves important symmetries of the system dynamics. We focuse on two algorithms of constructing higher-order exponential product formulas. The first is the fractal decomposition, where we construct higher-order formulas recursively. The second is to make use of the quantum analysis, where we compute higher-order correction terms directly. As interludes, we also have described the decomposition of symplectic integrators, the approximation of time-ordered exponentials, and the perturbational composition.
279 citations
TL;DR: 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 in this paper.
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.
248 citations
TL;DR: The paper proves that theset of context and the set of properties of a concept is a complete orthocomplemented lattice, and shows that the context lattice as well as the property lattice are non‐classical, i.e. quantum‐like, lattices.
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...
217 citations
TL;DR: In this paper, a nonlinear generalization of the well-known Kolmogorov's consistent theorem is used to construct filtrationconsistent nonlinear expectations via 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.
212 citations
TL;DR: In this paper, 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.
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...
200 citations
Journal Article•
TL;DR: In this paper, a generalized partial credit model (GDM) is proposed to estimate a general diagnostic model 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.
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.
197 citations
TL;DR: An overview on current approaches for the automation of generalization and data abstraction is given, and solutions for three generalization problems based on optimization techniques based on Neural Network techniques are presented.
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...
185 citations
TL;DR: In this article, 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).
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).
TL;DR: A generalization of learning vector quantization with three additional features: it directly integrates neighborhood cooperation, hence is less affected by local optima, and the method can be combined with any differentiable 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.
TL;DR: How theoretical category knowledge-specifically, knowledge of the causal relations that link the features of categories-supports the ability to infer the presence of unobserved features is established.
Posted Content•
TL;DR: A review of quantum algorithms for search problems can be found in this paper, where 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.
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.
TL;DR: Eight Avoiding Overfitting Techniques are presented, considering that these are methods for improving the generalization of ANNs and have been tested on two case studies—rainfall–runoff data from two drainage basins in the south of Italy—in order to gain insight into their properties.
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...
1 Jan 2005
TL;DR: This work automatically decomposes the generalization of an object into a sequence of elementary steps, which leads to smooth transitions between different object representations and is useful for incremental transmission of maps through limited bandwidth channels.
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.
TL;DR: In this article, the authors address the question of how many comparisons should be limited to in order to control increase in inconsistency yet sufficiently large to enable capture validity that can be improved by proposed changes in judgments using a gradient method.
TL;DR: In this paper, a generalization of an elegant divisor sum bound due to F. V. Atkinson is presented for Dirichlet series coefficients satisfying a suitable functional equation.
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.
31 Aug 2005
TL;DR: In this case, it is shown that the generalized approximation space is a topological space for any reflexive relation.
Abstract: The standard rough set theory has been introduced in 1982 [5]. In this paper we use a topological concepts to investigate a new definitions for the lower and upper approximation operators. This approach is a generalization for Pawlak approach and the generalizations in [2, 7, 10, 12, 13, 14, 15, 16]. Properties of the suggested concepts are obtained. Also comparison between our approach and previous approaches are given. In this case, we show that the generalized approximation space is a topological space for any reflexive relation.
1 Jan 2005
TL;DR: In this paper, the authors propose a method to solve the problem of homonymity in homonymization, i.e., homonym-of-individuals-in-homonymity.
Abstract: ION
TL;DR: In this paper, the authors calculate the full second-order radiation transfer function for Cosmic Microwave Background anisotropies on large angular scales in a flat universe filled with matter and cosmological constant.
Abstract: We calculate the full second-order radiation transfer function for Cosmic Microwave Background anisotropies on large angular scales in a flat universe filled with matter and cosmological constant. It includes (i) the second-order generalization of the Sachs-Wolfe effect, and of (ii) both the early and late Integrated Sachs-Wolfe effects, (iii) the contribution of the second-order tensor modes, and is valid for a generic set of initial conditions specifying the level of primordial non-Gaussianity.
TL;DR: This article proposes an equivalent number of degrees of freedom to be used in neural model selection criteria like the Akaike information criterion and the Bayes information criteria and in the unbiased estimation of the error variance and proves to be much smaller than the total number of parameters of the network usually adopted.
Abstract: Neural network modeling for small datasets can be justified from a theoretical point of view according to some of Bartlett's results showing that the generalization performance of a multilayer perc...
TL;DR: This work proposes by means of an axiomatic characterization a solution for such games in partition function form that will be a generalization of the Shapley value (1953).
Abstract: Lucas and Trall (1963) defined the games in partition function form as a generalization of the cooperative games with transferable utility. In our work we propose by means of an axiomatic characterization a solution for such games in partition function form. This solution will be a generalization of the Shapley value (1953).
TL;DR: It is shown that the p positive elements can be determined up to a constant number of misclassifications, bounded by the gap between the thresholds, and a two-phase strategy consisting of a Distill and a Compress phase is proposed.
TL;DR: In this paper, the authors developed the general framework of sensitivity analysis for equilibrium problems in the setting of vector topological normed space and obtained, in a similar spirit of Domokos [J. Math. Anal. Appl. 230 (1999) 382−389], the best lower bound for the continuity modulus despite of the properties of the boundary of Kλ.
TL;DR: In this paper, it was shown that the original version of the chemotaxis model has a macroscopic diffusion limit in both two and three dimensions when the equation of the attractant is of a parabolic type.
Abstract: We study a kinetic model for chemotaxis introduced by Othmer,
Dunbar, and Alt [23], which was motivated by earlier results of Alt, presented
in [1], [2]. In two papers by Chalub, Markowich, Perthame and Schmeiser,
it was rigorously shown that, in three dimensions, this kinetic model leads to
the classical Keller-Segel model as its drift-diffusion limit when the equation of
the chemo-attractant is of elliptic type [4], [5]. As an extension of these works
we prove that such kinetic models have a macroscopic diffusion limit in both
two and three dimensions also when the equation of the chemo-attractant is of
parabolic type, which is the original version of the chemotaxis model.
Journal Article•
TL;DR: This article conducted a qualitative study of 22 9-year-olds performing generalization on a task involving linear patterns and identified three different strategies: numerical, figural, and pragmatic, based on students' exhibited strategies, understanding of variables and representational fluency.
Abstract: San Jose State University, USA This is a qualitative study of 22 9 graders performing generalizations on a task involving linear patterns. Our research questions were: What enables/hinders students’ abilities to generalize a linear pattern? What strategies do successful students use to develop an explicit generalization? How do students make use of visual and numerical cues in developing a generalization? Do students use different representations equally? Can students connect different representations of a pattern with fluency? Twenty-three different strategies were identified falling into three types, numerical, figural, and pragmatic, based on students’ exhibited strategies, understanding of variables, and representational fluency.
TL;DR: It is shown that after training free-flying bees with a single, constant pair of patterns made of four quadrants, each displaying different oriented gratings, bees extract the orientation information corresponding to each quadrant and integrate it in a generic layout that preserved the spatial relationship between oriented edges.
31 Aug 2005
TL;DR: Four approaches to missing attribute values are discussed in this paper: lost values, ” do not care” conditions, restricted ”do not care" conditions, and attribute-concept values, which achieve a generalized rough set methodology.
Abstract: In incomplete data missing attribute values may be universally interpreted in several ways. Four approaches to missing attribute values are discussed in this paper: lost values, ”do not care” conditions, restricted ”do not care” conditions, and attribute-concept values. Rough set ideas, such as attribute-value pair blocks, characteristic sets, characteristic relations and generalization of lower and upper approximations are used in these four approaches. A generalized rough set methodology, achieved in the process, may be used for other applications as well. Additionally, this generalized methodology is compared with other extensions of rough set concepts.
TL;DR: This work extends existing theory on stability, namely how much changes in the training data influence the estimated models, and generalization performance of deterministic learning algorithms to the case of discrete-time reinforcement learning.
Abstract: We extend existing theory on stability, namely how much changes in the training data influence the estimated models, and generalization performance of deterministic learning algorithms to the case ...