Showing papers on "Generalization published in 2008"
TL;DR: In this article, the authors show how relative scales can be derived by making pairwise comparisons using numerical judgments from an absolute scale of numbers, when used to represent comparisons can be related and combined to define a cardinal scale of absolute numbers that is stronger than a ratio scale.
Abstract: According to the great mathematician Henri Lebesgue, making direct comparisons of objects with regard to a property is a fundamental mathematical process for deriving measurements. Measuring objects by using a known scale first then comparing the measurements works well for properties for which scales of measurement exist. The theme of this paper is that direct comparisons are necessary to establish measurements for intangible properties that have no scales of measurement. In that case the value derived for each element depends on what other elements it is compared with. We show how relative scales can be derived by making pairwise comparisons using numerical judgments from an absolute scale of numbers. Such measurements, when used to represent comparisons can be related and combined to define a cardinal scale of absolute numbers that is stronger than a ratio scale. They are necessary to use when intangible factors need to be added and multiplied among themselves and with tangible factors. To derive and synthesize relative scales systematically, the factors are arranged in a hierarchic or a network structure and measured according to the criteria represented within these structures. The process of making comparisons to derive scales of measurement is illustrated in two types of practical real life decisions, the Iran nuclear show-down with the West in this decade and building a Disney park in Hong Kong in 2005. It is then generalized to the case of making a continuum of comparisons by using Fredholm’s equation of the second kind whose solution gives rise to a functional equation. The Fourier transform of the solution of this equation in the complex domain is a sum of Dirac distributions demonstrating that proportionate response to stimuli is a process of firing and synthesis of firings as neurons in the brain do. The Fourier transform of the solution of the equation in the real domain leads to nearly inverse square responses to natural influences. Various generalizations and critiques of the approach are included.
1,218 citations
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TL;DR: In this article, a general framework for robust and efficient recovery of such signals from a given set of samples is developed. But this framework does not consider the problem of reconstructing an unknown signal from a series of samples.
Abstract: Traditional sampling theories consider the problem of reconstructing an unknown signal $x$ from a series of samples. A prevalent assumption which often guarantees recovery from the given measurements is that $x$ lies in a known subspace. Recently, there has been growing interest in nonlinear but structured signal models, in which $x$ lies in a union of subspaces. In this paper we develop a general framework for robust and efficient recovery of such signals from a given set of samples. More specifically, we treat the case in which $x$ lies in a sum of $k$ subspaces, chosen from a larger set of $m$ possibilities. The samples are modelled as inner products with an arbitrary set of sampling functions. To derive an efficient and robust recovery algorithm, we show that our problem can be formulated as that of recovering a block-sparse vector whose non-zero elements appear in fixed blocks. We then propose a mixed $\ell_2/\ell_1$ program for block sparse recovery. Our main result is an equivalence condition under which the proposed convex algorithm is guaranteed to recover the original signal. This result relies on the notion of block restricted isometry property (RIP), which is a generalization of the standard RIP used extensively in the context of compressed sensing. Based on RIP we also prove stability of our approach in the presence of noise and modelling errors.
A special case of our framework is that of recovering multiple measurement vectors (MMV) that share a joint sparsity pattern. Adapting our results to this context leads to new MMV recovery methods as well as equivalence conditions under which the entire set can be determined efficiently.
818 citations
TL;DR: This paper showed that successful generalization was associated with coupled changes in learning-phase activity in the hippocampus and midbrain (ventral tegmental area/substantia nigra).
572 citations
TL;DR: The authors proposed a new model of human concept learning based on Bayesian inference for a grammatically structured hypothesis space and compared the model predictions to human generalization judgments in several well-known category learning experiments, and found good agreement for both average and individual participant generalizations.
455 citations
TL;DR: In this article, a generalization of Gaiotto and Witten's theory of superconformal Chern-Simons supersymmetry was proposed. But the generalization was restricted to the case of the M-theory orbifold and it was shown that the theory can be seen as an orientifolding of the ABJM model.
Abstract: We explore further our recent generalization of the $\mathcal{N}=4$ superconformal Chern-Simons theories of Gaiotto and Witten. We find and construct explicitly theories of enhanced $\mathcal{N}=5$ or 6 supersymmetry, especially $\mathcal{N}=5$, $Sp(2M)\times O(N)$ and $\mathcal{N}=6$, $Sp(2M)\times O(2)$ theories. The $U(M)\times U(N)$ theory coincides with the $\mathcal{N}=6$ theory of Aharony, Bergman, Jafferis and Maldacena (ABJM). We argue that the $\mathcal{N}=5$ theory with $Sp(2N)\times O(2N)$ gauge group can be understood as an orientifolding of the ABJM model with $U(2N)\times U(2N)$ gauge group. We briefly discuss the Type IIB brane construction of the $\mathcal{N}=5$ theory and the geometry of the M-theory orbifold.
363 citations
18 Oct 2008
TL;DR: This chapter reviews methods for the assessment and comparison of Pareto set approximations based on a number of orthogonal criteria, including invariance to scaling, monotonicity and computational effort.
Abstract: This chapter reviews methods for the assessment and comparison of Pareto set approximations. Existing set quality measures from the literature are critically evaluated based on a number of orthogonal criteria, including invariance to scaling, monotonicity and computational effort. Statistical aspects of quality assessment are also considered in the chapter. Three main methods for the statistical treatment of Pareto set approximations deriving from stochastic generating methods are reviewed. The dominance ranking method is a generalization to partially-ordered sets of a standard non-parametric statistical test, allowing collections of Pareto set approximations from two or more stochastic optimizers to be directly compared statistically. The quality indicator method -- the dominant method in the literature -- maps each Pareto set approximation to a number, and performs statistics on the resulting distribution(s) of numbers. The attainment function method estimates the probability of attaining each goal in the objective space, and looks for significant differences between these probability density functions for different optimizers. All three methods are valid approaches to quality assessment, but give different information. We explain the scope and drawbacks of each approach and also consider some more advanced topics, including multiple testing issues, and using combinations of indicators. The chapter should be of interest to anyone concerned with generating and analysing Pareto set approximations.
348 citations
TL;DR: This paper examined how 15 third grade students (9-years old) come to produce and represent generalizations during the implementation of two lessons from a longitudinal study of early algebra and found that many students scan output values of f(n) as n increases, conceptualizing the function as a recursive sequence.
Abstract: We examine issues that arise in students’ making of generalizations about geometrical figures as they are introduced to linear functions. We focus on the concepts of patterns, function, and generalization in mathematics education in examining how 15 third grade students (9 years old) come to produce and represent generalizations during the implementation of two lessons from a longitudinal study of early algebra. Many students scan output values of f(n) as n increases, conceptualizing the function as a recursive sequence. If this instructional route is pursued, educators need to recognize how students’ conceptualizations of functions depart from the closed form expressions ultimately aimed for. Even more fundamentally, it is important to nurture a transition from empirical generalizations, based on conjectures regarding cases at hand, to theoretical generalizations that follow from operations on explicit statements about mathematical relations.
257 citations
TL;DR: The results suggest that the models with the prospect utility function can make generalizable predictions to new conditions, and different learning models are needed for making short-versus long-term predictions on simple gambling tasks.
245 citations
TL;DR: A new generalization of the one-dimensional differential transform method that will extend the application of the method to differential equations of fractional order is proposed, based on generalized Taylor’s formula and Caputo fractional derivative.
242 citations
TL;DR: In this article, the authors investigate the progressive manner in which students gain fluency with cultural algebraic modes of reflection and action in pattern generalizing tasks and propose a definition of algebraic generalization of patterns, which is used to distinguish between algebraic and arithmetic generalizations and some elementary naive forms of induction to which students often resort to solve pattern problems.
Abstract: The aim of this paper is to investigate the progressive manner in which students gain fluency with cultural algebraic modes of reflection and action in pattern generalizing tasks. The first section contains a short discussion of some epistemological aspects of generalization. Drawing on this section, a definition of algebraic generalization of patterns is suggested. This definition is used in the subsequent sections to distinguish between algebraic and arithmetic generalizations and some elementary naive forms of induction to which students often resort to solve pattern problems. The rest of the paper discusses the implementation of a teaching sequence in a Grade 7 class and focuses on the social, sign-mediated processes of objectification through which the students reached stable forms of algebraic reflection. The semiotic analysis puts into evidence two central processes of objectification—iconicity and contraction.
237 citations
4 Nov 2008
TL;DR: Wang et al. as mentioned in this paper proposed a generalization-based approach that applies to trajectories and sequences in general and proposed trajectory anonymization techniques to address time and space sensitive applications.
Abstract: Trajectory datasets are becoming more and more popular due to the massive usage of GPS and other location-based devices and services. In this paper, we address privacy issues regarding the identification of individuals in static trajectory datasets. We provide privacy protection by definig trajectory k-anonymity, meaning every released information refers to at least k users/trajectories. We propose a novel generalization-based approach that applies to trajectories and sequences in general. We also suggest the use of a simple random reconstruction of the original dataset from the anonymization, to overcome possible drawbacks of generalization approaches.We present a utility metric that maximizes the probability of a good representation and propose trajectory anonymization techniques to address time and space sensitive applications. The experimental results over synthetic trajectory datasets show the effectiveness of the proposed approach.
TL;DR: Support Vector Machine model, firmly based on the theory of statistical learning, is used in slope stability problem and gives better result than previously published result of ANN model for factor of safety prediction and stability status.
Abstract: Artificial Neural Network (ANN) such as backpropagation learning algorithm has been successfully used in slope stability problem. However, generalization ability of conventional ANN has some limitations. For this reason, Support Vector Machine (SVM) which is firmly based on the theory of statistical learning has been used in slope stability problem. An interesting property of this approach is that it is an approximate implementation of a structural risk minimization (SRM) induction principle that aims at minimizing a bound on the generalization error of a model, rather than minimizing only the mean square error over the data set. In this study, SVM predicts the factor of safety that has been modeled as a regression problem and stability status that has been modeled as a classification problem. For factor of safety prediction, SVM model gives better result than previously published result of ANN model. In case of stability status, SVM gives an accuracy of 85.71%.
25 Jun 2008
TL;DR: This paper presents a generalization of the nonlinear small-gain theorem for large-scale complex systems consisting of multiple input-to-output stable systems and includes as a special case the previous nonlinearsmall-gain theorems with two interconnected systems, and recent small- GainTheorems for networks of input- to-state stable subsystems.
Abstract: This paper presents a generalization of the nonlinear small-gain theorem for large-scale complex systems consisting of multiple input-to-output stable systems. It includes as a special case the previous nonlinear small-gain theorems with two interconnected systems, and recent small-gain theorems for networks of input-to-state stable subsystems. It is expected that this new small-gain theorem will find wide applications in the analysis and control synthesis of large-scale complex systems.
Proceedings Article•
8 Dec 2008TL;DR: These bounds hold in the scenario of dependent samples generated by a stationary β-mixing process, which is commonly adopted in many previous studies of non-i.i. i.d. settings and can be estimated from such finite samples and lead to tighter generalization bounds.
Abstract: This paper presents the first Rademacher complexity-based error bounds for non-i.i.d. settings, a generalization of similar existing bounds derived for the i.i.d. case. Our bounds hold in the scenario of dependent samples generated by a stationary β-mixing process, which is commonly adopted in many previous studies of non-i.i.d. settings. They benefit from the crucial advantages of Rademacher complexity over other measures of the complexity of hypothesis classes. In particular, they are data-dependent and measure the complexity of a class of hypotheses based on the training sample. The empirical Rademacher complexity can be estimated from such finite samples and lead to tighter generalization bounds. We also present the first margin bounds for kernel-based classification in this non-i.i.d. setting and briefly study their convergence.
7 Jul 2008
TL;DR: A fixedpoint algorithm for verifying safety properties of hybrid systems with differential equations whose right-hand sides are polynomials in the state variables is introduced and a saturation procedure that refines the system dynamics successively with differential invariants until safety becomes provable is introduced.
Abstract: We introduce a fixedpoint algorithm for verifying safety properties of hybrid systems with differential equations whose right-hand sides are polynomials in the state variables. In order to verify nontrivial systems without solving their differential equations and without numerical errors, we use a continuous generalization of induction, for which our algorithm computes the required differential invariants. As a means for combining local differential invariants into global system invariants in a sound way, our fixedpoint algorithm works with a compositional verification logic for hybrid systems. To improve the verification power, we further introduce a saturation procedurethat refines the system dynamics successively with differential invariants until safety becomes provable. By complementing our symbolic verification algorithm with a robust version of numerical falsification, we obtain a fast and sound verification procedure. We verify roundabout maneuvers in air traffic management and collision avoidance in train control.
TL;DR: A generalization of the original definition of rough sets and variable precision rough sets is introduced, based on the concept of absolute and relative rough membership, aimed at modeling data relationships expressed in terms of frequency distribution.
TL;DR: In this article, the authors studied the approximate controllability of the abstract evolution equations with nonlocal conditions in Hilbert spaces, and obtained sufficient conditions for the semilinear evolution equation.
Abstract: We study the approximate controllability for the abstract evolution equations with nonlocal conditions in Hilbert spaces. Assuming the approximate controllability of the corresponding linearized equation we obtain sufficient conditions for the approximate controllability of the semilinear evolution equation. The results we obtained are a generalization and continuation of the recent results on this issue. At the end, an example is given to show the application of our result.
TL;DR: In this paper, the authors propose selection criteria based on a fully Bayes formulation with a generalization of Zellner's $g$-prior which allows for $p>n.
Abstract: For the normal linear model variable selection problem, we propose selection criteria based on a fully Bayes formulation with a generalization of Zellner's $g$-prior which allows for $p>n$. A special case of the prior formulation is seen to yield tractable closed forms for marginal densities and Bayes factors which reveal new model evaluation characteristics of potential interest.
TL;DR: Two different models are presented on how protein domain combinations yield specific functions: one rule-based and one probabilistic, which were found to be better suited for incomplete training sets and more sensitive than a single-domain model on a large-scale dataset.
Abstract: Motivation: Computational assignment of protein function may be the single most vital application of bioinformatics in the post-genome era. These assignments are made based on various protein features, where one is the presence of identifiable domains. The relationship between protein domain content and function is important to investigate, to understand how domain combinations encode complex functions.
Results: Two different models are presented on how protein domain combinations yield specific functions: one rule-based and one probabilistic. We demonstrate how these are useful for Gene Ontology annotation transfer. The first is an intuitive generalization of the Pfam2GO mapping, and detects cases of strict functional implications of sets of domains. The second uses a probabilistic model to represent the relationship between domain content and annotation terms, and was found to be better suited for incomplete training sets. We implemented these models as predictors of Gene Ontology functional annotation terms. Both predictors were more accurate than conventional best BLAST-hit annotation transfer and more sensitive than a single-domain model on a large-scale dataset. We present a number of cases where combinations of Pfam-A protein domains predict functional terms that do not follow from the individual domains.
Availability: Scripts and documentation are available for download at http://sonnhammer.sbc.su.se/multipfam2go_source_docs.tar
Contact: Kristoffer.Forslund@sbc.su.se
Supplementary information:Supplementary data are available at Bioinformatics online.
TL;DR: This exploratory approach not only opens up an avenue to understand polyatomic reaction dynamics, even for motions at the molecular level in the fleeting transition-state region, but it also leads to a generalization of Polanyi's rules to reactions involving a polyatomic molecule.
Abstract: We report a comprehensive study of the quantum-state correlation property of product pairs from reactions of chlorine atoms with both the ground-state and the CH stretch-excited CHD3. In light of available ab initio theoretical results, this set of experimental data provides a conceptual framework to visualize the energy-flow pattern along the reaction path, to classify the activity of different vibrational modes in a reactive encounter, to gain deeper insight into the concept of vibrational adiabaticity, and to elucidate the intermode coupling in the transition-state region. This exploratory approach not only opens up an avenue to understand polyatomic reaction dynamics, even for motions at the molecular level in the fleeting transition-state region, but it also leads to a generalization of Polanyi's rules to reactions involving a polyatomic molecule.
TL;DR: A qualitative analysis of the solutions of three problems revealed two approaches to generalization: recursive-local and functional-global as discussed by the authors, which showed mental flexibility, shifting smoothly between pictorial, verbal and numerical representations and abandoning additive solution approaches in favor of more effective multiplicative strategies.
Abstract: This study focuses on the generalization methods used by talented pre-algebra students in solving linear and non-linear pattern problems. A qualitative analysis of the solutions of three problems revealed two approaches to generalization: recursive–local and functional–global. The students showed mental flexibility, shifting smoothly between pictorial, verbal and numerical representations and abandoning additive solution approaches in favor of more effective multiplicative strategies. Three forms of reflection aided generalization: reflection on commonalities in the pattern sequence’s structure, reflection on the generalization method, and reflection on the “tentative generalization” through verification of the pattern sequence. The latter indicates an intuitive grasp of the mathematical power of generalization. The students’ generalizations evinced algebraic thinking in the discovery of variables, constants and their mutual relations, and in the communication of these discoveries using invented algebraic notation. This study confirms the centrality of generalizations in mathematics and their potential as gateways to the world of algebra.
TL;DR: The authors discusses the content and structure of generalization involving figural patterns of middle school students, focusing on the extent to which they are capable of establishing and justifying complicated generalizations that entail possible overlap of aspects of the figures.
Abstract: This paper discusses the content and structure of generalization involving figural patterns of middle school students, focusing on the extent to which they are capable of establishing and justifying complicated generalizations that entail possible overlap of aspects of the figures. Findings from an ongoing 3-year longitudinal study of middle school students are used to extend the knowledge base in this area. Using pre-and post-interviews and videos of intervening teaching experiments, we specify three forms of generalization involving such figural linear patterns: constructive standard; constructive nonstandard; and deconstructive; and we classify these forms of generalization according to complexity based on student work. We document students’ cognitive tendency to shift from a figural to a numerical strategy in determining their figural-based patterns, and we observe the not always salutary consequences of such a shift in their representational fluency and inductive justifications.
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TL;DR: In this paper, a generalization of the concept of symmetric fuzzy measure based on a decomposition of the universal set in what we have called subsets of indifference is proposed, and some properties of these measures are studied, as well as their Choquet integral.
Abstract: In this paper we propose a generalization of the concept of symmetric fuzzy measure based in a decomposition of the universal set in what we have called subsets of indifference. Some properties of these measures are studied, as well as their Choquet integral. Finally, a degree of interaction between the subsets of indifference is defined.
TL;DR: Owing to the use of the Voronoi diagram the algorithm is parameter free and fully automatic and can be used in the generalization of point features arranged in clusters such as thematic dot maps and control points on cartographic maps.
TL;DR: It is argued that generalization is central to the enterprise of understanding categorization behavior, and some ways in which insights from machine learning can offer guidance are suggested.
Abstract: Exemplar theories of categorization depend on similarity for explaining subjects’ ability to generalize to new stimuli. A major criticism of exemplar theories concerns their lack of abstraction mechanisms and thus, seemingly, of generalization ability. Here, we use insights from machine learning to demonstrate that exemplar models can actually generalize very well. Kernel methods in machine learning are akin to exemplar models and are very successful in real-world applications. Their generalization performance depends crucially on the chosen similarity measure. Although similarity plays an important role in describing generalization behavior, it is not the only factor that controls generalization performance. In machine learning, kernel methods are often combined with regularization techniques in order to ensure good generalization. These same techniques are easily incorporated in exemplar models. We show that the generalized context model (Nosofsky, 1986) and ALCOVE (Kruschke, 1992) are closely related to a statistical model called kernel logistic regression. We argue that generalization is central to the enterprise of understanding categorization behavior, and we suggest some ways in which insights from machine learning can offer guidance.
Dissertation•
1 Jan 2008
TL;DR: Demaine and Barton as discussed by the authors gave an algorithm for computing a configuration above a given point on a given polynomial curve, running in time polylogarithmic in the size of the dense representation of the polynomial defining the curve, improving the previously known bounds of O(n) in two dimensions and O( n) in three dimensions.
Abstract: In 1876, A. B. Kempe presented a flawed proof of what is now called Kempe’s Universality Theorem: that the intersection of a closed disk with any curve in R defined by a polynomial equation can be drawn by a linkage. Kapovich and Millson published the first correct proof of this claim in 2002, but their argument relied on different, more complex constructions. We provide a corrected version of Kempe’s proof, using a novel contraparallelogram bracing. The resulting historical proof of Kempe’s Universality Theorem uses simpler gadgets than those of Kapovich and Millson. We use our two-dimensional proof of Kempe’s theorem to give simple proofs of two extensions of Kempe’s theorem first shown by King: a generalization to d dimensions and a characterization of the drawable subsets of R. Our results improve King’s by proving better continuity properties for the constructions. We prove that our construction requires only O(n) bars to draw a curve defined by a polynomial of degree n in d dimensions, improving the previously known bounds of O(n) in two dimensions and O(n) in three dimensions. We also prove a matching Ω(n) lower bound in the worst case. We give an algorithm for computing a configuration above a given point on a given polynomial curve, running in time polynomial in the size of the dense representation of the polynomial defining the curve. We use this algorithm to prove the coNP-hardness of testing the rigidity of a given configuration of a linkage. While this theorem has long been assumed in rigidity theory, we believe this to be the first published proof that this problem is computationally intractable. This thesis is joint work with Reid W. Barton and Erik D. Demaine. Thesis Supervisor: Erik D. Demaine Title: Esther & Harold Edgerton Associate Professor
5 Jul 2008
TL;DR: New theoretical insights are provided into these issues and a novel ν-SVC algorithm is proposed that has guaranteed generalization performance and convergence properties.
Abstract: The ν-support vector classification (ν-SVC) algorithm was shown to work well and provide intuitive interpretations, e.g., the parameter ν roughly specifies the fraction of support vectors. Although ν corresponds to a fraction, it cannot take the entire range between 0 and 1 in its original form. This problem was settled by a non-convex extension of ν-SVC and the extended method was experimentally shown to generalize better than original ν-SVC. However, its good generalization performance and convergence properties of the optimization algorithm have not been studied yet. In this paper, we provide new theoretical insights into these issues and propose a novel ν-SVC algorithm that has guaranteed generalization performance and convergence properties.
TL;DR: This article used the responses produced by a participant on one trial to generate the stimuli that either they or another participant will see on the next trial to reveal the inductive biases of these learners, as expressed in a prior probability distribution over hypotheses.
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TL;DR: In this paper, the initial-boundary value problems for an integrable generalization of the nonlinear Schrodinger equation formulated on the half-line were analyzed and the so-called linearizable boundary conditions, which in this case are of Robin type, were investigated.
Abstract: We analyze initial-boundary value problems for an integrable generalization of the nonlinear Schrodinger equation formulated on the half-line. In particular, we investigate the so-called linearizable boundary conditions, which in this case are of Robin type. Furthermore, we use a particular solution to verify explicitly all the steps needed for the solution of a well-posed problem.
TL;DR: This paper presents an approach to automated building grouping and generalization based on three principles of Gestalt theories, and six parameters selected to describe spatial patterns, distributions and relations of buildings.
Abstract: This paper presents an approach to automated building grouping and generalization. Three principles of Gestalt theories, i.e. proximity, similarity, and common directions, are employed as guidelines, and six parameters, i.e. minimum distance, area of visible scope, area ratio, edge number ratio, smallest minimum bounding rectangle (SMBR), directional Voronoi diagram (DVD), are selected to describe spatial patterns, distributions and relations of buildings. Based on these principles and parameters, an approach to building grouping and generalization is developed. First, buildings are triangulated based on Delaunay triangulation rules, by which topological adjacency relations between buildings are obtained and the six parameters are calculated and recorded. Every two topologically adjacent buildings form a potential group. Three criteria from previous experience and Gestalt principles are employed to tell whether a 2-building group is `strong,' `average' or `weak.' The `weak' groups are deleted from the group array. Secondly, the retained groups with common buildings are organized to form intermediate groups according to their relations. After this step, the intermediate groups with common buildings are aggregated or separated and the final groups are formed. Finally, appropriate operators/algorithms are selected for each group and the generalized buildings are achieved. This approach is fully automatic. As our experiments show, it can be used primarily in the generalization of buildings arranged in blocks.