Abstract: We propose a theory of eigenvalues, eigenvectors, singular values, and singular vectors for tensors based on a constrained variational approach much like the Rayleigh quotient for symmetric matrix eigenvalues. These notions are particularly useful in generalizing certain areas where the spectral theory of matrices has traditionally played an important role. For illustration, we will discuss a multilinear generalization of the Perron-Frobenius theorem.

Abstract: A central goal of statistical language modeling is to learn the joint probability function of sequences of words in a language. This is intrinsically difficult because of the curse of dimensionality: a word sequence on which the model will be tested is likely to be different from all the word sequences seen during training. Traditional but very successful approaches based on n-grams obtain generalization by concatenating very short overlapping sequences seen in the training set. We propose to fight the curse of dimensionality by learning a distributed representation for words which allows each training sentence to inform the model about an exponential number of semantically neighboring sentences. Generalization is obtained because a sequence of words that has never been seen before gets high probability if it is made of words that are similar (in the sense of having a nearby representation) to words forming an already seen sentence. Training such large models (with millions of parameters) within a reasonable time is itself a significant challenge. We report on several methods to speed-up both training and probability computation, as well as comparative experiments to evaluate the improvements brought by these techniques. We finally describe the incorporation of this new language model into a state-of-the-art speech recognizer of conversational speech.

Abstract: An explicit reciprocal transformation between a two-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

Abstract: Solutions of learning problems by Empirical Risk Minimization (ERM) – and almost-ERM when the minimizer does not exist – need to be consistent, so that they may be predictive. They also need to be well-posed in the sense of being stable, so that they might be used robustly. We propose a statistical form of stability, defined as leave-one-out (LOO) stability. We prove that for bounded loss classes LOO stability is (a) sufficient for generalization, that is convergence in probability of the empirical error to the expected error, for any algorithm satisfying it and, (b) necessary and sufficient for consistency of ERM. Thus LOO stability is a weak form of stability that represents a sufficient condition for generalization for symmetric learning algorithms while subsuming the classical conditions for consistency of ERM. In particular, we conclude that a certain form of well-posedness and consistency are equivalent for ERM.

Abstract: Plant–Pollinator Interactions: From Specialization to Generalization.—Nickolas M. Waser and Jeff Ollerton [editors]. 2006. University of Chicago Press, Chicago, Illinois. 445 pp. ISBN 0-226-87400-1. $45.00 (paper). In 1985 I was present at a lecture in which Nat Wheelwright proposed that the mutualism between seed-dispersing birds and fruit-bearing plants was characterized by lack of specialization in both partners. He gave convincing evidence for the paucity of obligate one-to-one specialization, and explained the reason why extreme specializations in these systems ought to be rare (Wheelwright and Orians 1982). In the discussion after the lecture, Pete Feinsinger expressed a similar view of plant–pollinator systems. He suggested that pollination systems shared with dispersal systems a dearth of tight dependence, one-to-one coevolution, and reciprocal specialization. For over two decades, I assumed that Wheelwright’s and Feinsinger’s perspective was orthodoxy; my reading of the animal–plant interactions literature did not dispel this notion. John Thompson (1994), one of the influential voices in the field, wrote: ‘‘...extreme specialization occurs even less commonly in free-living mutualisms (such as pollination and seed-dispersal [emphasis added]) than in predators and grazers’’ (p. 178). Hence, I was interested when I read the title of this edited volume: Plant–Pollinator Interactions: From Specialization to Generalization. Maybe it was time to revise my oldfashioned notions and embrace a new paradigm— perhaps one in which specialization in plant–pollinator systems was more important than I had believed. I was wrong. From the cover illustrations to the final page, the book is a thorough beating of what I thought was a very dead horse. A common theme that infuses most—albeit not all—chapters, is the perceived need to dispel the ‘‘...conventional wisdom shared by most evolutionary biologists... [that] ...specialization is advantageous and...a crucial feature of many pollination systems’’ (p. 145, chapter 7). Declaring that the book is a proclamation of a new paradigm that emphasizes generalization over specialization in pollination ecology is both tardy and unfortunate. One does not need a straw man to make the question of specialization and generalization in plant–pollinator systems interesting. Science often advances not as a result of new observations and experiments, but when someone clears up confusing and imprecise jargon. For example, Lavoisier’s clarification of chemical nomenclature revolutionized chemistry (Sánchez-Ron 2002). Arguably, the rapid growth of phylogenetic systematics is a consequence of Willy Hennig’s effort to purge this discipline’s language of ambiguity (Richter and Meier 1994). Waser and Ollerton’s (2006) book foregoes an opportunity to define the meaning, or rather the many possible meanings, of ‘‘specialized’’ and ‘‘generalized.’’ Waser and Ollerton justify their decision not to have a chapter that defines and elucidates these terms: ‘‘...such a chapter may imply that its authors have a corner on the proper definitions or wish to impose their view on other authors or readers of the book, neither of which is true’’ (p. xi). Although the tolerance of diversity of thought implied in such a view may be commendable under other circumstances, in a scientific text it is not. In 1790, Lavoisier (quoting Abbé de Condillac) noted that ‘‘We think only through the medium of words... The art of reasoning is nothing more than a language well arranged’’ (Lavoisier 1965:xiii). Failing to define scientific terms unambiguously condemns us to use them inaccurately. The chapters’ authors sometimes define the terms specialized and generalized, but more often leave it to the reader to divine the meaning of their words. The confusion that results can baffle readers, and can be damaging to readers who are only beginning to explore the ecological literature. It may give them the impression, perhaps sometimes justified, that ecologists are not conscientious about their vocabulary. This failure to define terms led some of the chapters’ authors to conflate unrelated concepts. For example, several chapters meld the idea of ‘‘pollination syndromes’’ with that of specialization, and imply that by rejecting one, the reader must reject the other (see Chapters 1, 4, 7, 10, and 14, and Waser et al. 1996, the original paper that initiated this mix-up). The hypothesis of ‘‘pollination syndromes’’ asserts that suites of certain floral traits, such as color, shape, timing of nectar production, and odors, associate the bearers of these traits with discrete groups of pollinators. Thus, plants pollinated by hummingbirds should be red, tubular, pendant, odorless, open during daylight, and secrete concentrated nectar. In contrast, plants pollinated by bats should be pale, open at night, smell like butyric acid, and secrete copious dilute nectar. Faegri and van der Pijl’s (1979) much-criticized, but still relevant, text lists putative pollination syndromes and their characteristics. However, syndromes and specialized mutualisms are not logically linked. This is best illustrated with an example. Imagine a group of plants characterized as birdpollinated; the bean-tree genus Erythrina is a good case in point. (Bruneau 1997). The number of avian pollinator species that visits each Erythrina species can range from one to very many. Furthermore,

Abstract: The meta-analysis of coefficient alpha across many studies is becoming more common in psychology by a methodology labeled reliability generalization. Existing reliability generalization studies have not used the sampling distribution of coefficient alpha for precision weighting and other common meta-analytic procedures. A framework is provided for a statistically grounded meta-analysis of coefficient alpha using its sampling distribution. Two empirical examples are offered to illustrate these methods, and limitations of reliability generalization are described.

Abstract: One of the most widely studied systems of argumentation is the one described by Dung in a paper from 1995. Unfortunately, this framework does not allow for joint attacks on arguments, which we argue must be required of any truly abstract argumentation framework. A few frameworks can be said to allow for such interactions among arguments, but for various reasons we believe that these are inadequate for modelling argumentation systems with joint attacks. In this paper we propose a generalization of the framework of Dung, which allows for sets of arguments to attack other arguments. We extend the semantics associated with the original framework to this generalization, and prove that all results in the paper by Dung have an equivalent in this more abstract framework.

Abstract: Biomechanical research into artistic gymnastics has grown substantially over the years. However, most research is still skill oriented with few tries at generalization. Consequently, our understanding of the principles and bases of the sport, although improved, is still marginal with gaps in knowledge about technique attributes throughout the sport. For that reason, this review begins with an attempt to identify important variables contributing to successful performance. The review is presented in clusters of work in similar apparatuses culminating in Tables offering an 'at a glance' summary of knowledge in each cluster. The last section of the review tries to give some direction to future biomechanical research in gymnastics in issues relating to data collection--two-dimensional or three-dimensional, image size, frame rate--and analysis, such as descriptive or explanatory, simulation and optimization, and statistical issues.

Abstract: Quantum theory can be regarded as a noncommutative generalization of classical probability. From this point of view, one expects quantum dynamics to be analogous to classical conditional probabilities. In this paper, a variant of the well-known isomorphism between completely positive maps and bipartite density operators is derived, which makes this connection much more explicit. This isomorphism is given an operational interpretation in terms of statistical correlations between ensemble preparation procedures and outcomes of measurements. Finally, the isomorphism is applied to elucidate the connection between no-cloning and no-broadcasting theorems and the monogamy of entanglement, and a simplified proof of the no-broadcasting theorem is obtained as a by-product.

Abstract: We present a canonical form for a natural and necessary generalization of the Lambert W function, natural in that it requires minimal mathematical definitions for this generalization, and necessary in that it provides a means of expressing solutions to a number of physical problems of fundamental nature. This generalization expresses the exact solutions for general-relativistic self-gravitating N-body systems in one spatial and one time dimension, and a previously unknown mathematical link between the (1+1) gravity problem and the Schrodinger wave equation.

Abstract: We describe an approach to the inductive synthesis of recursive equations from input/output-examples which is based on the classical two-step approach to induction of functional Lisp programs of Summers (1977). In a first step, I/O-examples are rewritten to traces which explain the outputs given the respective inputs based on a datatype theory. These traces can be integrated into one conditional expression which represents a non-recursive program. In a second step, this initial program term is generalized into recursive equations by searching for syntactical regularities in the term. Our approach extends the classical work in several aspects. The most important extensions are that we are able to induce a set of recursive equations in one synthesizing step, the equations may contain more than one recursive call, and additionally needed parameters are automatically introduced.

Abstract: A three-parameter generalization of the Weibull distribution is presented to deal with general situations in modeling survival process with various shapes in the hazard function. This generalized W...

Abstract: Recently much work has been done analyzing online machine learning algorithms in a worst case setting, where no probabilistic assumptions are made about the data. This is analogous to the H/sup /spl infin// setting used in adaptive linear filtering. Bregman divergences have become a standard tool for analyzing online machine learning algorithms. Using these divergences, we motivate a generalization of the least mean squared (LMS) algorithm. The loss bounds for these so-called p-norm algorithms involve other norms than the standard 2-norm. The bounds can be significantly better if a large proportion of the input variables are irrelevant, i.e., if the weight vector we are trying to learn is sparse. We also prove results for nonstationary targets. We only know how to apply kernel methods to the standard LMS algorithm (i.e., p=2). However, even in the general p-norm case, we can handle generalized linear models where the output of the system is a linear function combined with a nonlinear transfer function (e.g., the logistic sigmoid).

Abstract: In this paper we give an explicit solution to the rank constrained matrix approximation in Frobenius norm, which is a generalization of the classical approximation of an m by n matrix A by a matrix of rank k at most.

Abstract: A variety of techniques are described by which keyword sets and target audience profiles may be generalized in a systematic and effective way with reference to relationships between keywords, profiles, and the data of an underlying user population.

Abstract: A semi-simple tensor extension of the Poincare algebra is proposed for the arbitrary dimensions $D$. A supersymmetric also semi-simple generalization of this extension is constructed in the D=4 dimensions. This paper is dedicated to the memory of Anna Yakovlevna Gelyukh.

Abstract: A way to simulate the basic interactions between two individuals with different opinions, in the context of strategic game theory, is proposed. Various games are considered, which produce different kinds of opinion formation dynamics. First, by assuming that all individuals (players) are equals, we obtain the bounded confidence model of continuous opinion dynamics proposed by Deffuant et al. In such a model a tolerance threshold is defined, such that individuals with difference in opinion larger than the threshold can not interact. Then, we consider that the individuals have different inclinations to change opinion and different abilities in convincing the others. In this way, we obtain the so-called ``Stubborn individuals and Orators'' (SO) model, a generalization of the Deffuant et al. model, in which the threshold tolerance is different for every couple of individuals. We explore, by numerical simulations, the dynamics of the SO model, and we propose further generalizations that can be implemented.

Abstract: We consider the fractional generalization of nonholonomic constraints defined by equations with fractional derivatives and provide some examples. The corresponding equations of motion are derived using variational principle. We prove that fractional constraints can be used to describe the evolution of dynamical systems in which some coordinates and velocities are related to velocities through a power-law memory function.

Abstract: Active learning refers to algorithmic frameworks aimed at selecting training data points in order to reduce the number of required training data points and/or improve the generalization performance of a learning method. In this paper, we present an asymptotic analysis of active learning for generalized linear models. Our analysis holds under the common practical situation of model misspecification, and is based on realistic assumptions regarding the nature of the sampling distributions, which are usually neither independent nor identical. We derive unbiased estimators of generalization performance, as well as estimators of expected reduction in generalization error after adding a new training data point, that allow us to optimize its sampling distribution through a convex optimization problem. Our analysis naturally leads to an algorithm for sequential active learning which is applicable for all tasks supported by generalized linear models (e.g., binary classification, multi-class classification, regression) and can be applied in non-linear settings through the use of Mercer kernels.

Abstract: Maxwell models for nonlinear kinetic equations have many applications in physics, dynamics of granular gases, economy, etc. In the present manuscript we consider such models from a very general point of view, including those with arbitrary polynomial non-linearities and in any dimension space. It is shown that the whole class of generalized Maxwell models satisfies properties which one of them can be interpreted as an operator generalization of usual Lipschitz conditions. This property allows to describe in detail a behavior of solutions to the corresponding initial value problem. In particular, we prove in the most general case an existence of self similar solutions and study the convergence, in the sense of probability measures, of dynamically scaled solutions to the Cauchy problem to those self-similar solutions, as time goes to infinity. The properties of these self-similar solutions, leading to non classical equilibrium stable states, are studied in detail. We apply the results to three different specific problems related to the Boltzmann equation (with elastic and inelastic interactions) and show that all physically relevant properties of solutions follow directly from the general theory developed in this paper.

Abstract: We introduce a new class of functions called strongly θ-b-continuous function which is a generalization of both strongly θ-precontinuous functions [16]and strongly θ-semicontinuous functions [7]. Some characterizations and several properties concerning strongly θ-b-continuous functions are obtained.

Abstract: We introduce the concept of left APP-rings which is a generalization of left p.q.-Baer rings and right PP-rings, and investigate its properties. It is shown that the APP property is inherited by polynomial extensions and is a Morita invariant property.

Abstract: Herein, we present a canonical form for a natural and necessary generalization of the Lambert W function, natural in that it requires minimal mathematical definitions for this generalization, and necessary in that it provides a means of expressing solutions to a number of physical problems of fundamental nature. In particular, this generalization expresses the exact solutions for general-relativistic self-gravitating 2-body and 3-body systems in one spatial and one time dimension. It also expresses the solution to a previously unknown mathematical link between the lineal gravity problem and the Schroedinger equation.

Abstract: When neural networks (NNs) are used to identify tool conditions, the richness and size of training data are crucial. The training data set not only has to cover a wide range of cutting conditions, but also to capture the characteristics of the tool wear process. This data set imposes significant computing burdens, results in a complex identification model, and hampers the feasible application of NNs. In this paper, a training data selection method is proposed, and a systematic procedure is provided to perform this data selection. With this method, the generalization error surface is divided into three regions, and proper sampling factors are chosen for each region to prune the data points from the original training set. The quality of the training set is estimated by performance evaluation through decision making. In this work, SVM is used in the decision making method, and the generalization error is used as the performance evaluation criterion. The tradeoff between the generalization performance and the size of the training set is key to this selection. Experimental results have demonstrated that this selection strategy provides an effective and efficient training set, and the developed model based on this set is fast and reliable for tool condition identification.

Abstract: We consider a fractional generalization of gradient systems. We use differential forms and exterior derivatives of fractional orders. Examples of fractional gradient systems are considered. We describe the stationary states of these systems.