Showing papers on "Equivalence class published in 2013"

Enumerating Markov Equivalence Classes of Acyclic Digraph Models

Abstract: Graphical Markov models determined by acyclic digraphs (ADGs), also called directed acyclic graphs (DAGs), are widely studied in statistics, computer science (as Bayesian networks), operations research (as influence diagrams), and many related fields. Because different ADGs may determine the same Markov equivalence class, it long has been of interest to determine the efficiency gained in model specification and search by working directly with Markov equivalence classes of ADGs rather than with ADGs themselves. A computer program was written to enumerate the equivalence classes of ADG models as specified by Pearl & Verma's equivalence criterion. The program counted equivalence classes for models up to and including 10 vertices. The ratio of number of classes to ADGs appears to approach an asymptote of about 0.267. Classes were analyzed according to number of edges and class size. By edges, the distribution of number of classes approaches a Gaussian shape. By class size, classes of size 1 are most common, with the proportions for larger sizes initially decreasing but then following a more irregular pattern. The maximum number of classes generated by any undirected graph was found to increase approximately factorially. The program also includes a new variation of orderly algorithm for generating undirected graphs.

The Langlands quotient theorem for finite central extensions of p-adic groups

Abstract: In this paper, we prove the Langlands quotient theorem in the context of finite central extensions of connected, reductive p-adic groups.

Learning Equivalence Classes of Bayesian Networks Structures

Abstract: Approaches to learning Bayesian networks from data typically combine a scoring function with a heuristic search procedure. Given a Bayesian network structure, many of the scoring functions derived in the literature return a score for the entire equivalence class to which the structure belongs. When using such a scoring function, it is appropriate for the heuristic search algorithm to search over equivalence classes of Bayesian networks as opposed to individual structures. We present the general formulation of a search space for which the states of the search correspond to equivalence classes of structures. Using this space, any one of a number of heuristic search algorithms can easily be applied. We compare greedy search performance in the proposed search space to greedy search performance in a search space for which the states correspond to individual Bayesian network structures.

Fuzzy relation equations and subsystems of fuzzy transition systems

Abstract: In this paper we study subsystems, reverse subsystems and double subsystems of a fuzzy transition system. We characterize them in terms of fuzzy relation inequalities and equations, as eigen fuzzy sets of the fuzzy quasi-order Q"@d and the fuzzy equivalence E"@d generated by fuzzy transition relations, and as linear combinations of aftersets and foresets of Q"@d and equivalence classes of E"@d. We also show that subsystems, reverse subsystems and double subsystems of a fuzzy transition system T form both closure and opening systems in the lattice of fuzzy subsets of A, where A is the set of states of T, and we provide efficient procedures for computing related closures and openings of an arbitrary fuzzy subset of A. These procedures boil down to computing the fuzzy quasi-order Q"@d or the fuzzy equivalence E"@d, which can be efficiently computed using the well-known algorithms for computing the transitive closure of a fuzzy relation.

The bag-of-repeats representation of documents

Abstract: n-gram representations of documents may improve over a simple bag-of-word representation by relaxing the independence assumption of word and introducing context. However, this comes at a cost of adding features which are non-descriptive, and increasing the dimension of the vector space model exponentially. We present new representations that avoid both pitfalls. They are based on sound theoretical notions of stringology, and can be computed in optimal asymptotic time with algorithms using data structures from the suffix family. While maximal repeats have been used in the past for similar tasks, we show how another equivalence class of repeats -- largest-maximal repeats -- obtain similar or better results, with only a fraction of the features. This class acts as a minimal generative basis of all repeated substrings. We also report their use for topic modeling, showing easier to interpret models.

On Arithmetic Modular Categories

Abstract: Modular categories are important algebraic structures in a variety of subjects in mathematics and physics. We provide an explicit, motivated and elementary definition of a modular category over a field of characteristic 0 as an equivalence class of solutions to a set of polynomial equations. We conclude that within each class of solutions, there is one which consists entirely of algebraic numbers. These algebraic solutions make it possible to discuss defining algebraic number fields of modular categories and their Galois twists. One motivation for such a definition is an arithmetic theory of modular categories which plays an important role in their classification. Another is to facilitate implementation of computer-based tools to resolve computational and classification problems intractible by other means. We observe some basic properties of Galois twists of modular categories and make conjectures about their relation to the the intrinsic data of modular categories.

On the taxonomy of optimization problems under estimation of distribution algorithms

Abstract: Understanding the relationship between a search algorithm and the space of problems is a fundamental issue in the optimization field. In this paper, we lay the foundations to elaborate taxonomies of problems under estimation of distribution algorithms EDAs. By using an infinite population model and assuming that the selection operator is based on the rank of the solutions, we group optimization problems according to the behavior of the EDA. Throughout the definition of an equivalence relation between functions it is possible to partition the space of problems in equivalence classes in which the algorithm has the same behavior. We show that only the probabilistic model is able to generate different partitions of the set of possible problems and hence, it predetermines the number of different behaviors that the algorithm can exhibit. As a natural consequence of our definitions, all the objective functions are in the same equivalence class when the algorithm does not impose restrictions to the probabilistic model. The taxonomy of problems, which is also valid for finite populations, is studied in depth for a simple EDA that considers independence among the variables of the problem. We provide the sufficient and necessary condition to decide the equivalence between functions and then we develop the operators to describe and count the members of a class. In addition, we show the intrinsic relation between univariate EDAs and the neighborhood system induced by the Hamming distance by proving that all the functions in the same class have the same number of local optima and that they are in the same ranking positions. Finally, we carry out numerical simulations in order to analyze the different behaviors that the algorithm can exhibit for the functions defined over the search space .

Cardinality estimation using spanning trees

Abstract: A system, computer-implemented method, and computer-program product embodiments for determining a cardinality estimate for a query. A cardinality estimator identifies a predicate in a query, where the predicate is split into a plurality of equivalence classes. The cardinality estimator then generates a plurality of equivalence graphs from the plurality of equivalence classes, one equivalence graph for an equivalence class. Spanning trees are identified from the plurality of equivalence graphs, and the cardinality estimator then determines the cardinality estimate for the query from the spanning trees.

Geometric characterizations of big cycles

Abstract: A numerical equivalence class of k-cycles is said to be big if it lies in the interior of the closed cone generated by effective classes. We develop several geometric criteria that distinguish big classes from boundary classes. In particular, we construct for arbitrary cycle classes an analogue of the volume function for divisors.

Derived Equivalence Classification of the Cluster-Tilted Algebras of Dynkin Type E

Abstract: We obtain a complete derived equivalence classification of the cluster-tilted algebras of Dynkin type E. There are 67, 416, 1574 algebras in types E 6, E 7 and E 8 which turn out to fall into 6, 14, 15 derived equivalence classes, respectively. This classification can be achieved computationally and we outline an algorithm which has been implemented to carry out this task. We also make the classification explicit by giving standard forms for each derived equivalence class as well as complete lists of the algebras contained in each class; as these lists are quite long they are provided as supplementary material to this paper. From a structural point of view the remarkable outcome of our classification is that two cluster-tilted algebras of Dynkin type E are derived equivalent if and only if their Cartan matrices represent equivalent bilinear forms over the integers which in turn happens if and only if the two algebras are connected by a sequence of “good” mutations. This is reminiscent of the derived equivalence classification of cluster-tilted algebras of Dynkin type A, but quite different from the situation in Dynkin type D where a far-reaching classification has been obtained using similar methods as in the present paper but some very subtle questions are still open.

Four quotient set gems

Abstract: Our aim in this note is to present four remarkable facts about quotient sets. These observations seem to have been overlooked by the Monthly, despite its intense coverage of quotient sets over the years.

Automatically Propagating Updates in a Data Center

Abstract: Techniques, systems, and articles of manufacture for automatically propagating updates in a data center. A method includes dividing multiple virtual machines in a data center into one or more equivalence classes, wherein each equivalence class is based on a signature corresponding to an offline, online and/or manual update, automatically creating an offline and/or online virtual machine manifest for a first virtual machine of the multiple virtual machines based on one or more file system changes during the offline, online and/or manual update for the first virtual machine, wherein said manifest is applicable to each additional virtual machine in the same equivalence class as the first virtual machine, and applying the offline and/or online virtual machine manifest for the first virtual machine to the remaining multiple virtual machines in the same equivalence class to automatically update the remaining multiple virtual machines in the same equivalence class.

Algebras of acyclic cluster type: Tree type and type Ã

Abstract: In this paper, we study algebras of global dimension at most 2 whose generalized cluster category is equivalent to the cluster category of an acyclic quiver which is either a tree or of type A. We are particularly interested in their derived equivalence classification. We prove that each algebra which is cluster equivalent to a tree quiver is derived equivalent to the path algebra of this tree. Then we describe explicitly the algebras of cluster type A n for each possible orientation of A n . We give an explicit way to read off the derived equivalence class in which such an algebra lies, and we describe the Auslander-Reiten quiver of its derived category. Together, these results in particular provide a complete classification of algebras which are cluster equivalent to tame acyclic quivers.

On the bounded approximation property in Banach spaces

Abstract: We prove that the kernel of a quotient operator from an L 1-space onto a Banach space X with the Bounded Approximation Property (BAP) has the BAP. This completes earlier results of Lusky-case l 1-and Figiel, Johnson and Pelczynski-case X* separable. Given a Banach space X, we show that if the kernel of a quotient map from some L 1-space onto X has the BAP, then every kernel of every quotient map from any L 1-space onto X has the BAP. The dual result for L ∞-spaces also holds: if for some L ∞-space E some quotient E/X has the BAP, then for every L ∞-space E every quotient E/X has the BAP.

Non locality, topology, formal languages: new global tools to handle large data sets

Abstract: The basic idea that stems out of this work is that large sets of data can be handled through an organized set of mathematical and computational tools rooted in a global geometric vision of data space allowing to explore the structure and hidden information patterns thereof. Based on this perspective, the objective is naturally that of discovering and letting emerge, directly from probing the data space, the manifold hidden relations (patterns), e.g. correlations among facts, interactions among entities, relations among concepts and formally describing, in a semantic mining context, the discovered information. In this note, we propose an approach that exploits topological methods for classifying global information into equivalence classes and regular languages for describing the corresponding automaton as element an of hidden complex system.

Counting Permutations Modulo Pattern-Replacement Equivalences for Three-Letter Patterns

Abstract: We study a family of equivalence relations on $S_n$, the group of permutations on $n$ letters, created in a manner similar to that of the Knuth relation and the forgotten relation. For our purposes, two permutations are in the same equivalence class if one can be reached from the other through a series of pattern-replacements using patterns whose order permutations are in the same part of a predetermined partition of $S_c$. When the partition is of $S_3$ and has one nontrivial part and that part is of size greater than two, we provide formulas for the number of classes created in all cases left unresolved by past authros. When the partition is of $S_3$ and has two nontrivial parts, each of size two (as do the Knuth and forgotten relations), we enumerate the classes for 13 of the 14 unresolved cases. In two of these cases, enumerations arise which are the same as those yielded by the Knuth and forgotten relations. The reasons for this phenomenon are still largely a mystery.

Automated abstract planning with use of genetic algorithms

Abstract: The paper presents a new approach based on nature inspired algorithms to an abstract planning problem, which is a part of the web service composition problem. An abstract plan is defined as an equivalence class of sequences of the same service types that satisfy a user query. The objective of our genetic algorithm (GA) is to return representatives of abstract plans without generating all the equivalent sequences.

Equivalence Classes in $S_n$ for Three Families of Pattern-Replacement Relations

Abstract: We study a family of equivalence relations on $S_n$, the group of permutations on $n$ letters, created in a manner similar to that of the Knuth relation and the forgotten relation. For our purposes, two permutations are in the same equivalence class if one can be reached from the other through a series of pattern-replacements using patterns whose order permutations are in the same part of a predetermined partition of $S_c$. In particular, we are interested in the number of classes created in $S_n$ by each relation and in characterizing these classes. Imposing the condition that the partition of $S_c$ has one nontrivial part containing the cyclic shifts of a single permutation, we find enumerations for the number of nontrivial classes. When the permutation is the identity, we are able to compare the sizes of these classes and connect parts of the problem to Young tableaux and Catalan lattice paths. Imposing the condition that the partition has one nontrivial part containing all of the permutations in $S_c$ beginning with 1, we both enumerate and characterize the classes in $S_n$. We do the same for the partition that has two nontrivial parts, one containing all of the permutations in $S_c$ beginning with 1, and one containing all of the permutations in $S_c$ ending with 1.

Grouping and automatically propagating updates to equivalent online and offline virtual machines in a data center

Abstract: Techniques, systems, and articles of manufacture for automatically propagating updates in a data center. A method includes dividing multiple virtual machines in a data center into one or more equivalence classes, wherein each equivalence class is based on a signature corresponding to an offline, online and/or manual update, automatically creating an offline and/or online virtual machine manifest for a first virtual machine of the multiple virtual machines based on one or more file system changes during the offline, online and/or manual update for the first virtual machine, wherein said manifest is applicable to each additional virtual machine in the same equivalence class as the first virtual machine, and applying the offline and/or online virtual machine manifest for the first virtual machine to the remaining multiple virtual machines in the same equivalence class to automatically update the remaining multiple virtual machines in the same equivalence class.

Succinct Data Structures for Representing Equivalence Classes

Abstract: Given a partition of an n element set into equivalence classes, we consider time-space tradeoffs for representing it to support the query that asks whether two given elements are in the same equivalence class. This has various applications including for testing whether two vertices are in the same connected component in an undirected graph or in the same strongly connected component in a directed graph.

On the normed space of equivalence classes of fuzzy numbers.

Abstract: We study the norm induced by the supremum metric on the space of fuzzy numbers. And then we propose a method for constructing a norm on the quotient space of fuzzy numbers. This norm is very natural and works well with the induced metric on the quotient space.

Evolutionary Algorithms for Abstract Planning

Abstract: The paper presents a new approach based on evolutionary algorithms to an abstract planning problem, which is the first stage of the web service composition problem. An abstract plan is defined as an equivalence class of sequences of service types that satisfy a user query. Two sequences are equivalent if they are composed of the same service types, but not necessarily occurring in the same order. The objective of our genetic algorithm (GA) is to return representatives of abstract plans without generating all the equivalent sequences. Experimental results are presented and compared with these obtained using an SMT-solver, showing that GA finds solutions for very large sets of service types in a reasonable and shorter time.

A Hybrid Anytime Algorithm for the Constructiion of Causal Models From Sparse Data

Abstract: We present a hybrid constraint-based/Bayesian algorithm for learning causal networks in the presence of sparse data. The algorithm searches the space of equivalence classes of models (essential graphs) using a heuristic based on conventional constraint-based techniques. Each essential graph is then converted into a directed acyclic graph and scored using a Bayesian scoring metric. Two variants of the algorithm are developed and tested using data from randomly generated networks of sizes from 15 to 45 nodes with data sizes ranging from 250 to 2000 records. Both variations are compared to, and found to consistently outperform two variations of greedy search with restarts.

Efficient Identification of Equivalences in Dynamic Graphs and Pedigree Structures

Abstract: We propose a new framework for designing test and query functions for complex structures that vary across a given parameter such as genetic marker position. The operations we are interested in include equality testing, set operations, isolating unique states, duplication counting, or finding equivalence classes under identifiability constraints. A motivating application is locating equivalence classes in identity-by-descent (IBD) graphs, graph structures in pedigree analysis that change over genetic marker location. The nodes of these graphs are unlabeled and identified only by their connecting edges, a constraint easily handled by our approach. The general framework introduced is powerful enough to build a range of testing functions for IBD graphs, dynamic populations, and other structures using a minimal set of operations. The theoretical and algorithmic properties of our approach are analyzed and proved. Computational results on several simulations demonstrate the effectiveness of our approach.

A generalisation of the nonlinear small-gain theorem for systems with abstract initial conditions

Abstract: We consider the development of a general nonlinear small-gain theorem for systems with abstract initial conditions. Systems are defined in a set theoretic manner from input-output pairs on a doubly infinite time axis, and a general construction of the initial conditions (i.e. a state at time zero) is given in terms of an equivalence class of trajectories on the negative time axis. By using this formulation, an ISS-type nonlinear small-gain theorem is established with complete disconnection between the stability property and the existence, uniqueness properties. We provide an illustrative example.

Derivation of generalized test cases

Abstract: A first computer receives one or more pre-defined equivalence classes, wherein a pre-defined equivalence class comprises of one or more substantially equivalent values. The first computer receives a plurality of messages transmitted between a second computer and a third computer, wherein each message has at least one parameter and each parameter has at least one corresponding value. The first computer determines one or more parameters have one or more values that match one or more values of a pre-defined equivalence class. The first computer creates one or more value driven equivalence classes, wherein each value driven equivalence class comprises of one or more parameters and wherein each of the one or more parameters in each value driven equivalence class has the same corresponding value. The first computer creates a generalized test case, wherein the generalized test case includes at least the one or more value driven equivalence classes.

On the classification of nonsingular 2×2×2×2 hypercubes

Abstract: As a first step in the classification of nonsingular 2×2×2×2 hypercubes up to equivalence, we resolve the case where the base field is finite and the hypercubes can be written as a product of two 2×2×2 hypercubes. (Nonsingular hypercubes were introduced by D. Knuth in the context of semifields. Where semifields are related to hypercubes of dimension 3, this paper considers the next case, i.e., hypercubes of dimension 4.) We define the notion of ij-rank (with 1 ≤ i < j ≤ 4) and prove that a hypercube is the product of two 2×2×2 hypercubes if and only if its 12-rank is at most 2. We derive a ‘standard form’ for nonsingular 2×2×2×2 hypercubes of 12-rank less than 4 as a first step in the classification of such hypercubes up to equivalence. Our main result states that the equivalence class of a nonsingular 2×2×2×2 hypercube M of 12-rank 2 depends only on the value of an invariant δ0(M) which derives in a natural way from the Cayley hyperdeterminant det0M and another polynomial invariant det M of degree 4. As a corollary we prove that the number of equivalence classes is (q + 1)/2 or q/2 depending on whether the order q of the field is odd or even.

Building Quotient Cube with MapReduce in Hadoop

Abstract: In order to solve the problem that how to improve the efficiency of query and calculation in massive data, a method of building quotient cubes in Hadoop plateform which combined the advantage of the quotient cube and MapReduce model is proposed in this paper. At first, all cubes will be established and their aggregate value will be calculated in the Mapping stage. All the key/value pair formed in Mapping stage will be passed to Reducing stage. Equivalence partitioning will be carried out In this stage, and the minimum aggregation cube of each equivalence partitioning will be the key with its aggregate value. According to the minimum aggregation cubes, we can get the quotient cubes. In order to improve the speed of parallel computing and reduce network traffic, equivalence class division will be executed locally after the Map stage, it is named as combiner stage. In this paper, MapReduce model is used to improve the efficiency of building quotient cube because of its ability of parallel computing in a large amount of data. In addition, the experiment proved that, under certain circumstances, increasing the number of Mapper/Reducer task can reduce the building time effectively, and improve the construction efficiency.