Showing papers on "Hierarchy (mathematics) published in 2012"

A lasso for hierarchical interactions

Abstract: We add a set of convex constraints to the lasso to produce sparse interaction models that honor the hierarchy restriction that an interaction only be included in a model if one or both variables are marginally important. We give a precise characterization of the effect of this hierarchy constraint, prove that hierarchy holds with probability one and derive an unbiased estimate for the degrees of freedom of our estimator. A bound on this estimate reveals the amount of fitting "saved" by the hierarchy constraint. We distinguish between parameter sparsity - the number of nonzero coefficients - and practical sparsity - the number of raw variables one must measure to make a new prediction. Hierarchy focuses on the latter, which is more closely tied to important data collection concerns such as cost, time and effort. We develop an algorithm, available in the R package hierNet, and perform an empirical study of our method.

Hierarchy measure for complex networks.

Abstract: Nature, technology and society are full of complexity arising from the intricate web of the interactions among the units of the related systems (e.g., proteins, computers, people). Consequently, one of the most successful recent approaches to capturing the fundamental features of the structure and dynamics of complex systems has been the investigation of the networks associated with the above units (nodes) together with their relations (edges). Most complex systems have an inherently hierarchical organization and, correspondingly, the networks behind them also exhibit hierarchical features. Indeed, several papers have been devoted to describing this essential aspect of networks, however, without resulting in a widely accepted, converging concept concerning the quantitative characterization of the level of their hierarchy. Here we develop an approach and propose a quantity (measure) which is simple enough to be widely applicable, reveals a number of universal features of the organization of real-world networks and, as we demonstrate, is capable of capturing the essential features of the structure and the degree of hierarchy in a complex network. The measure we introduce is based on a generalization of the m-reach centrality, which we first extend to directed/partially directed graphs. Then, we define the global reaching centrality (GRC), which is the difference between the maximum and the average value of the generalized reach centralities over the network. We investigate the behavior of the GRC considering both a synthetic model with an adjustable level of hierarchy and real networks. Results for real networks show that our hierarchy measure is related to the controllability of the given system. We also propose a visualization procedure for large complex networks that can be used to obtain an overall qualitative picture about the nature of their hierarchical structure.

What is a Classification

Abstract: A classification is a hierarchy of objects that conforms to the following principles: 1. The classes (groups with members) of the hierarchy have a set of properties or rules that extend to every member of the class and to all of the subclasses of the class, to the exclusion of all other [unrelated] classes. A subclass is itself a type of class wherein the members have the defining class properties of the parent class plus some additional property(ies) specific for the subclass.

Optimality Conditions and Finite Convergence of Lasserre's Hierarchy

Abstract: Lasserre's hierarchy is a sequence of semidefinite relaxations for solving polynomial optimization problems globally. This paper studies the relationship between optimality conditions in nonlinear programming theory and finite convergence of Lasserre's hierarchy. Our main results are: i) Lasserre's hierarchy has finite convergence when the constraint qualification, strict complementarity and second order sufficiency conditions hold at every global minimizer, under the standard archimedean assumption; the proof uses a result of Marshall on boundary hessian conditions. ii) these optimality conditions are all satisfied at every local minimizer if a finite set of polynomials, which are in the coefficients of input polynomials, do not vanish at the input data (i.e., they hold in a Zariski open set). This implies that Lasserre's hierarchy has finite convergence generically.

An Infinite Latent Attribute Model for Network Data

Abstract: Latent variable models for network data extract a summary of the relational structure underlying an observed network. The simplest possible models subdivide nodes of the network into clusters; the probability of a link between any two nodes then depends only on their cluster assignment. Currently available models can be classified by whether clusters are disjoint or are allowed to overlap. These models can explain a "flat" clustering structure. Hierarchical Bayesian models provide a natural approach to capture more complex dependencies. We propose a model in which objects are characterised by a latent feature vector. Each feature is itself partitioned into disjoint groups (subclusters), corresponding to a second layer of hierarchy. In experimental comparisons, the model achieves significantly improved predictive performance on social and biological link prediction tasks. The results indicate that models with a single layer hierarchy over-simplify real networks.

Multiple Criteria Hierarchy Process in Robust Ordinal Regression

Abstract: A great majority of methods designed for Multiple Criteria Decision Aiding (MCDA) assume that all evaluation criteria are considered at the same level, however, it is often the case that a practical application is imposing a hierarchical structure of criteria. The hierarchy helps decomposing complex decision making problems into smaller and manageable subtasks, and thus, it is very attractive for users. To handle the hierarchy of criteria in MCDA, we propose a methodology called Multiple Criteria Hierarchy Process (MCHP) which permits consideration of preference relations with respect to a subset of criteria at any level of the hierarchy. MCHP can be applied to any MCDA method. In this paper, we apply MCHP to Robust Ordinal Regression (ROR) being a family of MCDA methods that takes into account all sets of parameters of an assumed preference model, which are compatible with preference information elicited by a Decision Maker (DM). As a result of ROR, one gets necessary and possible preference relations in the set of alternatives, which hold for all compatible sets of parameters or for at least one compatible set of parameters, respectively. Applying MCHP to ROR one gets to know not only necessary and possible preference relations with respect to the whole set of criteria, but also necessary and possible preference relations related to subsets of criteria at different levels of the hierarchy. We also show how MCHP can be extended to handle group decision and interactions among criteria.

Hierarchical POMDP Controller Optimization by Likelihood Maximization

Abstract: Planning can often be simpli ed by decomposing the task into smaller tasks arranged hierarchically. Charlin et al. [4] recently showed that the hierarchy discovery problem can be framed as a non-convex optimization problem. However, the inherent computational di culty of solving such an optimization problem makes it hard to scale to realworld problems. In another line of research, Toussaint et al. [18] developed a method to solve planning problems by maximumlikelihood estimation. In this paper, we show how the hierarchy discovery problem in partially observable domains can be tackled using a similar maximum likelihood approach. Our technique rst transforms the problem into a dynamic Bayesian network through which a hierarchical structure can naturally be discovered while optimizing the policy. Experimental results demonstrate that this approach scales better than previous techniques based on non-convex optimization.

Block type symmetry of bigraded Toda hierarchy

Abstract: In this paper, we define Orlov-Schulman's operators ML, MR, and then use them to construct the additional symmetries of the bigraded Toda hierarchy. We further show that these additional symmetries form an interesting infinite-dimensional Lie algebra known as a Block type Lie algebra, whose structure theory and representation theory have recently received much attention in the literature. By acting on two different spaces under the weak W-constraints, we find in particular two representations of this Block type Lie algebra.

A formal hierarchy of weak memory models

Abstract: We present in this paper a formal generic framework, implemented in the Coq proof assistant, for defining and reasoning about weak memory models. We first present the three axioms of our framework, with several examples as illustration and justification. Then we show how to implement several existing weak memory models in our framework, and prove formally that our implementation is equivalent to the native definition for each of these models.

Provably-Secure Time-Bound Hierarchical Key Assignment Schemes

Abstract: A time-bound hierarchical key assignment scheme is a method to assign time-dependent encryption keys to a set of classes in a partially ordered hierarchy, in such a way that each class can compute the keys of all classes lower down in the hierarchy, according to temporal constraints. In this paper we design and analyze time-bound hierarchical key assignment schemes which are provably-secure and efficient. We consider two different goals: security with respect to key indistinguishability and against key recovery. Moreover, we distinguish security against static and adaptive adversarial behaviors. We explore the relations between all possible combinations of security goals and adversarial behaviors and, in particular, we prove that security against adaptive adversaries is (polynomially) equivalent to security against static adversaries. Finally, we propose two different constructions for time-bound key assignment schemes. The first one is based on symmetric encryption schemes, whereas the second one makes use of bilinear maps. Both constructions support updates to the access hierarchy with local changes to the public information and without requiring any private information to be re-distributed.

The limits of determinacy in second-order arithmetic

Abstract: We establish the precise bounds for the amount of determinacy provable in second order arithmetic. We show that for every natural number n, second order arithmetic can prove that determinacy holds for Boolean combinations of n many 0 classes, but it cannot prove that all nite Boolean combinations of 0 classes are determined. More specically, we prove that 1+2-CA 0' n- 0-DET, but that 1+2-CA 0 n- 0-DET, where n- 0 is the nth level in the dierence hierarchy of 0 classes. We also show some conservativity results that imply that reversals for the theorems above are not possible. We prove that for every true 1 sentence T (as for instance n- 0 -DET ) and every n 2, 1 -CA0 +T + 1 -TI 0 1 -CA0 and 1 1 -CA0 + T + 1 -TI 0 1 -CA0.

Dispersionless bigraded toda hierarchy and its additional symmetry

Abstract: In this paper, we firstly give the definition of dispersionless bigraded Toda hierarchy (dBTH) and introduce some Sato theory on dBTH. Then we define Orlov–Schulman's , operator and give the additional Block symmetry of dBTH. Meanwhile we give tau function of dBTH and some related dispersionless bilinear equations.

Systems, Devices, and/or Methods for Managing Data

Abstract: Certain exemplary embodiments can provide a system, machine, device, manufacture, circuit, and/or user interface adapted for, and/or a method and/or machine-readable medium comprising machine-implementable instructions for, activities that can comprise: via a hierarchy processor implemented via a predetermined information device: for each of a plurality of records of a data structure, displaying data corresponding to a hierarchically displayable field according to a hierarchy location field containing a corresponding hierarchy location identifier indicative of a display location of that data in a displayable hierarchical data structure.

Anonymizing set-valued data by nonreciprocal recoding

Abstract: Today there is a strong interest in publishing set-valued data in a privacy-preserving manner. Such data associate individuals to sets of values (e.g., preferences, shopping items, symptoms, query logs). In addition, an individual can be associated with a sensitive label (e.g., marital status, religious or political conviction). Anonymizing such data implies ensuring that an adversary should not be able to (1) identify an individual's record, and (2) infer a sensitive label, if such exists. Existing research on this problem either perturbs the data, publishes them in disjoint groups disassociated from their sensitive labels, or generalizes their values by assuming the availability of a generalization hierarchy. In this paper, we propose a novel alternative. Our publication method also puts data in a generalized form, but does not require that published records form disjoint groups and does not assume a hierarchy either; instead, it employs generalized bitmaps and recasts data values in a nonreciprocal manner; formally, the bipartite graph from original to anonymized records does not have to be composed of disjoint complete subgraphs. We configure our schemes to provide popular privacy guarantees while resisting attacks proposed in recent research, and demonstrate experimentally that we gain a clear utility advantage over the previous state of the art.

An Infinite Latent Attribute Model for Network Data

Abstract: Latent variable models for network data extract a summary of the relational structure underlying an observed network. The simplest possible models subdivide nodes of the network into clusters; the probability of a link between any two nodes then depends only on their cluster assignment. Currently available models can be classified by whether clusters are disjoint or are allowed to overlap. These models can explain a "flat" clustering structure. Hierarchical Bayesian models provide a natural approach to capture more complex dependencies. We propose a model in which objects are characterised by a latent feature vector. Each feature is itself partitioned into disjoint groups (subclusters), corresponding to a second layer of hierarchy. In experimental comparisons, the model achieves significantly improved predictive performance on social and biological link prediction tasks. The results indicate that models with a single layer hierarchy over-simplify real networks.

The Hierarchy of Equivalence Relations on the Natural Numbers Under Computable Reducibility

Abstract: The notion of computable reducibility between equivalence relations on the natural numbers provides a natural computable analogue of Borel reducibility We investigate the computable reducibility hierarchy, comparing and contrasting it with the Borel reducibility hierarchy from descriptive set theory Meanwhile, the notion of computable reducibility appears well suited for an analysis of equivalence relations on the ce sets, and more specifically, on various classes of ce structures This is a rich context with many natural examples, such as the isomorphism relation on ce graphs or on computably presented groups Here, our exposition extends earlier work in the literature concerning the classification of computable structures An abundance of open questions remains

Investigating the role of hierarchy on the strength of composite materials: evidence of a crucial synergy between hierarchy and material mixing

Abstract: Natural materials are often organized in complex hierarchical architectures to optimize mechanical properties. Artificial bio-inspired materials, however, have thus far failed to successfully mimic how these architectures improve material characteristics, for example strength. Here, a method is proposed for evaluating the role of hierarchy on structural strength. To do this, we consider different hierarchical architectures of fiber bundles through analytical multiscale calculations based on a fiber bundle model at each hierarchical level. In general, we find that an increase in the number of hierarchy levels leads to a decrease in the strength of material. However, when a composite bundle with two different types of fibers is considered, an improvement in the mean strength is obtained for some specific hierarchical architectures, indicating that both hierarchy and material “mixing” are necessary ingredients to obtain improved mechanical properties. Results are promising for the improvement and “tuning” of the strength of bio-inspired materials.

The mathematical relationship between Zipf’s law and the hierarchical scaling law

Abstract: The empirical studies of city-size distribution show that Zipf’s law and the hierarchical scaling law are linked in many ways. The rank-size scaling and hierarchical scaling seem to be two different sides of the same coin, but their relationship has never been revealed by strict mathematical proof. In this paper, the Zipf’s distribution of cities is abstracted as a q -sequence. Based on this sequence, a self-similar hierarchy consisting of many levels is defined and the numbers of cities in different levels form a geometric sequence. An exponential distribution of the average size of cities is derived from the hierarchy. Thus we have two exponential functions, from which follows a hierarchical scaling equation. The results can be statistically verified by simple mathematical experiments and observational data of cities. A theoretical foundation is then laid for the conversion from Zipf’s law to the hierarchical scaling law, and the latter can show more information about city development than the former. Moreover, the self-similar hierarchy provides a new perspective for studying networks of cities as complex systems. A series of mathematical rules applied to cities such as the allometric growth law, the 2 n principle and Pareto’s law can be associated with one another by the hierarchical organization.

Second order logic or set theory

Abstract: We try to answer the question which is the “right” foundation of mathematics, second order logic or set theory. Since the former is usually thought of as a formal language and the latter as a first order theory, we have to rephrase the question. We formulate what we call the second order view and a competing set theory view, and then discuss the merits of both views. On the surface these two views seem to be in manifest conflict with each other. However, our conclusion is that it is very difficult to see any real difference between the two. We analyze a phenomenonwe call internal categoricity which extends the familiar categoricity results of second order logic to Henkin models and show that set theory enjoys the same kind of internal categoricity. Thus the existence of non-standard models, which is usually taken as a property of first order set theory, and categoricity, which is usually taken as a property of second order axiomatizations, can coherently coexist when put into their proper context. We also take a fresh look at complete second order axiomatizations and give a hierarchy result for second order characterizable structures. Finally we consider the problem of existence in mathematics from both points of view and find that second order logic depends on what we call large domain assumptions, which come quite close to the meaning of the axioms of set theory.

A multi-criteria decision-making methodology on the selection of facility location: fuzzy ANP

Abstract: Analytical ways to reach the best decisions are the most preferable issues in many business platforms. During the decision processes, besides the measurable variables, there exist qualitative variables, especially if the decision is based on a selection problem. Analytic hierarchy process (AHP) and analytic network process (ANP) are two of the best ways to decide among the complex criteria structure in different levels using qualitative variables. When there are interactions between the criteria in different levels of the hierarchy, then AHP cannot be used because of their one-way direction of hierarchy; the ANP has been developed for this kind of need. In this study, a fuzzy ANP method is developed for a multi-criteria facility location selection problem where the criteria set includes interactions with each other on the hierarchy structure. Besides the fuzzy ANP model development and implementation on facility location, sensitivity analysis was also originally performed to indicate the upper and lower bounds for the importance levels of alternative locations.

Information escaping the correlation hierarchy of the convergence field in the study of cosmological parameters.

Abstract: Using fits to numerical simulations, we show that the entire hierarchy of moments quickly ceases to provide a complete description of the convergence one-point probability density function leaving the linear regime. This suggests that the full N-point correlation function hierarchy of the convergence field becomes quickly generically incomplete and a very poor cosmological probe on nonlinear scales. At the scale of unit variance, only 5% of the Fisher information content of the one-point probability density function is still contained in its hierarchy of moments, making clear that information escaping the hierarchy is a far stronger effect than information propagating to higher order moments. It follows that the constraints on cosmological parameters achievable through extraction of the entire hierarchy become suboptimal by large amounts. A simple logarithmic mapping makes the moment hierarchy well suited again for parameter extraction.

An approach to separating the levels of hierarchical structure building in language and mathematics

Abstract: We aimed to dissociate two levels of hierarchical structure building in language and mathematics, namely ‘first-level’ (the build-up of hierarchical structure with externally given elements) and ‘second-level’ (the build-up of hierarchical structure with internally represented elements produced by first-level processes). Using functional magnetic resonance imaging, we investigated these processes in three domains: sentence comprehension, arithmetic calculation (using Reverse Polish notation, which gives two operands followed by an operator) and a working memory control task. All tasks required the build-up of hierarchical structures at the first- and second-level, resulting in a similar computational hierarchy across language and mathematics, as well as in a working memory control task. Using a novel method that estimates the difference in the integration cost for conditions of different trial durations, we found an anterior-to-posterior functional organization in the prefrontal cortex, according to the level of hierarchy. Common to all domains, the ventral premotor cortex (PMv) supports first-level hierarchy building, while the dorsal pars opercularis (POd) subserves second-level hierarchy building, with lower activation for language compared with the other two tasks. These results suggest that the POd and the PMv support domain-general mechanisms for hierarchical structure building, with the POd being uniquely efficient for language.

Multilevel structured additive regression

Abstract: Models with structured additive predictor provide a very broad and rich framework for complex regression modeling They can deal simultaneously with nonlinear covariate effects and time trends, unit- or cluster-specific heterogeneity, spatial heterogeneity and complex interactions between covariates of different type In this paper, we propose a hierarchical or multilevel version of regression models with structured additive predictor where the regression coefficients of a particular nonlinear term may obey another regression model with structured additive predictor In that sense, the model is composed of a hierarchy of complex structured additive regression models The proposed model may be regarded as an extended version of a multilevel model with nonlinear covariate terms in every level of the hierarchy The model framework is also the basis for generalized random slope modeling based on multiplicative random effects Inference is fully Bayesian and based on Markov chain Monte Carlo simulation techniques We provide an in depth description of several highly efficient sampling schemes that allow to estimate complex models with several hierarchy levels and a large number of observations within a couple of minutes (often even seconds) We demonstrate the practicability of the approach in a complex application on childhood undernutrition with large sample size and three hierarchy levels

The FO^2 alternation hierarchy is decidable

Abstract: We consider the two-variable fragment FO^2[<] of first-order logic over finite words. Numerous characterizations of this class are known. Th\'erien and Wilke have shown that it is decidable whether a given regular language is definable in FO^2[<]. From a practical point of view, as shown by Weis, FO^2[<] is interesting since its satisfiability problem is in NP. Restricting the number of quantifier alternations yields an infinite hierarchy inside the class of FO^2[<]-definable languages. We show that each level of this hierarchy is decidable. For this purpose, we relate each level of the hierarchy with a decidable variety of finite monoids. Our result implies that there are many different ways of climbing up the FO^2[<]-quantifier alternation hierarchy: deterministic and co-deterministic products, Mal'cev products with definite and reverse definite semigroups, iterated block products with J-trivial monoids, and some inductively defined omega-term identities. A combinatorial tool in the process of ascension is that of condensed rankers, a refinement of the rankers of Weis and Immerman and the turtle programs of Schwentick, Th\'erien, and Vollmer.

Using the Gene Ontology Hierarchy when Predicting Gene Function

Abstract: The problem of multilabel classification when the labels are related through a hierarchical categorization scheme occurs in many application domains such as computational biology. For example, this problem arises naturally when trying to automatically assign gene function using a controlled vocabularies like Gene Ontology. However, most existing approaches for predicting gene functions solve independent classification problems to predict genes that are involved in a given function category, independently of the rest. Here, we propose two simple methods for incorporating information about the hierarchical nature of the categorization scheme. In the first method, we use information about a gene's previous annotation to set an initial prior on its label. In a second approach, we extend a graph-based semi-supervised learning algorithm for predicting gene function in a hierarchy. We show that we can efficiently solve this problem by solving a linear system of equations. We compare these approaches with a previous label reconciliation-based approach. Results show that using the hierarchy information directly, compared to using reconciliation methods, improves gene function prediction.

On profiles and footprints --- relational semantics for petri nets

Abstract: Petri net systems have been successfully applied for modelling business processes and analysing their behavioural properties. In this domain, analysis techniques that are grounded on behavioural relations defined between pairs of transitions emerged recently. However, different use cases motivated different definitions of behavioural relation sets. This paper focusses on two prominent examples, namely behavioural profiles and behavioural footprints. We show that both represent different ends of a spectrum of relation sets for Petri net systems, each inducing a different equivalence class. As such, we provide a generalisation of the known relation sets. We illustrate that different relation sets complement each other for general systems, but form an abstraction hierarchy for distinguished net classes. For these net classes, namely S-WF-systems and sound free-choice WF-systems, we also prove a close relation between the structure and the relational semantics. Finally, we discuss implications of our results for the field of business process modelling and analysis.

Macro-class selection for hierarchical k-nn classification of inertial sensor data

Abstract: Quality classifiers can be difficult to implement on the limited resources of an embedded system, especially if the data contains many confusing classes. This can be overcome by using a hierarchical set of classifiers in which specialized feature sets are used at each node to distinguish within the macro-classes defined by the hierarchy. This method exploits the fact that similar classes according to one feature set may be dissimilar according to another, allowing normally confused classes to be grouped and handled separately. However, determining these macro-classes of similarity is not straightforward when the selected feature set has yet to be determined. In this paper, we present a new greedy forward selection algorithm to simultaneously determine good macro-classes and the features that best distinguish them. The algorithm is tested on two human activity recognition datasets: CMU-MMAC (29 classes), and a custom dataset collected from a commodity smartphone for this paper (9 classes). In both datasets, we employ statistical features obtained from on-body IMU sensors. Classification accuracy using the selected macro-classes was increased 69% and 12% respectively over our non-hierarchical baselines.

A Dynamic Interpretation of the CPS Hierarchy

Abstract: The CPS hierarchy of control operators shift i /reset i of Danvy and Filinski is a natural generalization of the shift and reset static control operators that allow for abstracting delimited control in a structured and CPS-guided manner In this article we show that a dynamic variant of shift/reset, known as shift 0/reset 0, where the discipline of static access to the stack of delimited continuations is relaxed, can fully express the CPS hierarchy This result demonstrates the expressive power of shift 0 /reset 0 and it offers a new perspective on practical applications of the CPS hierarchy