Showing papers on "Hierarchy (mathematics) published in 2017"

WordSup: Exploiting Word Annotations for Character Based Text Detection

Abstract: Imagery texts are usually organized as a hierarchy of several visual elements, i.e. characters, words, text lines and text blocks. Among these elements, character is the most basic one for various languages such as Western, Chinese, Japanese, mathematical expression and etc. It is natural and convenient to construct a common text detection engine based on character detectors. However, training character detectors requires a vast of location annotated characters, which are expensive to obtain. Actually, the existing real text datasets are mostly annotated in word or line level. To remedy this dilemma, we propose a weakly supervised framework that can utilize word annotations, either in tight quadrangles or the more loose bounding boxes, for character detector training. When applied in scene text detection, we are thus able to train a robust character detector by exploiting word annotations in the rich large-scale real scene text datasets, e.g. ICDAR15 [19] and COCO-text [39]. The character detector acts as a key role in the pipeline of our text detection engine. It achieves the state-of-the-art performance on several challenging scene text detection benchmarks. We also demonstrate the flexibility of our pipeline by various scenarios, including deformed text detection and math expression recognition.

A robust ranking method extending ELECTRE III to hierarchy of interacting criteria, imprecise weights and stochastic analysis

Abstract: A great majority of methods designed for Multiple Criteria Decision Aiding (MCDA) assume that all assessment criteria are considered at the same level, however, decision problems encountered in practice often impose a hierarchical structure of criteria. The hierarchy helps to decompose complex decision problems into smaller and manageable subtasks, and thus, it is very attractive for computational efficiency and explanatory purposes. To handle the hierarchy of criteria in MCDA, a methodology called Multiple Criteria Hierarchy Process (MCHP), has been recently proposed. MCHP permits to consider preference relations with respect to a subset of criteria at any level of the hierarchy. Here, we propose to apply MCHP to the ELECTRE III ranking method adapted to handle three types of interaction effects between criteria: mutual-weakening, mutual-strengthening and antagonistic effect. We also involve in MCHP an imprecise elicitation of criteria weights, generalizing a technique called the SRF method. In order to explore the plurality of rankings obtained by the ELECTRE III method for possible sets of criteria weights, we apply the Stochastic Multiobjective Acceptability Analysis (SMAA) that permits to draw robust conclusions in terms of rankings and preference relations at each level of the hierarchy of criteria. The novelty of the whole methodology consists of a joint consideration of hierarchical assessments of alternatives performances on interacting criteria, imprecise criteria weights, and robust analysis of ranking recommendations resulting from ELECTRE III. An example regarding the multiple criteria ranking of some European universities will show how to apply the proposed methodology on a decision problem.

A bounded degree SOS hierarchy for polynomial optimization

Abstract: We consider a new hierarchy of semidefinite relaxations for the general polynomial optimization problem $(P):\:f^{\ast}=\min \{\,f(x):x\in K\,\}$ on a compact basic semi-algebraic set $K\subset\R^n$. This hierarchy combines some advantages of the standard LP-relaxations associated with Krivine's positivity certificate and some advantages of the standard SOS-hierarchy. In particular it has the following attractive features: (a) In contrast to the standard SOS-hierarchy, for each relaxation in the hierarchy, the size of the matrix associated with the semidefinite constraint is the same and fixed in advance by the user. (b) In contrast to the LP-hierarchy, finite convergence occurs at the first step of the hierarchy for an important class of convex problems. Finally (c) some important techniques related to the use of point evaluations for declaring a polynomial to be zero and to the use of rank-one matrices make an efficient implementation possible. Preliminary results on a sample of non convex problems are encouraging.

Mean Field Control Hierarchy

Abstract: In this paper we model the role of a government of a large population as a mean field optimal control problem. Such control problems are constrained by a PDE of continuity-type, governing the dynamics of the probability distribution of the agent population. We show the existence of mean field optimal controls both in the stochastic and deterministic setting. We derive rigorously the first order optimality conditions useful for numerical computation of mean field optimal controls. We introduce a novel approximating hierarchy of sub-optimal controls based on a Boltzmann approach, whose computation requires a very moderate numerical complexity with respect to the one of the optimal control. We provide numerical experiments for models in opinion formation comparing the behavior of the control hierarchy.

Solutions of the nonlocal nonlinear Schr\"odinger hierarchy via reduction

Abstract: In this letter we propose an approach to obtain solutions for the nonlocal nonlinear Schr\"{o}dinger hierarchy from the known ones of the Ablowitz-Kaup-Newell-Segur hierarchy by reduction. These solutions are presented in terms of double Wronskian and some of them are new.The approach is general and can be used for other systems with double Wronskian solutions which admit local and nonlocal reductions.

Hierarchical Segmentation Using Tree-Based Shape Spaces

Abstract: Current trends in image segmentation are to compute a hierarchy of image segmentations from fine to coarse. A classical approach to obtain a single meaningful image partition from a given hierarchy is to cut it in an optimal way, following the seminal approach of the scale-set theory. While interesting in many cases, the resulting segmentation, being a non-horizontal cut, is limited by the structure of the hierarchy. In this paper, we propose a novel approach that acts by transforming an input hierarchy into a new saliency map. It relies on the notion of shape space: a graph representation of a set of regions extracted from the image. Each region is characterized with an attribute describing it. We weigh the boundaries of a subset of meaningful regions (local minima) in the shape space by extinction values based on the attribute. This extinction-based saliency map represents a new hierarchy of segmentations highlighting regions having some specific characteristics. Each threshold of this map represents a segmentation which is generally different from any cut of the original hierarchy. This new approach thus enlarges the set of possible partition results that can be extracted from a given hierarchy. Qualitative and quantitative illustrations demonstrate the usefulness of the proposed method.

Diagonal gates in the Clifford hierarchy

Abstract: The Clifford hierarchy is a set of gates that appears in the theory of fault-tolerant quantum computation, but its precise structure remains elusive. We give a complete characterization of the diagonal gates in the Clifford hierarchy for prime-dimensional qudits. They turn out to be $p^{m}$-th roots of unity raised to polynomial functions of the basis state to which they are applied, and we determine which level of the Clifford hierarchy a given gate sits in based on $m$ and the degree of the polynomial.

Coherent Probabilistic Forecasts for Hierarchical Time Series

Abstract: Many applications require forecasts for a hierarchy comprising a set of time series along with aggregates of subsets of these series Although forecasts can be produced independently for each series in the hierarchy, typically this does not lead to coherent forecasts -- the property that forecasts add up appropriately across the hierarchy State-of-the-art hierarchical forecasting methods usually reconcile the independently generated forecasts to satisfy the aggregation constraints A fundamental limitation of prior research is that it has considered only the problem of forecasting the mean of each time series We consider the situation where probabilistic forecasts are needed for each series in the hierarchy We define forecast coherency in this setting, and propose an algorithm to compute predictive distributions for each series in the hierarchy Our algorithm has the advantage of synthesizing information from different levels in the hierarchy through a sparse forecast combination and a probabilistic hierarchical aggregation We evaluate the accuracy of our forecasting algorithm on both simulated data and large-scale electricity smart meter data The results show consistent performance gains compared to state-of-the art methods

Operational framework for quantum measurement simulability

Abstract: We introduce a framework for simulating quantum measurements based on classical processing of a set of accessible measurements. Well-known concepts such as joint measurability and projective simulability naturally emerge as particular cases of our framework, but our study also leads to novel results and questions. First, a generalisation of joint measurability is derived, which yields a hierarchy for the incompatibility of sets of measurements. A similar hierarchy is defined based on the number of outcomes necessary to perform a simulation of a given measurement. This general approach also allows us to identify connections between different kinds of simulability and, in particular, we characterise the qubit measurements that are projective-simulable in terms of joint measurability. Finally, we discuss how our framework can be interpreted in the context of resource theories.

Hierarchical multi-label classification with chained neural networks

Abstract: In classification tasks, an object usually belongs to one class within a set of disjoint classes In more complex tasks, an object can belong to more than one class, in what is conventionally termed multi-label classification Moreover, there are cases in which the set of classes are organised in a hierarchical fashion, and an object must be associated to a single path in this hierarchy, defining the so-called hierarchical classification Finally, in even more complex scenarios, the classes are organised in a hierarchical structure and the object can be associated to multiple paths of this hierarchy, defining the problem investigated in this article: hierarchical multi-label classification (HMC) We address a typical problem of HMC, which is protein function prediction, and for that we propose an approach that chains multiple neural networks, performing both local and global optimisation in order to provide the final prediction: one or multiple paths in the hierarchy of classes We experiment with four variations of this chaining process, and we compare these strategies with the state-of-the-art HMC algorithms for protein function prediction, showing that our novel approach significantly outperforms these methods

Lasserre hierarchy for large scale polynomial optimization in real and complex variables

Abstract: We propose general notions to deal with large scale polynomial optimization problems and demonstrate their efficiency on a key industrial problem of the twenty first century, namely the optimal power flow problem. These notions enable us to find global minimizers on instances with up to 4,500 variables and 14,500 constraints. First, we generalize the Lasserre hierarchy from real to complex to numbers in order to enhance its tractability when dealing with complex polynomial optimization. Complex numbers are typically used to represent oscillatory phenomena, which are omnipresent in physical systems. Using the notion of hyponormality in operator theory, we provide a finite convergence criterion which generalizes the Curto-Fialkow conditions of the real Lasserre hierarchy. Second, we introduce the multi-ordered Lasserre hierarchy in order to exploit sparsity in polynomial optimization problems (in real or complex variables) while preserving global convergence. It is based on two ideas: 1) to use a different relaxation order for each constraint, and 2) to iteratively seek a closest measure to the truncated moment data until a measure matches the truncated data. Third and last, we exhibit a block diagonal structure of the Lasserre hierarchy in the presence of commonly encountered symmetries. To the best of our knowledge, the Lasserre hierarchy was previously limited to small scale problems, while we solve a large scale industrial problem with thousands of variables and constraints to global optimality.

Ruling out Higher-Order Interference from Purity Principles

Abstract: As first noted by Rafael Sorkin, there is a limit to quantum interference. The interference pattern formed in a multi-slit experiment is a function of the interference patterns formed between pairs of slits; there are no genuinely new features resulting from considering three slits instead of two. Sorkin has introduced a hierarchy of mathematically conceivable higher-order interference behaviours, where classical theory lies at the first level of this hierarchy and quantum theory theory at the second. Informally, the order in this hierarchy corresponds to the number of slits on which the interference pattern has an irreducible dependence. Many authors have wondered why quantum interference is limited to the second level of this hierarchy. Does the existence of higher-order interference violate some natural physical principle that we believe should be fundamental? In the current work we show that such principles can be found which limit interference behaviour to second-order, or “quantum-like”, interference, but that do not restrict us to the entire quantum formalism. We work within the operational framework of generalised probabilistic theories, and prove that any theory satisfying Causality, Purity Preservation, Pure Sharpness, and Purification—four principles that formalise the fundamental character of purity in nature—exhibits at most second-order interference. Hence these theories are, at least conceptually, very “close” to quantum theory. Along the way we show that systems in such theories correspond to Euclidean Jordan algebras. Hence, they are self-dual and, moreover, multi-slit experiments in such theories are described by pure projectors.

Context-Associative Hierarchical Memory Model for Human Activity Recognition and Prediction

Abstract: Human activity recognition is a challenging high-level vision task, for which multiple factors, such as subject, object, and their diverse interactions, have to be considered and modeled. Current learning-based methods are limited in the capability to integrate human-level concepts into an easily extensible computational framework. Inspired by the existing human memory model, we present a context-associative approach to recognize activity with human-object interaction. The proposed system can recognize incoming visual content based on the previous experienced activities. The high-level activity is parsed into consecutive subactivities, and we build a context cluster to model the temporal relations. The semantic attributes of the subactivity are organized by a concept hierarchy. Based on the hierarchy, a series of similarity functions are defined to turn the recognition computing into retrievals over the contextual memory, similar to the auto-associative characteristics of human memory. Partially matching in retrieval and stored memory make the activity prediction possible. The dynamical evolution of the brain memory is mimicked to allow decay and reinforcement of the input information, providing a natural way to maintain data and save computational time. We evaluate our approach on three data sets: CAD-120, MHOI, and OPPORTUNITY. The proposed method demonstrates promising results compared with other state-of-the-art techniques.

New solitary wave solutions to the (2+1)-dimensional Calogero–Bogoyavlenskii–Schiff and the Kadomtsev–Petviashvili hierarchy equations

Abstract: In this paper, with the help of Wolfram Mathematica 9 we employ the powerful sine-Gordon expansion method in investigating the solution structures of the two well known nonlinear evolution equations, namely; Calogero–Bogoyavlenskii–Schiff and Kadomtsev–Petviashvili hierarchy equations We obtain new solutions with complex, hyperbolic and trigonometric function structures All the obtained solutions in this paper verified their corresponding equations We also plot the three- and two-dimensional graphics of all the obtained solutions in this paper by using the same program in Wolfram Mathematica 9 We finally submit a comprehensive conclusion

An Improved Semidefinite Programming Hierarchy for Testing Entanglement

Abstract: We present a stronger version of the Doherty–Parrilo–Spedalieri (DPS) hierarchy of approximations for the set of separable states. Unlike DPS, our hierarchy converges exactly at a finite number of rounds for any fixed input dimension. This yields an algorithm for separability testing that is singly exponential in dimension and polylogarithmic in accuracy. Our analysis makes use of tools from algebraic geometry, but our algorithm is elementary and differs from DPS only by one simple additional collection of constraints.

Coherent Probabilistic Forecasts for Hierarchical Time Series.

Abstract: Many applications require forecasts for a hierarchy comprising a set of time series along with aggregates of subsets of these series. Hierarchical forecasting require not only good prediction accuracy at each level of the hierarchy, but also the coherency between different levels — the property that forecasts add up appropriately across the hierarchy. A fundamental limitation of prior research is the focus on forecasting the mean of each time series. We consider the situation where probabilistic forecasts are needed for each series in the hierarchy, and propose an algorithm to compute predictive distributions rather than mean forecasts only. Our algorithm has the advantage of synthesizing information from different levels in the hierarchy through a sparse forecast combination and a probabilistic hierarchical aggregation. We evaluate the accuracy of our forecasting algorithm on both simulated data and large-scale electricity smart meter data. The results show consistent performance gains compared to state-of-the art methods.

An integrable coupling hierarchy of Dirac integrable hierarchy, its Liouville integrability and Darboux transformation

Abstract: An integrable coupling hierarchy of Dirac integrable hierarchy is presented by means of zero curvature representation. A Hamiltonian operator involving two parameters is introduced, and it is used to derive a pair of Hamiltonian operators. A bi-Hamiltonian structure of the obtained integrable coupling hierarchy is constructed with the aid of Magri pattern of biHamiltonian formulation. Moreover, we prove the Liouville integrability of the obtained integrable coupling hierarchy and establish a Darboux transformation of the integrable coupling. As an application, an exact solution of the integrable coupling of Dirac equation is given. c ©2017 All rights reserved.

Solutions of local and nonlocal equations reduced from the AKNS hierarchy

Abstract: In the paper possible local and nonlocal reductions of the Ablowitz-Kaup-Newell-Suger (AKNS) hierarchy are collected, including the Korteweg-de Vries (KdV) hierarchy, modified KdV hierarchy and their nonlocal versions, nonlinear Schrodinger hierarchy and their nonlocal versions, sine-Gordon equation in nonpotential form and its nonlocal forms. A reduction technique for solutions is employed, by which exact solutions in double Wronskian form are obtained for these reduced equations from those double Wronskian solutions of the AKNS hierarchy. As examples of dynamics we illustrate new interaction of two-soliton solutions of the reverse-$t$ nonlinear Schrodinger equation. Although as a single soliton it is always stationary, two solitons travel along completely symmetric trajectories in $\{x,t\}$ plane and their amplitudes are affected by phase parameters. Asymptotic analysis is given as demonstration. The approach and relation described in this paper are systematic and general and can be used to other nonlocal equations.

Semidefinite Approximations of Reachable Sets for Discrete-time Polynomial Systems

Abstract: We consider the problem of approximating the reachable set of a discrete-time polynomial system from a semialgebraic set of initial conditions under general semialgebraic set constraints. Assuming inclusion in a given simple set like a box or an ellipsoid, we provide a method to compute certified outer approximations of the reachable set. The proposed method consists of building a hierarchy of relaxations for an infinite-dimensional moment problem. Under certain assumptions, the optimal value of this problem is the volume of the reachable set and the optimum solution is the restriction of the Lebesgue measure on this set. Then, one can outer approximate the reachable set as closely as desired with a hierarchy of super level sets of increasing degree polynomials. For each fixed degree, finding the coefficients of the polynomial boils down to computing the optimal solution of a convex semidefinite program. When the degree of the polynomial approximation tends to infinity, we provide strong convergence guarantees of the super level sets to the reachable set. We also present some application examples together with numerical results.

Type-2 Fuzzy Sets

Abstract: This chapter formally introduces type-2 fuzzy sets and is the backbone for the rest of this book. It includes a lot of new terminologies. Coverage includes: the concept of a type-2 fuzzy set, definitions of general type-2 fuzzy sets and associated concepts, definitions of interval type-2 fuzzy sets and associated concepts, examples of two popular footprints of uncertainty, interval type-2 fuzzy numbers, a hierarchy of different kinds of type-2 fuzzy sets, mathematical representations of type-2 fuzzy sets including the vertical slice, wavy slice, and horizontal slice representations, which mathematical representations are most useful for optimal design applications, how to represent non-type-2 fuzzy sets as type-2 fuzzy sets, returning to linguistic labels for type-2 fuzzy sets, and multivariable MFs. 24 examples are used to illustrate the important concepts.

KP hierarchy for the cyclic quiver

Abstract: We introduce a generalisation of the KP hierarchy, closely related to the cyclic quiver and the Cherednik algebra Hk(Zm). This hierarchy depends on m parameters (one of which can be eliminated), with the usual KP hierarchy corresponding to the m = 1 case. Generalising the result of Wilson [Invent. Math. 133(1), 1–41 (1998)], we show that our hierarchy admits solutions parameterised by suitable quiver varieties. The pole dynamics for these solutions is shown to be governed by the classical Calogero–Moser system for the wreath-product Zm≀Sn and its new spin version. These results are further extended to the case of the multi-component hierarchy.

PRF-ODH: Relations, Instantiations, and Impossibility Results.

Abstract: The pseudorandom-function oracle-Diffie–Hellman (PRF-ODH) assumption has been introduced recently to analyze a variety of DH-based key exchange protocols, including TLS 1.2 and the TLS 1.3 candidates, as well as the extended access control (EAC) protocol. Remarkably, the assumption comes in different flavors in these settings and none of them has been scrutinized comprehensively yet. In this paper here we therefore present a systematic study of the different PRF-ODH variants in the literature. In particular, we analyze their strengths relative to each other, carving out that the variants form a hierarchy. We further investigate the boundaries between instantiating the assumptions in the standard model and the random oracle model. While we show that even the strongest variant is achievable in the random oracle model under the strong Diffie–Hellman assumption, we provide a negative result showing that it is implausible to instantiate even the weaker variants in the standard model via algebraic black-box reductions to common cryptographic problems.

A Hierarchical Multi-Label Classification Algorithm for Gene Function Prediction

Abstract: Gene function prediction is a complicated and challenging hierarchical multi-label classification (HMC) task, in which genes may have many functions at the same time and these functions are organized in a hierarchy. This paper proposed a novel HMC algorithm for solving this problem based on the Gene Ontology (GO), the hierarchy of which is a directed acyclic graph (DAG) and is more difficult to tackle. In the proposed algorithm, the HMC task is firstly changed into a set of binary classification tasks. Then, two measures are implemented in the algorithm to enhance the HMC performance by considering the hierarchy structure during the learning procedures. Firstly, negative instances selecting policy associated with the SMOTE approach are proposed to alleviate the imbalanced data set problem. Secondly, a nodes interaction method is introduced to combine the results of binary classifiers. It can guarantee that the predictions are consistent with the hierarchy constraint. The experiments on eight benchmark yeast data sets annotated by the Gene Ontology show the promising performance of the proposed algorithm compared with other state-of-the-art algorithms.

A Multilevel Proximal Gradient Algorithm for a Class of Composite Optimization Problems

Abstract: Composite optimization models consist of the minimization of the sum of a smooth (not necessarily convex) function and a nonsmooth convex function. Such models arise in many applications where, in addition to the composite nature of the objective function, a hierarchy of models is readily available. It is common to take advantage of this hierarchy of models by first solving a low fidelity model and then using the solution as a starting point to a high fidelity model. We adopt an optimization point of view and show how to take advantage of the availability of a hierarchy of models in a consistent manner. We do not use the low fidelity model just for the computation of promising starting points but also for the computation of search directions. We establish the convergence and convergence rate of the proposed algorithm. Our numerical experiments on large scale image restoration problems and the transition path problem suggest that, for certain classes of problems, the proposed algorithm is significantly fas...