Showing papers on "Hierarchy (mathematics) published in 2015"

Order-Embeddings of Images and Language

Abstract: Hypernymy, textual entailment, and image captioning can be seen as special cases of a single visual-semantic hierarchy over words, sentences, and images. In this paper we advocate for explicitly modeling the partial order structure of this hierarchy. Towards this goal, we introduce a general method for learning ordered representations, and show how it can be applied to a variety of tasks involving images and language. We show that the resulting representations improve performance over current approaches for hypernym prediction and image-caption retrieval.

DASH-N: Joint Hierarchical Domain Adaptation and Feature Learning

Abstract: Complex visual data contain discriminative structures that are difficult to be fully captured by any single feature descriptor. While recent work on domain adaptation focuses on adapting a single hand-crafted feature, it is important to perform adaptation of a hierarchy of features to exploit the richness of visual data. We propose a novel framework for domain adaptation using a sparse and hierarchical network (DASH-N). Our method jointly learns a hierarchy of features together with transformations that rectify the mismatch between different domains. The building block of DASH-N is the latent sparse representation. It employs a dimensionality reduction step that can prevent the data dimension from increasing too fast as one traverses deeper into the hierarchy. The experimental results show that our method compares favorably with the competing state-of-the-art methods. In addition, it is shown that a multi-layer DASH-N performs better than a single-layer DASH-N.

Unconditional Uniqueness for the Cubic Gross-Pitaevskii Hierarchy via Quantum de Finetti

Abstract: We present a new, simpler proof of the unconditional uniqueness of solutions to the cubic Gross-Pitaevskii hierarchy in . One of the main tools in our analysis is the quantum de Finetti theorem. Our uniqueness result is equivalent to the one established in the celebrated works of Erdős, Schlein, and Yau. © 2015 Wiley Periodicals, Inc.

Entity Hierarchy Embedding

Abstract: Existing distributed representations are limited in utilizing structured knowledge to improve semantic relatedness modeling. We propose a principled framework of embedding entities that integrates hierarchical information from large-scale knowledge bases. The novel embedding model associates each category node of the hierarchy with a distance metric. To capture structured semantics, the entity similarity of context prediction are measured under the aggregated metrics of relevant categories along all inter-entity paths. We show that both the entity vectors and category distance metrics encode meaningful semantics. Experiments in entity linking and entity search show superiority of the proposed method.

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.

Electre-iii-h

Abstract: A new ELECTRE-III method is proposed handling hierarchical structure of criteria.ELECTRE-III-H generates a partial pre-order of alternatives in each node of the hierarchy.Concordance and discordance indices are adapted to aggregation of partial pre-orders.The pseudo-criteria concept is extended on intermediate criteria of the hierarchy.A real-case study about the evaluation of the quality of websites is presented. This paper proposes a method for ranking a set of alternatives evaluated using multiple and conflicting criteria that are organised in a hierarchical structure. The hierarchy permits the decision maker to identify different intermediate sub-problems of interest. In that way, the analysis of the criteria is done according to the subsets defined in the hierarchy, and following the precedence relations in a bottom-up approach. To deal with this type of hierarchical structures, an extension of the ELECTRE-III method, called ELECTRE-III-H, is presented. As all methods of ELECTRE family, this one also relies on building a binary outranking relation on the set of alternatives on the basis of concordance and discordance tests. The exploitation of this outranking relation generates a partial pre-order, establishing an indifference, preference or incomparability relation for each pair of alternatives. The idea of a bottom-up application of the classical ELECTRE-III method to sub-problems involving subsets of criteria at the intermediate levels of the hierarchy is infeasible because the evaluations of alternatives by criteria aggregating some sub-criteria have the form of partial pre-orders, and not complete pre-orders. Thus, we propose a new procedure for building outranking relations from a set of partial pre-orders, as well as a mechanism for propagating these pre-orders upwards in the hierarchy. With this method, the decision maker is able to analyse the problem in a decomposed way and gain information from the outputs obtained at intermediate levels. In addition, ELECTRE-III-H gives the decision maker the possibility to define a local preference model at each node of the hierarchy, according to his objectives and sub-problem characteristics. We show an application of this method to rank websites of tourist destination brands evaluated using a hierarchy with 4 levels.

On the Space and Circuit Complexity of Parameterized Problems: Classes and Completeness

Abstract: The parameterized complexity of a problem is generally considered "settled" once it has been shown to be fixed-parameter tractable or to be complete for a class in a parameterized hierarchy such as the weft hierarchy. Several natural parameterized problems have, however, resisted such a classification. In the present paper we argue that, in some cases, this is due to the fact that the parameterized complexity of these problems can be better understood in terms of their parameterized space or parameterized circuit complexity. This includes well-studied, natural problems like the feedback vertex set problem, the associative generability problem, or the longest common subsequence problem. We show that these problems lie in and may even be complete for different parameterized space classes, leading to new insights into the problems' complexity. The classes we study are defined in terms of different forms of bounded nondeterminism and simultaneous time---space bounds.

Moment/Sum-of-Squares Hierarchy for Complex Polynomial Optimization

Abstract: We consider the problem of finding the global optimum of a real-valued complex polynomial on a compact set defined by real-valued complex polynomial inequalities. It reduces to solving a sequence of complex semidefinite programming relaxations that grow tighter and tighter thanks to D'Angelo's and Putinar's Positivstellenstatz discovered in 2008. In other words, the Lasserre hierarchy may be transposed to complex numbers. We propose a method for exploiting sparsity and apply the complex hierarchy to problems with several thousand complex variables. These problems consist of computing optimal power flows in the European high-voltage transmission network.

Integrable equations of the infinite nonlinear Schrödinger equation hierarchy with time variable coefficients

Abstract: We present an infinite nonlinear Schrodinger equation hierarchy of integrable equations, together with the recurrence relations defining it. To demonstrate integrability, we present the Lax pairs for the whole hierarchy, specify its Darboux transformations and provide several examples of solutions. These resulting wavefunctions are given in exact analytical form. We then show that the Lax pair and Darboux transformation formalisms still apply in this scheme when the coefficients in the hierarchy depend on the propagation variable (e.g., time). This extension thus allows for the construction of complicated solutions within a greatly diversified domain of generalised nonlinear systems.

A semidefinite programming hierarchy for packing problems in discrete geometry

Abstract: Packing problems in discrete geometry can be modeled as finding independent sets in infinite graphs where one is interested in independent sets which are as large as possible. For finite graphs one popular way to compute upper bounds for the maximal size of an independent set is to use Lasserre's semidefinite programming hierarchy. We generalize this approach to infinite graphs. For this we introduce topological packing graphs as an abstraction for infinite graphs coming from packing problems in discrete geometry. We show that our hierarchy converges to the independence number.

Shapes In a Box: Disassembling 3D Objects for Efficient Packing and Fabrication

Abstract: Modern 3D printing technologies and the upcoming mass-customization paradigm call for efficient methods to produce and distribute arbitrarily shaped 3D objects. This paper introduces an original algorithm to split a 3D model in parts that can be efficiently packed within a box, with the objective of reassembling them after delivery. The first step consists in the creation of a hierarchy of possible parts that can be tightly packed within their minimum bounding boxes. In a second step, the hierarchy is exploited to extract the single segmentation whose parts can be most tightly packed. The fact that shape packing is an NP-complete problem justifies the use of heuristics and approximated solutions whose efficacy and efficiency must be assessed. Extensive experimentation demonstrates that our algorithm produces satisfactory results for arbitrarily shaped objects while being comparable to ad hoc methods when specific shapes are considered.

Hierarchical Granular Clustering: An Emergence of Information Granules of Higher Type and Higher Order

Abstract: In this study, we introduce a concept of hierarchical granular clustering and establish its algorithmic framework. We show that the proposed model naturally gives rise to information granules that are both of higher order and higher type, offering a compelling justification behind their emergence. In a concise way, we can capture the overall architecture of information granules as a hierarchy exhibiting conceptual layers of increasing abstraction: numeric data → information granules → information granules of type-2, order-2 → … information granules of higher type/order. The elevated type of information granules is reflective of the visible hierarchical facet of processing and the inherent diversity of the individual locally revealed structures in data. While the concept and the methodology deliver some general settings, the detailed algorithmic aspects are discussed in detail when using fuzzy clustering realized by means of fuzzy c-means. Furthermore, for illustrative purposes, we mainly focus on interval-valued fuzzy sets and granular interval fuzzy sets arising at the higher level of the hierarchy. Higher type fuzzy sets are formed with the help of the principle of justifiable granularity. The conceptually sound hierarchy is established in a general way, which makes it equally applicable to various formalisms of representation of information granules. Experiments are reported for synthetic and publicly available datasets.

Open intersection numbers, matrix models and MKP hierarchy

Abstract: In this paper we conjecture that the generating function of the intersection numbers on the moduli spaces of Riemann surfaces with boundary, constructed recently by R. Pandharipande, J. Solomon and R. Tessler and extended by A. Buryak, is a tau-function of the KP integrable hierarchy. Moreover, it is given by a simple modification of the Kontsevich matrix integral so that the generating functions of open and closed intersection numbers are described by the MKP integrable hierarchy. Virasoro constraints for the open intersection numbers naturally follow from the matrix integral representation.

The importance of the label hierarchy in hierarchical multi-label classification

Abstract: We address the task of hierarchical multi-label classification (HMC). HMC is a task of structured output prediction where the classes are organized into a hierarchy and an instance may belong to multiple classes. In many problems, such as gene function prediction or prediction of ecological community structure, classes inherently follow these constraints. The potential for application of HMC was recognized by many researchers and several such methods were proposed and demonstrated to achieve good predictive performances in the past. However, there is no clear understanding when is favorable to consider such relationships (hierarchical and multi-label) among classes, and when this presents unnecessary burden for classification methods. To this end, we perform a detailed comparative study over 8 datasets that have HMC properties. We investigate two important influences in HMC: the multiple labels per example and the information about the hierarchy. More specifically, we consider four machine learning tasks: multi-label classification, hierarchical multi-label classification, single-label classification and hierarchical single-label classification. To construct the predictive models, we use predictive clustering trees (a generalized form of decision trees), which are able to tackle each of the modelling tasks listed. Moreover, we investigate whether the influence of the hierarchy and the multiple labels carries over for ensemble models. For each of the tasks, we construct a single tree and two ensembles (random forest and bagging). The results reveal that the hierarchy and the multiple labels do help to obtain a better single tree model, while this is not preserved for the ensemble models.

The hierarchy of local minimums in polynomial optimization

Abstract: This paper studies the hierarchy of local minimums of a polynomial in the vector space $$\mathbb {R}^n$$Rn. For this purpose, we first compute $$H$$H-minimums, for which the first and second order necessary optimality conditions are satisfied. To compute each $$H$$H-minimum, we construct a sequence of semidefinite relaxations, based on optimality conditions. We prove that each constructed sequence has finite convergence, under some generic conditions. A procedure for computing all local minimums is given. When there are equality constraints, we have similar results for computing the hierarchy of critical values and the hierarchy of local minimums.

Low-level vision by consensus in a spatial hierarchy of regions

Abstract: We introduce a multi-scale framework for low-level vision, where the goal is estimating physical scene values from image data—such as depth from stereo image pairs. The framework uses a dense, overlapping set of image regions at multiple scales and a “local model,” such as a slanted-plane model for stereo disparity, that is expected to be valid piecewise across the visual field. Estimation is cast as optimization over a dichotomous mixture of variables, simultaneously determining which regions are inliers with respect to the local model (binary variables) and the correct co-ordinates in the local model space for each inlying region (continuous variables). When the regions are organized into a multi-scale hierarchy, optimization can occur in an efficient and parallel architecture, where distributed computational units iteratively perform calculations and share information through sparse connections between parents and children. The framework performs well on a standard benchmark for binocular stereo, and it produces a distributional scene representation that is appropriate for combining with higher-level reasoning and other low-level cues.

Multiscale network generation

Abstract: Relationships between entities in complex systems could be represented using the paradigm of networks. The network representation can then reveal the evolution, structure and dynamics of those systems. Frequently, obtaining the required scientific data about the networks is expensive or infeasible. In other words, the amount of available empirical data is insufficient for simulation, validation, verification, and other scientific tasks. In these situations, empirical data should be augmented by synthetic data generated from models in such a way that properties of the system are preserved in the synthetic data, even when those properties are unique to the system or not fully known, but existing methods only reproduce a limited set of specified network properties. Here we introduce a novel strategy for synthesizing artificial networks that can realistically model a variety of network properties and that is termed multiscale network generation, or as a specific algorithm, MUSKETEER. This strategy first creates a hierarchy of aggregated representations of the original network, and then reformulates the network generation problem at all levels of this hierarchy in order to take into account properties at multiple scales of the system. The network is then edited at any or all scales, depending on the desired variability in the ensemble of synthetic networks. We find that for many complex networks taken from real-world systems, the strategy is able to preserve important properties with little statistical bias while achieving high degree of variability and arbitrary difference from the original.

Self-Organization of Spatio-Temporal Hierarchy via Learning of Dynamic Visual Image Patterns on Action Sequences

Abstract: It is well known that the visual cortex efficiently processes high-dimensional spatial information by using a hierarchical structure. Recently, computational models that were inspired by the spatial hierarchy of the visual cortex have shown remarkable performance in image recognition. Up to now, however, most biological and computational modeling studies have mainly focused on the spatial domain and do not discuss temporal domain processing of the visual cortex. Several studies on the visual cortex and other brain areas associated with motor control support that the brain also uses its hierarchical structure as a processing mechanism for temporal information. Based on the success of previous computational models using spatial hierarchy and temporal hierarchy observed in the brain, the current report introduces a novel neural network model for the recognition of dynamic visual image patterns based solely on the learning of exemplars. This model is characterized by the application of both spatial and temporal constraints on local neural activities, resulting in the self-organization of a spatio-temporal hierarchy necessary for the recognition of complex dynamic visual image patterns. The evaluation with the Weizmann dataset in recognition of a set of prototypical human movement patterns showed that the proposed model is significantly robust in recognizing dynamically occluded visual patterns compared to other baseline models. Furthermore, an evaluation test for the recognition of concatenated sequences of those prototypical movement patterns indicated that the model is endowed with a remarkable capability for the contextual recognition of long-range dynamic visual image patterns.

Integrability of the Frobenius algebra-valued Kadomtsev-Petviashvili hierarchy

Abstract: We introduce a Frobenius algebra-valued Kadomtsev-Petviashvili (KP) hierarchy and show the existence of Frobenius algebra-valued τ-function for this hierarchy. In addition, we construct its Hamiltonian structures by using the Adler-Dickey-Gelfand method. As a byproduct of these constructions, we show that the coupled KP hierarchy, defined by Casati and Ortenzi [J. Geom. Phys. 56, 418-449 (2006)], has at least n-“basic” different local bi-Hamiltonian structures. Finally, via the construction of the second Hamiltonian structures, we obtain some local matrix, or Frobenius algebra-valued, generalizations of classical W-algebras.

Efficiency measurement for hierarchical network systems

Abstract: The conventional data envelopment analysis (DEA) models for measuring the relative efficiency of a set of decision making units (DMUs), without considering the operations of the component processes, often produce misleading results, and network models have thus been recommended. This paper discusses the development of a network DEA model for systems with a hierarchical structure. It is shown that the hierarchical structure is equivalent to a parallel structure, with the components being the units at the bottom of the hierarchy. Due to the characteristics of a parallel system, the efficiency of a hierarchical system is thus a weighted average of those of the units at the bottom of the hierarchy. A hypothetic example shows that the proposed model is able to distinguish the order of the efficient DMUs evaluated by the conventional DEA model. Moreover, it provides the efficiencies of the functions of the DMU, which enables managers to identify areas of weakness, and thus better focus efforts to improve overall performance.

Hierarchical private/shared classification: The key to simple and efficient coherence for clustered cache hierarchies

Abstract: Hierarchical clustered cache designs are becoming an appealing alternative for mulucores. Grouping cores and their caches in clusters reduces network congestion by localizing traffic among several hierarchical levels, potentially enabling much higher scalability. While such architectures can be formed recursively by replicating a base design pattern, keeping the whole hierarchy coherent requires more effort and consideration. The reason is that, in hierarchical coherence, even basic operations must be recursive. As a consequence, intermediate-level caches behave both as directories and as leaf caches. This leads to an explosion of states, protocol-races, and protocol complexity. While there have been previous efforts to extend directory-based coherence to hierarchical designs their increased complexity and verification cost is a serious impediment to their adoption. We aim to address these concerns by encapsulating all hierarchical complexity in a simple function: that of determining when a data block is shared entirely within a cluster (sub-tree of the hierarchy) and is private from the outside. This allows us to eliminate complex recursive operations that span the hierarchy and instead employ simple coherence mechanisms such as self-invalidation and write-through — now restricted to operate within the cluster where a data block is shared. We examine two inclusivity options and discuss the relation of our approach to the recently proposed Hierarchical-Race-Free (HRF) memory models. Finally, comparisons to a hierarchical directory-based MOESI, VIPS-M, and TokenCMP protocols show that, despite its simplicity our approach results in competitive performance and decreased network traffic.

A unifying hierarchy of valuations with complements and substitutes

Abstract: We introduce a new hierarchy over monotone set functions, that we refer to as MPH (Maximum over Positive Hyper-graphs). Levels of the hierarchy correspond to the degree of complementarity in a given function. The highest level of the hierarchy, MPH-m (where m is the total number of items) captures all monotone functions. The lowest level, MPH-1, captures all monotone submodular functions, and more generally, the class of functions known as XOS. Every monotone function that has a positive hypergraph representation of rank κ (in the sense defined by Abraham, Babaioff, Dughmi and Roughgarden [EC 2012]) is in MPH-κ. Every monotone function that has supermodular degree κ (in the sense defined by Feige and Izsak [ITCS 2013]) is in MPH-(κ+1). In both cases, the converse direction does not hold, even in an approximate sense. We present additional results that demonstrate the expressiveness power of MPH-κ. One can obtain good approximation ratios for some natural optimization problems, provided that functions are required to lie in low levels of the MPH hierarchy. We present two such applications. One shows that the maximum welfare problem can be approximated within a ratio of κ + 1 if all players hold valuation functions in MPH-κ. The other is an upper bound of 2κ on the price of anarchy of simultaneous first price auctions.

A geometric reasoning approach to hierarchical representation for B-rep model retrieval

Abstract: 3D solid model similarity is dependency of many intelligent design applications, such as design reuse, part management, case-based reasoning, and cost estimation. Matching and comparing its intrinsic boundary representation (B-rep) is a critical issue to retrieval. In this paper, we proposed a geometric reasoning approach to generate hierarchy for B-rep model retrieval. We extracted the winged-edge data structure to support series algorithms for underlying geometric reasoning, which is mainly composed of 3 steps to build hierarchy: partitioning, assembling and simplifying. This hierarchical representation is featured with level of detail ( L O D ) retaining geometric and topological information which is proved to be efficient in both global and partial retrieval. Our approach is based on the standard for the exchange of product information ( S T E P ) , which is suitable for data exchange between heterogeneous CAD systems. The result of case studies from prototype implementation demonstrates its effectiveness and efficiency. Hierarchy of B-rep model is generated by geometric reasoning automatically.This shape descriptor eases global and partial retrieval at level of detail.Interactive prototype system gives why and how similar features are matched.

An approach for determining and measuring network hierarchy applied to comparing the phosphorylome and the regulome

Abstract: Many biological networks naturally form a hierarchy with a preponderance of downward information flow. In this study, we define a score to quantify the degree of hierarchy in a network and develop a simulated-annealing algorithm to maximize the hierarchical score globally over a network. We apply our algorithm to determine the hierarchical structure of the phosphorylome in detail and investigate the correlation between its hierarchy and kinase properties. We also compare it to the regulatory network, finding that the phosphorylome is more hierarchical than the regulome.

An Average-Case Depth Hierarchy Theorem for Boolean Circuits

Abstract: We prove an average-case depth hierarchy theorem for Boolean circuits over the standard basis of AND, OR, and NOT gates. Our hierarchy theorem says that for every d a#x2265; 2, there is an explicit n-variable Boolean function f, computed by a linear-size depth-d formula, which is such that any depth-(d -- 1) circuit that agrees with f on (1/2 + on(1)) fraction of all inputs must have size exp(na#x03A9;(1/d)). This answers an open question posed by Has tad in his Ph.D. Thesis [Has86b]. Our average-case depth hierarchy theorem implies that the polynomial hierarchy is infinite relative to a random oracle with probability 1, confirming a conjecture of Has tad [Has86a], Cai [Cai86], and Babai [Bab87]. We also use our result to show that there is no "approximate converse" to the results of Linial, Mansour, Nisan [LMN93] and Boppana [Bop97] on the total influence of constant-depth circuits, thus answering a question posed by Kalai [Kal12] and Hatami [Hat14]. A key ingredient in our proof is a notion of random projections which generalize random restrictions.

On the Hardest Problem Formulations for the 0/1 Lasserre Hierarchy

Abstract: The Lasserre/Sum-of-Squares (SoS) hierarchy is a systematic procedure for constructing a sequence of increasingly tight semidefinite relaxations. It is known that the hierarchy converges to the 0/1 polytope in n levels and captures the convex relaxations used in the best available approximation algorithms for a wide variety of optimization problems. In this paper we characterize the set of 0/1 integer linear problems and unconstrained 0/1 polynomial optimization problems that can still have an integrality gap at level n-1. These problems are the hardest for the Lasserre hierarchy in this sense.