Showing papers on "Hierarchy (mathematics) published in 2021"
09 Feb 2021
61 citations
TL;DR: In this paper, the TSSOS hierarchy was proposed for solving polynomial optimization problems, which is a follow-up and a complement to [J. Wang, V. Magron and J. B. Lasserre, preprint, arXiv:1912.08899, 2019].
Abstract: This work is a follow-up and a complement to [J. Wang, V. Magron and J. B. Lasserre, preprint, arXiv:1912.08899, 2019] where the TSSOS hierarchy was proposed for solving polynomial optimization pro...
57 citations
TL;DR: In this paper, the authors exploit term sparsity of the input polynomials to obtain a new converging hierarchy of semidefinite polynomial optimization problems, which is then used to solve the problem.
Abstract: This paper is concerned with polynomial optimization problems. We show how to exploit term (or monomial) sparsity of the input polynomials to obtain a new converging hierarchy of semidefinite progr...
49 citations
TL;DR: Zhang et al. as discussed by the authors proposed a part-relation aware human parser (PRHP), which precisely describes three kinds of human part relations, i.e., decomposition, composition and dependency, by three distinct relation networks.
Abstract: Human parsing is for pixel-wise human semantic understanding. As human bodies are hierarchically structured, how to model human structures is the central theme in this task. We start with analyzing three inference processes over the human hierarchy: direct inference (directly predicting human semantic parts using image information), bottom-up inference (assembling knowledge from constituent parts), and top-down inference (leveraging context from parent nodes). We then formulate the problem as a compositional neural information fusion (CNIF) framework, which assembles the information from the three inference processes in a conditional manner, i.e., considering the confidence of the sources. Based on CNIF, we further present a part-relation aware human parser (PRHP), which precisely describes three kinds of human part relations, i.e., decomposition, composition, and dependency, by three distinct relation networks. Expressive relation information can be captured by imposing the parameters in the relation networks to satisfy specific geometric characteristics of different relations. By assimilating generic message-passing networks with their edge-typed, convolutional counterparts, PRHP performs iterative reasoning over the human hierarchy. With these efforts, PRHP provides a more general and powerful form of CNIF, and lays the foundation for more sophisticated and flexible human relation patterns of reasoning. Experiments on five datasets demonstrate that our two human parsers outperform the state-of-the-arts in all cases.
46 citations
TL;DR: A weight-determining method is developed, which considers three kinds of information including the subjective weights and two kinds of objective weights, and a consensus model is set up to manage the GDM problems with self-confident DHLPRs based on the priority ordering theory.
Abstract: Group decision making (GDM) can be defined as an environment where there exist a set of possible alternatives and a set of individuals (experts, judgements, etc.). Preference relation is one of the most widely used preference representation structures in GDM. Considering that the self-confidence degree is an important part to express preference information, and double hierarchy linguistic preference relation (DHLPR) is a cognitive complex linguistic information representation tool to express complex linguistic information, this paper presents a novel preference relation named as self-confident DHLPR. In addition, a weight-determining method is developed, which considers three kinds of information including the subjective weights and two kinds of objective weights. Furthermore, a consensus model is set up to manage the GDM problems with self-confident DHLPRs based on the priority ordering theory. The effectiveness of the proposed consensus model is illustrated by a case study concerning the selection of optimal hospitals in the field of Telemedicine. Finally, a simulation experiment is devised to testify the proposed consensus model and then some comparisons with other consensus reaching models are provided from three different angles including the number of iterations, the consensus success ratio and the distance between the original and adjusted preferences.
34 citations
01 Jan 2021
TL;DR: In this paper, in-depth analyses on the structural properties of the hierarchy are conducted, and well-designed local update techniques are proposed that significantly outperform the baselines on runtime by up to 3 orders of magnitude.
Abstract: The model of k-core and its decomposition have been applied in various areas, such as social networks, the world wide web, and biology. A graph can be decomposed into an elegant k-core hierarchy to...
23 citations
TL;DR: In this paper, the successive applications of three elementary gauge transformation operators with i = 1,2,3 for the mKP hierarchy in Kupershmidt-Kiso version were investigated.
Abstract: In this paper, we firstly investigate the successive applications of three elementary gauge transformation operators Ti with i = 1,2,3 for the mKP hierarchy in Kupershmidt-Kiso version, and find th...
20 citations
TL;DR: The purpose of this paper is to review all the researches of DHLTS and its extensions, as well as proposing some challenges in the future.
Abstract: Double hierarchy linguistic term set (DHLTS) is a powerful tool when expressing the real thoughts of experts and handling complex linguistic information considering that it divides complex linguistic information into two simple linguistic hierarchies in which the first hierarchy linguistic term set (LTS) is the main linguistic hierarchy and the second hierarchy LTS is the linguistic feature or detailed supplementary of each linguistic term in the first hierarchy LTS. Some extensions of DHLTS have been developed, such as the double hierarchy hesitant fuzzy LTS, the unbalanced DHLTS, the linguistic preference ordering, the double hierarchy linguistic preference relation, and the double hierarchy hesitant fuzzy linguistic preference relation. In recent years, DHLTS and its extensions have been researched by scholars in lots of fields, including the extended concepts, the operational laws, the comparative methods, the measure methods, the consistency and consensus methods, the decision‐making methods, the applications, and so forth. Therefore, the purpose of this paper is to review all the researches of DHLTS and its extensions, as well as proposing some challenges in the future.
20 citations
06 Jun 2021
TL;DR: In this paper, the Weisfeiler-Lehman algorithm (WL) is generalized to a hierarchy of higher-order tests, known as k-WL, which is used to characterize the expressive power of graph neural networks.
Abstract: Graph neural networks are designed to learn functions on graphs. Typically, the relevant target functions are invariant with respect to actions by permutations. Therefore the design of some graph neural network architectures has been inspired by graph-isomorphism algorithms.The classical Weisfeiler-Lehman algorithm (WL)—a graph-isomorphism test based on color refinement—became relevant to the study of graph neural networks. The WL test can be generalized to a hierarchy of higher-order tests, known as k-WL. This hierarchy has been used to characterize the expressive power of graph neural networks, and to inspire the design of graph neural network architectures.A few variants of the WL hierarchy appear in the literature. The goal of this short note is pedagogical and practical: We explain the differences between the WL and folklore-WL formulations, with pointers to existing discussions in the literature. We illuminate the differences between the formulations by visualizing an example.
19 citations
TL;DR: In this paper, theta function representations of algebro-geometric solutions for the Fokas-Lenells (FL) hierarchy were provided through the degeneration of associated Riemann surfaces.
Abstract: This paper is dedicated to provide theta function representations of algebro-geometric solutions for the Fokas- Lenells (FL) hierarchy through studying an algebro-geometric initial value problem. Further, we reduce these solutions into n-dark solutions through the degeneration of associated Riemann surfaces.
18 citations
TL;DR: An optimized implementation of a hierarchy of k-point bounds extending the Delsarte–Goethals–Seidel linear programming 2-point bound and the Bachoc–Vallentin semidefinite programming 3- point bound for spherical codes allows for the maximum number of equiangular lines in Euclidean space with a fixed common angle.
Abstract: We propose a hierarchy of k-point bounds extending the Delsarte–Goethals–Seidel linear programming 2-point bound and the Bachoc–Vallentin semidefinite programming 3-point bound for spherical codes. An optimized implementation of this hierarchy allows us to compute 4, 5, and 6-point bounds for the maximum number of equiangular lines in Euclidean space with a fixed common angle.
TL;DR: This paper proposes an approach to linguistic three‐way decision making problem with double hierarchy linguistic term evaluation information, which takes into account different decision attitudes of decision makers and deduced by the minimum‐loss principle.
Abstract: This paper proposes an approach to linguistic three-way decision making problem with double hierarchy linguistic term evaluation information. Double hierarchy linguistic term set consists of the first hierarchy and second hierarchy linguistic term set, which can describe uncertainty and fuzziness more flexibly. First, the Hamacher operational rules, score function and distance measure of double hierarchy linguistic elements are defined. Next, we construct the double hierarchy linguistic decision-theoretic rough set model. And the conditional probability is calculated based on the double hierarchy linguistic term environment with grey relational analysis, which makes the process of decisions more rational. Then the loss functions are aggregated by the double hierarchy linguistic Hamacher weighted averaging operator, which takes into account different decision attitudes of decision makers. And the results of decision are deduced by the minimum-loss principle. Finally, a case study about the selection of cooperation companies during the COVID-19 is used to demonstrate the practicability of our proposed method. © 2021 Wiley Periodicals LLC
TL;DR: In this article, the authors construct Darboux transformations of the supersymmetric BKP hierarchy, which can generate new solutions from seed solutions by using bosonic eigenfunctions.
Abstract: In this paper, we construct Darboux transformations of the supersymmetric BKP(SBKP) hierarchy. These Darboux transformations can generate new solutions from seed solutions by using bosonic eigenfunctions.
TL;DR: In this paper, the bilinear identities for the wave functions of an extended Kadomtsev-Petviashvili (KP) hierarchy with particular extended flows were constructed.
Abstract: In this paper, we construct the bilinear identities for the wave functions of an extended Kadomtsev-Petviashvili (KP) hierarchy, which is the KP hierarchy with particular extended flows. By introducing an auxiliary parameter, whose flow corresponds to the so-called squared eigenfunction symmetry of KP hierarchy, we find the tau-function for this extended KP hierarchy. It is shown that the bilinear identities will generate all the Hirota's bilinear equations for the zero-curvature forms of the extended KP hierarchy, which includes two types of KP equation with self-consistent sources (KPSCS). The Hirota's bilinear equations obtained in this paper for the KPSCS are in different forms by comparing with the existing results.
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TL;DR: In this article, it was shown that any τ-function of the Korteweg-de Vries (KdV) hierarchy also solves the type B Kadomtsev-Petviashvili (BKP) hierarchy after a simple rescaling of times.
Abstract: In this note, we prove that any τ-function of the Korteweg-de Vries (KdV) hierarchy also solves the type B Kadomtsev-Petviashvili (BKP) hierarchy after a simple rescaling of times.
05 Apr 2021
TL;DR: In this article, the authors introduce a novel algorithm called Planning-Augmented Hierarchical Reinforcement Learning (PAHRL) which translates the concept of hybrid planning/RL to such problems with implicitly defined goal using a hierarchical framework, and divide the original task, formulated as a Markov Decision Process (MDP), into a hierarchy of shorter horizon MDPs.
Abstract: Planning algorithms are powerful at solving long-horizon decision-making problems but require that environment dynamics are known Model-free reinforcement learning has recently been merged with graph-based planning to increase the robustness of trained policies in state-space navigation problems Recent ideas suggest to use planning in order to provide intermediate waypoints guiding the policy in long-horizon tasks Yet, it is not always practical to describe a problem in the setting of state-to-state navigation Often, the goal is defined by one or multiple disjoint sets of valid states or implicitly using an abstract task description Building upon previous efforts, we introduce a novel algorithm called Planning-Augmented Hierarchical Reinforcement Learning (PAHRL) which translates the concept of hybrid planning/RL to such problems with implicitly defined goal Using a hierarchical framework, we divide the original task, formulated as a Markov Decision Process (MDP), into a hierarchy of shorter horizon MDPs Actor-critic agents are trained in parallel for each level of the hierarchy During testing, a planner then determines useful subgoals on a state graph constructed at the bottom level of the hierarchy The effectiveness of our approach is demonstrated for a set of continuous control problems in simulation including robot arm reaching tasks and the manipulation of a deformable object
TL;DR: In this article, the exact solution of Burgers' hierarchy of nonlinear evolution equations is considered and a formula of the nth solution is provided, and to demonstrate its application, the solution to several members of the nonlinear hierarchy.
Abstract: This paper considers the exact solution of Burgers’ hierarchy of nonlinear evolution equations. We construct the general nth conservation law of the hierarchy and prove that these expressions may be transformed into ordinary differential equations. In particular, a coordinate transformation leads to the systematic reduction of the conservation law properties of the Burgers’ hierarchy. Such an approach yields a nonlinear equation, where a second transformation is derived to linearize the expression. Consequently, this approach describes a procedure for finding the exact solutions of the hierarchy. A formula of the nth solution is provided, and to demonstrate its application, we discuss the solution to several members of the nonlinear hierarchy.
TL;DR: In this article, a deep long short-term memory (DLSTM) based auto-encoder (AE) was proposed to solve the problem of hierarchical time series forecasting.
Abstract: Hierarchical time series is a set of data sequences organized by aggregation constraints to represent many real-world applications in research and the industry. Forecasting of hierarchical time series is a challenging and time-consuming problem owing to ensuring the forecasting consistency among the hierarchy levels based on their dimensional features. The excellent empirical performance of our Deep Long Short-Term Memory (DLSTM) approach on various forecasting tasks motivated us to extend it to solve the forecasting problem through hierarchical architectures. Toward this target, we develop the DLSTM model in auto-encoder (AE) fashion and take full advantage of the hierarchical architecture for better time series forecasting. DLSTM-AE works as an alternative approach to traditional and machine learning approaches that have been used to manipulate hierarchical forecasting. However, training a DLSTM in hierarchical architectures requires updating the weight vectors for each LSTM cell, which is time-consuming and requires a large amount of data through several dimensions. Transfer learning can mitigate this problem by training first the time series at the bottom level of the hierarchy using the proposed DLSTM-AE approach. Then, we transfer the learned features to perform synchronous training for the time series of the upper levels of the hierarchy. To demonstrate the efficiency of the proposed approach, we compare its performance with existing approaches using two case studies related to the energy and tourism domains. An evaluation of all approaches was based on two criteria, namely, the forecasting accuracy and the ability to produce coherent forecasts through through the hierarchy. In both case studies, the proposed approach attained the highest accuracy results among all counterparts and produced more coherent forecasts.
07 Jun 2021
TL;DR: In this article, the authors devise a method for the simulation of generic local open quantum systems on quantum computers and verify experimentally the existence of a hierarchy of decay timescales.
Abstract: The authors devise a method for the simulation of generic local open quantum systems on quantum computers and verify experimentally the existence of a hierarchy of decay timescales.
TL;DR: The Hodge-FVH correspondence established a relationship between the special cubic Hodge integrals and an integrable hierarchy, which is called the fractional Volterra hierarchy as discussed by the authors.
Abstract: The Hodge-FVH correspondence establishes a relationship between the special cubic Hodge integrals and an integrable hierarchy, which is called the fractional Volterra hierarchy. In this paper we prove this correspondence. As an application of this result, we prove a gap condition for certain special cubic Hodge integrals and give an algorithm for computing the coefficients that appear in the gap condition.
TL;DR: In this article, a N = 2 supersymmetric CKP (SCKP) hierarchy is defined and additional symmetries are constructed, which constitute a C type S W 1 + ∞ Lie algebra.
25 May 2021
TL;DR: In this paper, the problem of computing controlled invariant sets for controllable discrete-time linear systems is revisited, where the problem is lifted to a higher dimensional space, where a closed-form expression for a set whose projection back onto the original space is proven to be controlled is computed.
Abstract: We revisit the problem of computing controlled invariant sets for controllable discrete-time linear systems. Inspired by previous work by the authors, our main idea works in two moves: the problem is lifted to a higher dimensional space, where we provide a closed-form expression for a set whose projection back onto the original space is proven to be controlled invariant. We propose two methods in which the key insight is computing controlled invariant sets by considering periodic control policies. The first method considers hyperboxes that are rendered recurrent and essentially improves computational performance of the authors' previous work, while computing the same sets. The second method relaxes the assumption of recurrent hyper-boxes and yields substantially larger controlled invariant sets as shown in case studies. These methods do not rely on iterative computations and their scalability is illustrated in several examples, which show that none of the methods is strictly better than the other.
01 Nov 2021
TL;DR: In this paper, a hierarchy of reduced-order models is developed to approximate the aggregate frequency dynamics of coherent generators, which provides increasing accuracy in the approximation of the aggregate system response, outperforming existing aggregation techniques.
Abstract: We introduce a novel framework to approximate the aggregate frequency dynamics of coherent generators. By leveraging recent results on dynamics concentration of tightly connected networks, and frequency weighted balanced truncation, a hierarchy of reduced-order models is developed. This hierarchy provides increasing accuracy in the approximation of the aggregate system response, outperforming existing aggregation techniques.
TL;DR: In this paper, two kinds of gauge transformations for the constrained modified KP hierarchy in Kupershmidt-Kiso version are investigated, and the corresponding corresponding gauge transformations are required t...
Abstract: In this paper, we mainly investigate two kinds of gauge transformations for the constrained modified KP hierarchy in Kupershmidt-Kiso version. The corresponding gauge transformations are required t...
TL;DR: A generalization of the recently published interval outranking approach to deal with complex decision-making problems, which allows an easy setting of weights, and other model parameters as interval numbers, and to set outranking relations associated with each non-elementary criterion.
TL;DR: Mulase as mentioned in this paper solved the Cauchy problem of the Kadomtsev-Petviashvili (KP) hierarchy in an algebraic category in "Solvability of the super KP equation and a generalization of the Birkhoff decomposition".
Abstract: Mulase solved the Cauchy problem of the Kadomtsev-Petviashvili (KP) hierarchy in an algebraic category in “Solvability of the super KP equation and a generalization of the Birkhoff decomposition” (...
TL;DR: In this paper, three types of gauge transformation operators for the q-mKP hierarchy are studied and the successive applications of these operators are derived, and the corresponding communities between them are also investigated.
Abstract: In this paper, we mainly study three types of gauge transformation operators for the q-mKP hierarchy. The successive applications of these gauge transformation operators are derived. And the corresponding communities between them are also investigated.
TL;DR: In this paper, the authors proposed a new concept of mining frequent weighted utility itemsets in hierarchical quantitative databases (HQDBs) using the extended dynamic bit vector structure with large integer elements.
Abstract: Mining frequent itemsets in traditional databases and quantitative databases (QDBs) has drawn many researchers’ interest. Although many studies have been conducted on this topic, a major limitation of these studies is that they ignore the relationships between items. However, in real-life datasets, items are often related to each other through a generalization/specialization relationship. To consider the relationships and discover a more generalized form of patterns, this study proposes a new concept of mining frequent weighted utility itemsets in hierarchical quantitative databases (HQDBs). In this kind of databases, items are organized in a hierarchy. Using the extended dynamic bit vector structure with large integer elements, two efficient algorithms named MINE_FWUIS and FAST_MINE_FWUIS are developed. The empirical evaluations in terms of processing time between MINE_FWUIS and FAST_MINE_FWUIS are conducted. The experimental results indicate that FAST_MINE_FWUIS is recommended for mining frequent weighted utility itemsets in hierarchical QDBs.
TL;DR: In this article, the bilinear identities of the constrained modified KP hierarchy were investigated. But they mainly investigate an equivalent form of the KP hierarchy, and they do not deal with the relation between the KP hierarchies.
Abstract: In this paper, we mainly investigate an equivalent form of the constrained modified KP hierarchy: the bilinear identities. By introducing two auxiliary functions ρ and σ, the corresponding identiti...
TL;DR: The authors proposed a divide-and-conquer approach that automatically induces a hierarchy of generative spaces from a dataset of semantic concepts and texts, based on a notion of similarity of partial knowledge graphs.