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Two Channel Filter Banks on Arbitrary Graphs With Positive Semi Definite Variation Operators

IEEE TRANSACTIONS ON SIGNAL PROCESSING 2023年 71卷 917-932页

作者： Pavez, Eduardo Girault, Benjamin Ortega, Antonio Chou, Philip A. Univ Southern Calif Dept Elect & Comp Engn Los Angeles CA 90089 USA Univ Rennes ENSAI CNRS CREST UMR F-9194 Rennes France Google Res Seattle WA 98013 USA

We propose novel two-channel filter banks for signals on graphs. Our designs can be applied to arbitrary graphs, given a positive semi definite variation operator, while using arbitrary vertex partitions for downsampling. The proposed generalized filter banks (GFBs) also satisfy several desirable properties including perfect reconstruction and critical sampling, while having efficient implementations. Our results generalize previous approaches that were only valid for the normalized Laplacian of bipartite graphs. Our approach is based on novel graph Fourier transforms (GFTs) given by the generalized eigenvectors of the variation operator. These GFTs are orthogonal in an alternative inner product space which depends on the downsampling and variation operators. Our key theoretical contribution is showing that the spectral folding property of the normalized Laplacian of bipartite graphs, at the core of bipartite filter bank theory, can be generalized for the proposed GFT if the inner product matrix is chosen properly. In addition, we study vertex domain and spectral domain properties of GFBs and illustrate their probabilistic interpretation using Gaussian graphical models. While GFBs can be defined given any choice of a vertex partition for downsampling, we propose an algorithm to optimize these partitions with a criterion that favors balanced partitions with large graph cuts, which are shown to lead to efficient and stable GFB implementations. Our numerical experiments show that partition-optimized GFBs can be implemented efficiently on 3D point clouds with hundreds of thousands of points (nodes), while also improving the color signal representation quality over competing state-of-the-art approaches.

关键词： Laplace equations Bipartite graph Three-dimensional displays Point cloud compression Partitioning algorithms Filter banks approximation algorithms Two-channel filter banks graph Fourier transform graph signal multiresolution representation

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DECONET: An Unfolding Network for Analysis-Based Compressed Sensing With Generalization Error Bounds

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2023年 71卷 1938-1951页

作者： Kouni, Vicky Panagakis, Yannis Natl & Kapodistrian Univ Athens Dept Informat & Telecommun Athens 15772 Greece

We present a new deep unfolding network for analysis-sparsity-based Compressed Sensing. The proposed network coined Decoding Network (DECONET) jointly learns a decoder that reconstructs vectors from their incomplete, noisy measurements and a redundant sparsifying analysis operator, which is shared across the layers of DECONET. Moreover, we formulate the hypothesis class of DECONET and estimate its associated Rademacher complexity. Then, we use this estimate to deliver meaningful upper bounds for the generalization error of DECONET. Finally, the validity of our theoretical results is assessed and comparisons to state-of-the-art unfolding networks are made, on both synthetic and real-world datasets. Experimental results indicate that our proposed network outperforms the baselines, consistently for all datasets, and its behaviour complies with our theoretical findings.

关键词： Signal processing algorithms Decoding approximation algorithms Transforms Analytical models Optimization Neural networks Compressed sensing analysis sparsity unfolding network generalization error rademacher complexity

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Heterogeneous Multi-resource Planning and Allocation Under Stochastic Demand

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INFORMS JOURNAL ON COMPUTING 2023年第5期35卷 929-951页

作者： Baxter, Arden Keskinocak, Pinar Singh, Mohit Georgia Inst Technol H Milton Stewart Sch Ind & Syst Engn Atlanta GA 30332 USA Georgia Inst Technol Ctr Hlth & Humanitarian Syst Atlanta GA 30332 USA

We study the capacity planning and allocation decisions for multiple heterogeneous resources, considering potential demand scenarios, where each demand requests a subset of the available resource types simultaneously at a specified time, location, and duration (smRmD). We model this problem as a two-stage stochastic integer program and consider two variants for the objective function: (a) maximize the expected reward of demands met over all scenarios, subject to a budget B for resources, and (b) maximize the expected reward of demands met over all scenarios minus the cost of resources. Contributions of this work include (i) a thorough complexity analysis of smRmD and its variants, (ii) analysis of structural properties, (iii) development of various approximation algorithms using the unique structural properties of smRmD and its variants, and (iv) an extensive computational study to explore the ease with which exact and approximate solutions may be found.

关键词： stochastic integer programming resource allocation resource planning approximation algorithms

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Convergence of Decentralized Actor-Critic Algorithm in General-Sum Markov Games

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IEEE CONTROL SYSTEMS LETTERS 2024年 8卷 2643-2648页

作者： Maheshwari, Chinmay Wu, Manxi Sastry, Shankar Univ Calif Berkeley Dept EECS Berkeley CA 94709 USA Univ Calif Berkeley Dept Civil & Environm Engn Berkeley CA 94709 USA

Markov games provide a powerful framework for modeling strategic multi-agent interactions in dynamic environments. Traditionally, convergence properties of decentralized learning algorithms in these settings have been established only for special cases, such as Markov zero-sum and potential games, which do not fully capture real-world interactions. In this letter, we address this gap by studying the asymptotic properties of learning algorithms in general-sum Markov games. In particular, we focus on a decentralized algorithm where each agent adopts an actor-critic learning dynamic with asynchronous step sizes. This decentralized approach enables agents to operate independently, without requiring knowledge of others' strategies or payoffs. We introduce the concept of a Markov Near-Potential Function (MNPF) and demonstrate that it serves as an approximate Lyapunov function for the policy updates in the decentralized learning dynamics, which allows us to characterize the convergent set of strategies. We further strengthen our result under specific regularity conditions and with finite Nash equilibria.

关键词： Games Convergence Nash equilibrium Heuristic algorithms approximation algorithms Trajectory Lyapunov methods Vectors Stochastic processes Standards Markov games decentralized learning Markov near-potential functions actor-critic algorithm

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Applications of the Partial Fraction Decomposition to Pole Variation Rate for Root Locus Construction

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IEEE CONTROL SYSTEMS LETTERS 2024年 8卷 2925-2930页

作者： Tebaldi, Davide Zanasi, Roberto Univ Modena & Reggio Emilia Dept Engn Enzo Ferrari I-41125 Modena Italy

The applications of the partial fraction decomposition in control and systems engineering are several. In this letter, we use the partial fraction decomposition to address the pole variation rate problem, namely to study the rate of variation of the system poles when the control parameter changes and when the system is subject to variations of its own parameters, which has led to the proposal of a new algorithm for the construction of the root locus. The new algorithm is proven to be much more efficient in terms of execution time than the dedicated MATLAB function, while providing the same output results.

关键词： Mathematical models MATLAB Vectors approximation algorithms Proposals Polynomials Poles and zeros Electric motors Control theory Software algorithms Partial fraction decomposition pole variation rate root locus root locus construction

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Practical algorithms with Guaranteed approximation Ratio for TTP with Maximum Tour Length Two

arXiv

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arXiv 2022年

作者： Zhao, Jingyang Xiao, Mingyu University of Electronic Science and Technology of China China

The Traveling Tournament Problem (TTP) is a hard but interesting sports scheduling problem inspired by Major League Baseball, which is to design a double round-robin schedule such that each pair of teams plays one game in each other’s home venue, minimizing the total distance traveled by all n teams (n is even). In this paper, we consider TTP-2, i.e., TTP under the constraint that at most two consecutive home games or away games are allowed for each team. We propose practical algorithms for TTP-2 with improved approximation ratios. Due to the different structural properties of the problem, all known algorithms for TTP-2 are different for n/2 being odd and even, and our algorithms are also different for these two cases. For even n/2, our approximation ratio is 1 + 3/n, improving the previous result of 1 + 4/n. For odd n/2, our approximation ratio is 1 + 5/n, improving the previous result of 3/2 + 6/n. In practice, our algorithms are easy to implement. Experiments on well-known benchmark sets show that our algorithms beat previously known solutions for all instances with an average improvement of 5.66%. © 2022, CC BY.

关键词： approximation algorithms

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Efficient Asset Allocation to Dynamic Requests for Support

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IEEE CONTROL SYSTEMS LETTERS 2024年 8卷 2577-2582页

作者： Elliott, D. Sawyer Finn, Connor Hicks, Greg Johns Hopkins Univ Appl Phys Lab Air & Missile Def Sect Laurel MD 20723 USA

This letter details two novel algorithms for computing asset allocations given dynamic requests for support. Using spatial and temporal discretization, the first algorithm casts the allocation problem as a lexicographic integer-linear program (ILP) that is efficiently solved using an ILP solver. Improving computational efficiency, the second algorithm replaces the ILP solver with a novel dual-tactic linear program solver. Provided proofs, complexity analyses, and numerical analyses demonstrate the computational efficiency and convergence of both algorithms to performant solutions. Discussion of application to broad domains such as defense, conservation, and resource management for businesses is provided.

关键词： Resource management Heuristic algorithms Optimization Dynamic scheduling approximation algorithms Surveillance Physics Space missions Robots Research and development Resource allocation asset allocation linear programming optimization integer-linear programming

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Asymptotic Errors for Teacher-Student Convex Generalized Linear Models (Or: How to Prove Kabashima's Replica Formula)

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IEEE TRANSACTIONS ON INFORMATION THEORY 2023年第3期69卷 1824-1852页

作者： Gerbelot, Cedric Abbara, Alia Krzakala, Florent Univ PSL Sorbonne Univ Lab Phys Ecole normale Super CNRS F-75005 Ens Paris France NYU Courant Inst Math Sci New York NY 10012 USA Lab Phys Ecole normalesu perieure EPFL F-75005 Ens Paris France Lab Computat Biol & Theoret Biophys EPFL CH-1015 Lausanne Switzerland IdePh Lab EPFL CH-1015 Lausanne Switzerland

There has been a recent surge of interest in the study of asymptotic reconstruction performance in various cases of generalized linear estimation problems in the teacher-student setting, especially for the case of i.i.d standard normal matrices. Here, we go beyond these matrices, and prove an analytical formula for the reconstruction performance of convex generalized linear models with rotationally-invariant data matrices with arbitrary bounded spectrum, rigorously confirming, under suitable assumptions, a conjecture originally derived using the replica method from statistical physics. The proof is achieved by leveraging on message passing algorithms and the statistical properties of their iterates, allowing to characterize the asymptotic empirical distribution of the estimator. For sufficiently strongly convex problems, we show that the two-layer vector approximate message passing algorithm (2-MLVAMP) converges, where the convergence analysis is done by checking the stability of an equivalent dynamical system, which gives the result for such problems. We then show that, under a concentration assumption, an analytical continuation may be carried out to extend the result to convex (non-strongly) problems. We illustrate our claim with numerical examples on mainstream learning methods such as sparse logistic regression and linear support vector classifiers, showing excellent agreement between moderate size simulation and the asymptotic prediction.

关键词： Heuristic algorithms approximation algorithms Estimation Convergence Signal processing algorithms Mathematical models Machine learning Parametric statistics estimation error message passing expectation-maximization algorithms optimization convergence convex functions linear matrix inequalities

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An Improvement of Reed’s Treewidth approximation

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Journal of Graph algorithms and Applications 2022年第2期26卷 257-282页

作者： Belbasi, Mahdi Fürer, Martin Computer Science and Engineering Department Pennsylvania State University University Park PA United States

We present a new approximation algorithm for the treewidth problem which finds an upper bound on the treewidth and constructs a corresponding tree de-composition as well. Our algorithm is a faster variation of Reed’s classical algorithm. For the benefit of the reader, and to be able to compare these two algorithms, we start with a detailed time analysis of Reed’s algorithm. We fill in many details that have been omitted in Reed’s paper. Computing tree decompositions parameterized by the treewidth k is fixed parameter tractable (FPT), meaning that there are algorithms running in time O(f(k)g(n)) where f is a computable function, and g(n) is polynomial in n, where n is the number of vertices. An analysis of Reed’s algorithm shows f(k) = 2O(klogk) and g(n) = n log n for a 5-approximation. Reed simply claims time O(n log n) for bounded k for his constant factor approximation algorithm, but the bound of 2Ω(klogk)n log n is well known. From a practical point of view, we notice that the time of Reed’s algorithm also contains a term of O(k2224kn log n), which for small k is much worse than the asymptotically leading term of 2O(klogk)n log n. We analyze f(k) more precisely, because the purpose of this paper is to improve the running times for all reasonably small values of k. Our algorithm runs in O(f(k)n log n) too, but with a much smaller dependence on k. In our case, f(k) = 2O(k). This algorithm is simple and fast, especially for small values of k. We should mention that Bodlaender et al. [2016] have an algorithm with a linear dependence on n, and Korhonen [2021] obtains the much better approximation ratio of 2, while the current paper achieves a better dependence on k. © 2022, Brown University. All rights reserved.

关键词： approximation algorithms

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On solving the 2L-CVRP using an adaptive chemical reaction algorithm: postal transportation real-case

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INTERNATIONAL JOURNAL OF SYSTEM ASSURANCE ENGINEERING AND MANAGEMENT 2024年 1-25页

作者： Dahmani, Nadia Sbai, Ines Tlili, Takwa Krichen, Saoussen Zayed Univ Coll Technol Innovat Abu Dhabi U Arab Emirates Univ Tunis Inst Super Gest Tunis LARODEC Lab Tunis Tunisia Univ Manouba Inst Super Arts Multimedia Manouba ISAMM Tunis Tunisia

The postal sector plays a crucial role in enhancing and advancing services for businesses and citizens through its diverse services. Hence, optimizing the routing system collecting and transporting letters and parcels is a vital element within a well-rounded delivery management system. We model the problem as a Capacitated vehicle routing problem (CVRP) with two-dimensional loading constraints (2L-CVRP). This involves designing a set of routes that start and end at a central depot. Moreover, items in each vehicle trip must satisfy the two-dimensional orthogonal packing constraints. The main objective is to optimize the total transportation costs using a homogeneous vehicle fleet. Due to the NP-hardness of the 2L-CVRP, we proposed an adaptive chemical reaction optimization (ACRO) metaheuristic to generate potential solutions. The algorithm adjusts its parameters and is intelligent search strategies during the optimization process based on the characteristics of the problem. Consequently, the algorithm can exploit and explore new regions of the search space. We compared our results with state-of-the-art meta-heuristics using 2L-CVRP benchmark instances from the literature. The results showed competitive solutions regarding the optimal ones. The empirical results, derived from benchmark datasets comprising a total of 180 instancesrove the high competitiveness of the proposed ACRO. It achieves a 67% success rate out of 36 instances for class 1 and a 59% success rate out of 144 instances for class 2-5 in terms of obtained solutions. In addition to benchmarking, we considered a real-world case study from the Tunisian Post Office. The ACRO results outperform the scenario adopted by the post office.

关键词： Postal transportation Capacitated vehicle routing problem approximation algorithms Adaptive chemical reaction optimization Two-dimensional loading constraints

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