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LSRN: A PARALLEL ITERATIVE SOLVER FOR STRONGLY OVER- OR UNDERDETERMINED SYSTEMS

SIAM JOURNAL ON SCIENTIFIC COMPUTING 2014年第2期36卷 C95-C118页

作者： Meng, Xiangrui Saunders, Michael A. Mahoney, Michael W. Stanford Univ ICME Stanford CA 94305 USA Stanford Univ Dept Management Sci & Engn Syst Optimizat Lab Stanford CA 94305 USA Stanford Univ Dept Math Stanford CA 94305 USA

We describe a parallel iterative least squares solver named LSRN that is based on random normal projection. LSRN computes the min-length solution to min(x is an element of Rn) vertical bar vertical bar Ax - b vertical bar vertical bar(2), where A is an element of R-mxn with m >> n or m << n, and where A may be rank-deficient. Tikhonov regularization may also be included. Since A is involved only in matrix-matrix and matrix-vector multiplications, it can be a dense or sparse matrix or a linear operator, and LSRN automatically speeds up when A is sparse or a fast linear operator. The preconditioning phase consists of a random normal projection, which is embarrassingly parallel, and a singular value decomposition of size inverted right perpendicular gamma min(m, n) inverted left perpendicular x min(m, n), where gamma is moderately larger than 1, e.g., gamma = 2. We prove that the preconditioned system is well-conditioned, with a strong concentration result on the extreme singular values, and hence that the number of iterations is fully predictable when we apply LSQR or the Chebyshev semi-iterative method. As we demonstrate, the Chebyshev method is particularly efficient for solving large problems on clusters with high communication cost. Numerical results show that on a shared-memory machine, LSRN is very competitive with LAPACK's DGELSD and a fast randomized least squares solver called Blendenpik on large dense problems, and it outperforms the least squares solver from SuiteSparseQR on sparse problems without sparsity patterns that can be exploited to reduce fill-in. Further experiments show that LSRN scales well on an Amazon Elastic Compute Cloud cluster.

关键词： linear least squares overdetermined system underdetermined system rank-deficient minimum-length solution LAPACK sparse matrix iterative method preconditioning LSQR Chebyshev semi-iterative method Tikhonov regularization ridge regression parallel computing random projection random sampling random matrix randomized algorithm

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A MAPREDUCE algorithm FOR MINIMUM VERTEX COVER PROBLEMS AND ITS RANDOMIZATION

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COMPUTING AND INFORMATICS 2020年第5期39卷 952-972页

作者： Nakamura, Morikazu Kinjo, Daiki Yoshida, Takeo Univ Ryukyus Fac Engn Nishihara Okinawa 9030213 Japan

MapReduce is a programming paradigm for large-scale distributed information processing. This paper proposes a MapReduce algorithm for the minimum vertex cover problem, which is known to be NP-hard. The MapReduce algorithm can efficiently obtain a minimal vertex cover in a small number of rounds. We show the effectiveness of the algorithm through experimental evaluation and comparison with exact and approximate algorithms which demonstrates a high quality in a small number of MapReduce rounds. We also confirm from experimentation that the algorithm has good scalability allowing high-quality solutions under restricted computation times due to increased graph size. Moreover, we extend our algorithm to randomized one to obtain a good expected approximate ratio.

关键词： MapReduce minimum vertex cover Hadoop optimization algorithm design randomized algorithm

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Generative modeling via tensor train sketching

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APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS 2023年 67卷

作者： Hur, YoonHaeng Hoskins, Jeremy G. Lindsey, Michael Stoudenmire, E. M. Khoo, Yuehaw Univ Chicago Dept Stat Chicago IL 60637 USA NYU Courant Inst Math Sci Dept Math New York NY USA Flatiron Inst Ctr Computat Quantum Phys New York NY USA

In this paper, we introduce a sketching algorithm for constructing a tensor train representation of a probability density from its samples. Our method deviates from the standard recursive SVD-based procedure for constructing a tensor train. Instead, we formulate and solve a sequence of small linear systems for the individual tensor train cores. This approach can avoid the curse of dimensionality that threatens both the algorithmic and sample complexities of the recovery problem. Specifically, for Markov models under natural conditions, we prove that the tensor cores can be recovered with a sample complexity that scales logarithmically in the dimensionality. Finally, we illustrate the performance of the method with several numerical experiments.& COPY;2023 Elsevier Inc. All rights reserved.

关键词： Tensor decompositions Tensor train randomized algorithm Generative modeling High-dimensional function approximation

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Sublinear message bounds of authenticated implicit Byzantine agreement

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THEORETICAL COMPUTER SCIENCE 2024年 1022卷

作者： Kumar, Manish Molla, Anisur Rahaman IIT Madras Dept Comp Sci & Engn Chennai 600036 India Indian Stat Inst Cryptol & Secur Res Unit Kolkata 700108 India

This paper studies the message complexity of authenticated Byzantine agreement (BA) in synchronous, fully-connected distributed networks under an honest majority. We focus on the so-called implicit Byzantine agreement problem where each node starts with an input value and at the end a non-empty subset of the honest nodes should agree on a common input value by satisfying the BA properties (i.e., there can be undecided nodes)(3). We show that a sublinear (in n, number of nodes) message complexity BA protocol under honest majority is possible in the standard PKI model when the nodes have access to an unbiased global coin and hash function. In particular, we present a randomized Byzantine agreement algorithm which, with high probability achieves implicit agreement, uses O(root n) messages, and runs in O(1) rounds while tolerating (1/2-& varepsilon;)n Byzantine nodes for any fixed & varepsilon;>0, the notation O hides a O(polylogn) factor(4). The algorithm requires standard cryptographic setup PKI and hash function with a static Byzantine adversary. The algorithm works in the CONGEST model and each node does not need to know the identity of its neighbors, i.e., works in the KT0 model. The message complexity (and also the time complexity) of our algorithm is optimal up to a polylog n factor, as we show a Omega(n) lower bound on the message complexity. We further extend the result to Byzantine subset agreement, where a non-empty subset of nodes should agree on a common value. Lastly, we analyze several relevant results that follow from the construction of the main result. To the best of our knowledge, this is the first sublinear message complexity result of Byzantine agreement. A quadratic message lower bound is known for any deterministic BA protocol (due to Dolev-Reischuk [JACM 1985]). The existing randomized BA protocols have at least quadratic message complexity in the honest majority setting. Our result shows the power of a global coin in achieving significant improv

关键词： Distributed algorithm randomized algorithm Byzantine agreement Message complexity Global coin Cryptographic assumptions

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HOUSEHOLDER QR FACTORIZATION WITH RANDOMIZATION FOR COLUMN PIVOTING (HQRRP)

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2017年第2期39卷 C96-C115页

作者： Martinsson, Per-Gunnar Orti, Gregorio Quintana Heavner, Nathan van de Geijn, Robert Univ Colorado Boulder Dept Appl Math Boulder CO 80309 USA Univ Jaume 1 Dept Ingn & Ciencia Comp Castellon de La Plana 12071 Spain Univ Texas Austin Dept Comp Sci Austin TX 78712 USA Univ Texas Austin Inst Computat Engn & Sci Austin TX 78712 USA

A fundamental problem when adding column pivoting to the Householder QR factorization is that only about half of the computation can be cast in terms of high performing matrix matrix multiplications, which greatly limits the benefits that can be derived from so-called blocking of algorithms. This paper describes a technique for selecting groups of pivot vectors by means of randomized projections. It is demonstrated that the asymptotic flop count for the proposed method is 2mn(2) (2/3)n(3) for an m x n matrix, identical to that of the best classical unblocked Householder QR factorization algorithm (with or without pivoting). Experiments demonstrate acceleration in speed of close to an order of magnitude relative to the GEQP3 function in LAPACK, when executed on a modern CPU with multiple cores. Further, experiments demonstrate that the quality of the randomized pivot selection strategy is roughly the same as that of classical column pivoting. The described algorithm is made available under open source license and can be used with LAPACK or libflame.

关键词： QR factorization Householder QR pivoting strategy randomized algorithm blocked algorithm

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A hybrid stochastic interpolation and compression method for kernel matrices

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JOURNAL OF COMPUTATIONAL PHYSICS 2023年 494卷 112491页

作者： Chen, Duan Univ North Carolina Charlotte Dept Math & Stat Charlotte NC 28223 USA

Kernel functions play a pivotal role in a wide range of scientific computing and machine learning problems, but they ofter result in dense kernel matrices that impose great challenges in computational costs at large scale. To address this issue, we develop a set of fast kernel matrix compressing algorithms, which can reduce computation cost of matrix operations in the related applications. The foundation of these algorithms is the polyharmonic spline interpolation, which encompass a set of radial basis functions that allow flexible choices of interpolating nodes, and a set of polynomial basis functions that guarantee the solvability and convergence of the interpolation. With these properties, original data points in the interacting kernel function can be randomly sampled with great flexibility, so the proposed method is suitable for complicated data structures, such as high -dimensionality, random distribution, or manifold. To further boost the algorithm accuracy and efficiency, our scheme incorporates a QR sampling strategy, and combined with a recently developed fast stochastic SVD to form a hybrid method. If the overall number of degree of freedom is N, then the compressing algorithm has complexity of O (N) for low -rank matrices, and O (N log N) for general matrices with a hierarchical structure. Numerical results for data on various domains and different kernel functions validate the accuracy and efficiency of the proposed method.(c) 2023 Elsevier Inc. All rights reserved.

关键词： Fast kernel compressing randomized algorithm Matrix approximation Polyharmonic spline interpolation Hybrid method

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An efficient probability-based VNS algorithm for delivery territory design

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COMPUTERS & OPERATIONS RESEARCH 2024年 170卷

作者： Aly, Ahmed Gabor, Adriana F. Mladenovic, Nenad Sleptchenko, Andrei Khalifa Univ Dept Math Abu Dhabi U Arab Emirates Khalifa Univ Dept Management Sci & Engn Abu Dhabi U Arab Emirates Khalifa Univ Res Ctr Digital Supply Chain & Operat Management Abu Dhabi U Arab Emirates

This paper deals with the Delivery Territory Design Problem (DTDP), a districting problem that often occurs in delivery operations. The goal of the problem is to construct clusters of nodes (territories) such that the maximum diameter of a territory is minimized, while the territories designed are balanced w.r.t. some performance measures. We propose to solve the DTDP using a Probabilistic Variable Neighborhood Search (ProbVNS) algorithm based on two local search procedures: a tailored randomized shake procedure that targets both a reduction of infeasibility and diversification, and a deterministic local search based on a linear combination of objective and constraint violation. In addition to searching in different neighborhoods, the ProbVNS also changes the search direction by exploring different penalties for violating constraints. Numerical experiments show that ProbVNS outperforms a recent GRASP with the Path-Relinking (PR) algorithm proposed in the literature in terms of feasibility and objective value. In the tested instances, ProbVNS obtained a lower infeasibility measure in 90% of the instances. For these instances, the average decrease in the objective value was 8.3%, with a maximum decrease of 51%. Finally, the running times of ProbVNS are, on average, 2.7 times lower than those of PR.

关键词： Territory design Clustering Variable neighborhood search (VNS) randomized algorithm

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An O(N log N) hierarchical random compression method for kernel matrices by sampling partial matrix entries

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JOURNAL OF COMPUTATIONAL PHYSICS 2019年 397卷

作者： Chen, Duan Cai, Wei Univ North Carolina Charlotte Dept Math & Stat Charlotte NC 28223 USA Southern Methodist Univ Dept Math Dallas TX 75275 USA

In this paper, we propose an O(N log N) hierarchical random compression method (HRCM) for kernel matrix compressing, which only requires sampling O(N log N) entries of a matrix. The HRCM combines the hierarchical framework of the H-matrix and a randomized sampling technique of column and row spaces for far-field interaction kernel matrix. We show that a uniform column/row sampling of a far-field kernel matrix, thus without the need and associated cost to pre-compute a costly sampling distribution, will give a low-rank compression of such low-rank matrix, independent of the matrix size and only dependent on the separation of the source and target locations. This far-field random compression technique is then implemented at each level of the hierarchical decomposition for general kernel matrices, resulting in an O(N log N) random compression method. Error and complexity analysis for the HRCM are included. Numerical results for electrostatic and low frequency Helmholtz wave kernels have validated the efficiency and accuracy of the proposed method in comparison of direct O(N-2) summations. (C) 2019 Elsevier Inc. All rights reserved.

关键词： Fast kernel summation randomized algorithm Matrix approximation Singular value decomposition H-matrix Hierarchical algorithms

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Exact simulation of diffusion first exit times: algorithm acceleration.

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JOURNAL OF MACHINE LEARNING RESEARCH 2022年 23卷

作者： Herrmann, Samuel Zucca, Cristina Univ Bourgogne Franche Inst Math Bourgogne IMB UMR 5584 CNRS F-21000 Dijon France Univ Torino Dept Math G Peano Via Carlo Alberto 10 I-10123 Turin Italy

In order to describe or estimate different quantities related to a specific random variable, it is of prime interest to numerically generate such a variate. In specific situations, the exact generation of random variables might be either momentarily unavailable or too expensive in terms of computation time. It therefore needs to be replaced by an approximation procedure. As was previously the case, the ambitious exact simulation of first exit times for diffusion processes was unreachable though it concerns many applications in different fields like mathematical finance, neuroscience or reliability. The usual way to describe first exit times was to use discretization schemes, that are of course approximation procedures. Recently, Herrmann and Zucca (Herrmann and Zucca, 2020) proposed a new algorithm, the so-called GDET-algorithm (General Diffusion Exit Time), which permits to simulate exactly the first exit time for one-dimensional diffusions. The only drawback of exact simulation methods using an acceptance-rejection sampling is their time consumption. In this paper the authors highlight an acceleration procedure for the GDET-algorithm based on a multi-armed bandit model. The efficiency of this acceleration is pointed out through numerical examples.

关键词： Exit time Brownian motion diffusion processes rejection sampling exact simulation multi-armed bandit randomized algorithm

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Fast Neighborhood Rendezvous

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IEICE TRANSACTIONS ON INFORMATION AND SYSTEMS 2022年第3期E105D卷 597-610页

作者： Eguchi, Ryota Kitamura, Naoki Izumi, Taisuke Nagoya Inst Technol Nagoya Aichi 4668555 Japan Osaka Univ Suita Osaka 5650871 Japan

In the rendezvous problem, two computing entities (called agents) located at different vertices in a graph have to meet at the same vertex. In this paper, we consider the synchronous neighborhood rendezvous problem, where the agents are initially located at two adjacent vertices. While this problem can be trivially solved in O(Delta) rounds (Delta is the maximum degree of the graph), it is highly challenging to reveal whether that problem can be solved in o(Delta) rounds, even assuming the rich computational capability of agents. The only known result is that the time complexity of O(root n) rounds is achievable if the graph is complete and agents are probabilistic, asymmetric, and can use whiteboards placed at vertices. Our main contribution is to clarify the situation (with respect to computational models and graph classes) admitting such a sublinear-time rendezvous algorithm. More precisely, we present two algorithms achieving fast rendezvous additionally assuming bounded minimum degree, unique vertex identifier, accessibility to neighborhood IDs, and randomization. The first algorithm runs within (O) over tilde( root n Delta/delta + n/delta) rounds for graphs of the minimum degree larger than root n, where n is the number of vertices in the graph, and delta is the minimum degree of the graph. The second algorithm assumes that the largest vertex ID is O(n), and achieves (O) over tilde (n/root delta)-round time complexity without using whiteboards. These algorithms attain o(Delta)-round complexity in the case of delta = omega(root n log n) and delta = omega(n(2/3) log(4/3) n) respectively. We also prove that four unconventional assumptions of our algorithm, bounded minimum degree, accessibility to neighborhood IDs, initial distance one, and randomization are all inherently necessary for attaining fast rendezvous. That is, one can obtain the Omega(n)-round lower bound if either one of them is removed.

关键词： rendezvous randomized algorithm sublinear-time algorithm whiteboards

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