Preconditioned Spectral Clustering for Stochastic Block Partition Streaming Graph Challenge
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Recursive Approach in Sparse Matrix LU Factorization
51 Recursive approach in sparse matrix LU factorization Jack Dongarra, Victor Eijkhout and the resulting matrix is often guaranteed to be positive Piotr Łuszczek∗ definite or close to it. However, when the linear sys- University of Tennessee, Department of Computer tem matrix is strongly unsymmetric or indefinite, as Science, Knoxville, TN 37996-3450, USA is the case with matrices originating from systems of Tel.: +865 974 8295; Fax: +865 974 8296 ordinary differential equations or the indefinite matri- ces arising from shift-invert techniques in eigenvalue methods, one has to revert to direct methods which are This paper describes a recursive method for the LU factoriza- the focus of this paper. tion of sparse matrices. The recursive formulation of com- In direct methods, Gaussian elimination with partial mon linear algebra codes has been proven very successful in pivoting is performed to find a solution of Eq. (1). Most dense matrix computations. An extension of the recursive commonly, the factored form of A is given by means technique for sparse matrices is presented. Performance re- L U P Q sults given here show that the recursive approach may per- of matrices , , and such that: form comparable to leading software packages for sparse ma- LU = PAQ, (2) trix factorization in terms of execution time, memory usage, and error estimates of the solution. where: – L is a lower triangular matrix with unitary diago- nal, 1. Introduction – U is an upper triangular matrix with arbitrary di- agonal, Typically, a system of linear equations has the form: – P and Q are row and column permutation matri- Ax = b, (1) ces, respectively (each row and column of these matrices contains single a non-zero entry which is A n n A ∈ n×n x where is by real matrix ( R ), and 1, and the following holds: PPT = QQT = I, b n b, x ∈ n and are -dimensional real vectors ( R ). -
Abstract 1 Introduction
Implementation in ScaLAPACK of Divide-and-Conquer Algorithms for Banded and Tridiagonal Linear Systems A. Cleary Department of Computer Science University of Tennessee J. Dongarra Department of Computer Science University of Tennessee Mathematical Sciences Section Oak Ridge National Laboratory Abstract Described hereare the design and implementation of a family of algorithms for a variety of classes of narrow ly banded linear systems. The classes of matrices include symmetric and positive de - nite, nonsymmetric but diagonal ly dominant, and general nonsymmetric; and, al l these types are addressed for both general band and tridiagonal matrices. The family of algorithms captures the general avor of existing divide-and-conquer algorithms for banded matrices in that they have three distinct phases, the rst and last of which arecompletely paral lel, and the second of which is the par- al lel bottleneck. The algorithms have been modi ed so that they have the desirable property that they are the same mathematical ly as existing factorizations Cholesky, Gaussian elimination of suitably reordered matrices. This approach represents a departure in the nonsymmetric case from existing methods, but has the practical bene ts of a smal ler and more easily hand led reduced system. All codes implement a block odd-even reduction for the reduced system that al lows the algorithm to scale far better than existing codes that use variants of sequential solution methods for the reduced system. A cross section of results is displayed that supports the predicted performance results for the algo- rithms. Comparison with existing dense-type methods shows that for areas of the problem parameter space with low bandwidth and/or high number of processors, the family of algorithms described here is superior. -
Overview of Iterative Linear System Solver Packages
Overview of Iterative Linear System Solver Packages Victor Eijkhout July, 1998 Abstract Description and comparison of several packages for the iterative solu- tion of linear systems of equations. 1 1 Intro duction There are several freely available packages for the iterative solution of linear systems of equations, typically derived from partial di erential equation prob- lems. In this rep ort I will give a brief description of a numberofpackages, and giveaninventory of their features and de ning characteristics. The most imp ortant features of the packages are which iterative metho ds and preconditioners supply; the most relevant de ning characteristics are the interface they present to the user's data structures, and their implementation language. 2 2 Discussion Iterative metho ds are sub ject to several design decisions that a ect ease of use of the software and the resulting p erformance. In this section I will give a global discussion of the issues involved, and how certain p oints are addressed in the packages under review. 2.1 Preconditioners A go o d preconditioner is necessary for the convergence of iterative metho ds as the problem to b e solved b ecomes more dicult. Go o d preconditioners are hard to design, and this esp ecially holds true in the case of parallel pro cessing. Here is a short inventory of the various kinds of preconditioners found in the packages reviewed. 2.1.1 Ab out incomplete factorisation preconditioners Incomplete factorisations are among the most successful preconditioners devel- op ed for single-pro cessor computers. Unfortunately, since they are implicit in nature, they cannot immediately b e used on parallel architectures. -
Segment-Based Clustering of Hyperspectral Images Using Tree-Based Data Partitioning Structures
algorithms Article Segment-Based Clustering of Hyperspectral Images Using Tree-Based Data Partitioning Structures Mohamed Ismail and Milica Orlandi´c* Department of Electronic Systems , NTNU: Norwegian University of Science and Technology (NTNU), 7491 Trondheim, Norway; [email protected] * Correspondence: [email protected] Received: 2 October 2020; Accepted: 8 December 2020; Published: 10 December 2020 Abstract: Hyperspectral image classification has been increasingly used in the field of remote sensing. In this study, a new clustering framework for large-scale hyperspectral image (HSI) classification is proposed. The proposed four-step classification scheme explores how to effectively use the global spectral information and local spatial structure of hyperspectral data for HSI classification. Initially, multidimensional Watershed is used for pre-segmentation. Region-based hierarchical hyperspectral image segmentation is based on the construction of Binary partition trees (BPT). Each segmented region is modeled while using first-order parametric modelling, which is then followed by a region merging stage using HSI regional spectral properties in order to obtain a BPT representation. The tree is then pruned to obtain a more compact representation. In addition, principal component analysis (PCA) is utilized for HSI feature extraction, so that the extracted features are further incorporated into the BPT. Finally, an efficient variant of k-means clustering algorithm, called filtering algorithm, is deployed on the created BPT structure, producing the final cluster map. The proposed method is tested over eight publicly available hyperspectral scenes with ground truth data and it is further compared with other clustering frameworks. The extensive experimental analysis demonstrates the efficacy of the proposed method. -
Accelerating the LOBPCG Method on Gpus Using a Blocked Sparse Matrix Vector Product
Accelerating the LOBPCG method on GPUs using a blocked Sparse Matrix Vector Product Hartwig Anzt and Stanimire Tomov and Jack Dongarra Innovative Computing Lab University of Tennessee Knoxville, USA Email: [email protected], [email protected], [email protected] Abstract— the computing power of today’s supercomputers, often accel- erated by coprocessors like graphics processing units (GPUs), This paper presents a heterogeneous CPU-GPU algorithm design and optimized implementation for an entire sparse iter- becomes challenging. ative eigensolver – the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) – starting from low-level GPU While there exist numerous efforts to adapt iterative lin- data structures and kernels to the higher-level algorithmic choices ear solvers like Krylov subspace methods to coprocessor and overall heterogeneous design. Most notably, the eigensolver technology, sparse eigensolvers have so far remained out- leverages the high-performance of a new GPU kernel developed side the main focus. A possible explanation is that many for the simultaneous multiplication of a sparse matrix and a of those combine sparse and dense linear algebra routines, set of vectors (SpMM). This is a building block that serves which makes porting them to accelerators more difficult. Aside as a backbone for not only block-Krylov, but also for other from the power method, algorithms based on the Krylov methods relying on blocking for acceleration in general. The subspace idea are among the most commonly used general heterogeneous LOBPCG developed here reveals the potential of eigensolvers [1]. When targeting symmetric positive definite this type of eigensolver by highly optimizing all of its components, eigenvalue problems, the recently developed Locally Optimal and can be viewed as a benchmark for other SpMM-dependent applications. -
Image Segmentation Based on Multiscale Fast Spectral Clustering Chongyang Zhang, Guofeng Zhu, Minxin Chen, Hong Chen, Chenjian Wu
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. XX, NO. X, XX 2018 1 Image Segmentation Based on Multiscale Fast Spectral Clustering Chongyang Zhang, Guofeng Zhu, Minxin Chen, Hong Chen, Chenjian Wu Abstract—In recent years, spectral clustering has become one In recent years, researchers have proposed various ap- of the most popular clustering algorithms for image segmenta- proaches to large-scale image segmentation. The approaches tion. However, it has restricted applicability to large-scale images are based on three following strategies: constructing a sparse due to its high computational complexity. In this paper, we first propose a novel algorithm called Fast Spectral Clustering similarity matrix, using Nystrom¨ approximation and using based on quad-tree decomposition. The algorithm focuses on representative points. The following approaches are based the spectral clustering at superpixel level and its computational on constructing a sparse similarity matrix. In 2000, Shi and 3 complexity is O(n log n) + O(m) + O(m 2 ); its memory cost Malik [18] constructed the similarity matrix of the image by is O(m), where n and m are the numbers of pixels and the using the k-nearest neighbor sparse strategy to reduce the superpixels of a image. Then we propose Multiscale Fast Spectral complexity of constructing the similarity matrix to O(n) and Clustering by improving Fast Spectral Clustering, which is based to reduce the complexity of eigen-decomposing the Laplacian on the hierarchical structure of the quad-tree. The computational 3 complexity of Multiscale Fast Spectral Clustering is O(n log n) matrix to O(n 2 ) by using the Lanczos algorithm. -
Learning on Hypergraphs: Spectral Theory and Clustering
Learning on Hypergraphs: Spectral Theory and Clustering Pan Li, Olgica Milenkovic Coordinated Science Laboratory University of Illinois at Urbana-Champaign March 12, 2019 Learning on Graphs Graphs are indispensable mathematical data models capturing pairwise interactions: k-nn network social network publication network Important learning on graphs problems: clustering (community detection), semi-supervised/active learning, representation learning (graph embedding) etc. Beyond Pairwise Relations A graph models pairwise relations. Recent work has shown that high-order relations can be significantly more informative: Examples include: Understanding the organization of networks (Benson, Gleich and Leskovec'16) Determining the topological connectivity between data points (Zhou, Huang, Sch}olkopf'07). Graphs with high-order relations can be modeled as hypergraphs (formally defined later). Meta-graphs, meta-paths in heterogeneous information networks. Algorithmic methods for analyzing high-order relations and learning problems are still under development. Beyond Pairwise Relations Functional units in social and biological networks. High-order network motifs: Motif (Benson’16) Microfauna Pelagic fishes Crabs & Benthic fishes Macroinvertebrates Algorithmic methods for analyzing high-order relations and learning problems are still under development. Beyond Pairwise Relations Functional units in social and biological networks. Meta-graphs, meta-paths in heterogeneous information networks. (Zhou, Yu, Han'11) Beyond Pairwise Relations Functional units in social and biological networks. Meta-graphs, meta-paths in heterogeneous information networks. Algorithmic methods for analyzing high-order relations and learning problems are still under development. Review of Graph Clustering: Notation and Terminology Graph Clustering Task: Cluster the vertices that are \densely" connected by edges. Graph Partitioning and Conductance A (weighted) graph G = (V ; E; w): for e 2 E, we is the weight. -
Over-Scalapack.Pdf
1 2 The Situation: Parallel scienti c applications are typically Overview of ScaLAPACK written from scratch, or manually adapted from sequen- tial programs Jack Dongarra using simple SPMD mo del in Fortran or C Computer Science Department UniversityofTennessee with explicit message passing primitives and Mathematical Sciences Section Which means Oak Ridge National Lab oratory similar comp onents are co ded over and over again, co des are dicult to develop and maintain http://w ww .ne tl ib. org /ut k/p e opl e/J ackDo ngarra.html debugging particularly unpleasant... dicult to reuse comp onents not really designed for long-term software solution 3 4 What should math software lo ok like? The NetSolve Client Many more p ossibilities than shared memory Virtual Software Library Multiplicityofinterfaces ! Ease of use Possible approaches { Minimum change to status quo Programming interfaces Message passing and ob ject oriented { Reusable templates { requires sophisticated user Cinterface FORTRAN interface { Problem solving environments Integrated systems a la Matlab, Mathematica use with heterogeneous platform Interactiveinterfaces { Computational server, like netlib/xnetlib f2c Non-graphic interfaces Some users want high p erf., access to details {MATLAB interface Others sacri ce sp eed to hide details { UNIX-Shell interface Not enough p enetration of libraries into hardest appli- Graphic interfaces cations on vector machines { TK/TCL interface Need to lo ok at applications and rethink mathematical software { HotJava Browser -
Fast Approximate Spectral Clustering
Fast Approximate Spectral Clustering Donghui Yan Ling Huang Michael I. Jordan University of California Intel Research Lab University of California Berkeley, CA 94720 Berkeley, CA 94704 Berkeley, CA 94720 [email protected] [email protected] [email protected] ABSTRACT variety of methods have been developed over the past several decades Spectral clustering refers to a flexible class of clustering proce- to solve clustering problems [15, 19]. A relatively recent area dures that can produce high-quality clusterings on small data sets of focus has been spectral clustering, a class of methods based but which has limited applicability to large-scale problems due to on eigendecompositions of affinity, dissimilarity or kernel matri- its computational complexity of O(n3) in general, with n the num- ces [20, 28, 31]. Whereas many clustering methods are strongly ber of data points. We extend the range of spectral clustering by tied to Euclidean geometry, making explicit or implicit assump- developing a general framework for fast approximate spectral clus- tions that clusters form convex regions in Euclidean space, spectral tering in which a distortion-minimizing local transformation is first methods are more flexible, capturing a wider range of geometries. applied to the data. This framework is based on a theoretical anal- They often yield superior empirical performance when compared ysis that provides a statistical characterization of the effect of local to competing algorithms such as k-means, and they have been suc- distortion on the mis-clustering rate. We develop two concrete in- cessfully deployed in numerous applications in areas such as com- stances of our general framework, one based on local k-means clus- puter vision, bioinformatics, and robotics. -
An Algorithmic Introduction to Clustering
AN ALGORITHMIC INTRODUCTION TO CLUSTERING APREPRINT Bernardo Gonzalez Department of Computer Science and Engineering UCSC [email protected] June 11, 2020 The purpose of this document is to provide an easy introductory guide to clustering algorithms. Basic knowledge and exposure to probability (random variables, conditional probability, Bayes’ theorem, independence, Gaussian distribution), matrix calculus (matrix and vector derivatives), linear algebra and analysis of algorithm (specifically, time complexity analysis) is assumed. Starting with Gaussian Mixture Models (GMM), this guide will visit different algorithms like the well-known k-means, DBSCAN and Spectral Clustering (SC) algorithms, in a connected and hopefully understandable way for the reader. The first three sections (Introduction, GMM and k-means) are based on [1]. The fourth section (SC) is based on [2] and [5]. Fifth section (DBSCAN) is based on [6]. Sixth section (Mean Shift) is, to the best of author’s knowledge, original work. Seventh section is dedicated to conclusions and future work. Traditionally, these five algorithms are considered completely unrelated and they are considered members of different families of clustering algorithms: • Model-based algorithms: the data are viewed as coming from a mixture of probability distributions, each of which represents a different cluster. A Gaussian Mixture Model is considered a member of this family • Centroid-based algorithms: any data point in a cluster is represented by the central vector of that cluster, which need not be a part of the dataset taken. k-means is considered a member of this family • Graph-based algorithms: the data is represented using a graph, and the clustering procedure leverage Graph theory tools to create a partition of this graph. -
Solving Symmetric Semi-Definite (Ill-Conditioned)
Solving Symmetric Semi-definite (ill-conditioned) Generalized Eigenvalue Problems Zhaojun Bai University of California, Davis Berkeley, August 19, 2016 Symmetric definite generalized eigenvalue problem I Symmetric definite generalized eigenvalue problem Ax = λBx where AT = A and BT = B > 0 I Eigen-decomposition AX = BXΛ where Λ = diag(λ1; λ2; : : : ; λn) X = (x1; x2; : : : ; xn) XT BX = I: I Assume λ1 ≤ λ2 ≤ · · · ≤ λn LAPACK solvers I LAPACK routines xSYGV, xSYGVD, xSYGVX are based on the following algorithm (Wilkinson'65): 1. compute the Cholesky factorization B = GGT 2. compute C = G−1AG−T 3. compute symmetric eigen-decomposition QT CQ = Λ 4. set X = G−T Q I xSYGV[D,X] could be numerically unstable if B is ill-conditioned: −1 jλbi − λij . p(n)(kB k2kAk2 + cond(B)jλbij) · and −1 1=2 kB k2kAk2(cond(B)) + cond(B)jλbij θ(xbi; xi) . p(n) · specgapi I User's choice between the inversion of ill-conditioned Cholesky decomposition and the QZ algorithm that destroys symmetry Algorithms to address the ill-conditioning 1. Fix-Heiberger'72 (Parlett'80): explicit reduction 2. Chang-Chung Chang'74: SQZ method (QZ by Moler and Stewart'73) 3. Bunse-Gerstner'84: MDR method 4. Chandrasekaran'00: \proper pivoting scheme" 5. Davies-Higham-Tisseur'01: Cholesky+Jacobi 6. Working notes by Kahan'11 and Moler'14 This talk Three approaches: 1. A LAPACK-style implementation of Fix-Heiberger algorithm 2. An algebraic reformulation 3. Locally accelerated block preconditioned steepest descent (LABPSD) This talk Three approaches: 1. A LAPACK-style implementation of Fix-Heiberger algorithm Status: beta-release 2. -
On the Performance and Energy Efficiency of Sparse Linear Algebra on Gpus
Original Article The International Journal of High Performance Computing Applications 1–16 On the performance and energy efficiency Ó The Author(s) 2016 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav of sparse linear algebra on GPUs DOI: 10.1177/1094342016672081 hpc.sagepub.com Hartwig Anzt1, Stanimire Tomov1 and Jack Dongarra1,2,3 Abstract In this paper we unveil some performance and energy efficiency frontiers for sparse computations on GPU-based super- computers. We compare the resource efficiency of different sparse matrix–vector products (SpMV) taken from libraries such as cuSPARSE and MAGMA for GPU and Intel’s MKL for multicore CPUs, and develop a GPU sparse matrix–matrix product (SpMM) implementation that handles the simultaneous multiplication of a sparse matrix with a set of vectors in block-wise fashion. While a typical sparse computation such as the SpMV reaches only a fraction of the peak of current GPUs, we show that the SpMM succeeds in exceeding the memory-bound limitations of the SpMV. We integrate this kernel into a GPU-accelerated Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) eigensolver. LOBPCG is chosen as a benchmark algorithm for this study as it combines an interesting mix of sparse and dense linear algebra operations that is typical for complex simulation applications, and allows for hardware-aware optimizations. In a detailed analysis we compare the performance and energy efficiency against a multi-threaded CPU counterpart. The reported performance and energy efficiency results are indicative of sparse computations on supercomputers. Keywords sparse eigensolver, LOBPCG, GPU supercomputer, energy efficiency, blocked sparse matrix–vector product 1Introduction GPU-based supercomputers.