DOCSLIB.ORG

A High Performance Block Eigensolver for Nuclear Configuration Interaction Calculations Hasan Metin Aktulga, Md

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

1 A High Performance Block Eigensolver for Nuclear Configuration Interaction Calculations Hasan Metin Aktulga, Md. Afibuzzaman, Samuel Williams, Aydın Buluc¸, Meiyue Shao, Chao Yang, Esmond G. Ng, Pieter Maris, James P. Vary Abstract—As on-node parallelism increases and the performance gap between the processor and the memory system widens, achieving high performance in large-scale scientific applications requires an architecture-aware design of algorithms and solvers. We focus on the eigenvalue problem arising in nuclear Configuration Interaction (CI) calculations, where a few extreme eigenpairs of a sparse symmetric matrix are needed. We consider a block iterative eigensolver whose main computational kernels are the multiplication of a sparse matrix with multiple vectors (SpMM), and tall-skinny matrix operations. We present techniques to significantly improve the SpMM and the transpose operation SpMMT by using the compressed sparse blocks (CSB) format. We achieve 3–4× speedup on the requisite operations over good implementations with the commonly used compressed sparse row (CSR) format. We develop a performance model that allows us to correctly estimate the performance of our SpMM kernel implementations, and we identify cache bandwidth as a potential performance bottleneck beyond DRAM. We also analyze and optimize the performance of LOBPCG kernels (inner product and linear combinations on multiple vectors) and show up to 15× speedup over using high performance BLAS libraries for these operations. The resulting high performance LOBPCG solver achieves 1.4× to 1.8× speedup over the existing Lanczos solver on a series of CI computations on high-end multicore architectures (Intel Xeons). We also analyze the performance of our techniques on an Intel Xeon Phi Knights Corner (KNC) processor. Keywords—Sparse Matrix Multiplication; Block Eigensolver; Configuration Interaction; Extended Roofline Model; Tall-Skinny Matrices F 1 INTRODUCTION of interest, a partial diagonalization of the large CI H^ 2 N×N The choice of numerical algorithms and how efficiently many-body Hamiltonian R is sufficient. More they can be implemented on high performance computer formally, we are interested in finding a small number of (HPC) systems critically affect the time-to-solution for extreme eigenpairs of a large, sparse, symmetric matrix: large-scale scientific applications. Several new numerical Hx^ i = λxi; i = 1; : : : ; m; m N: (1) techniques or adaptations of existing ones that can better leverage the massive parallelism available on modern Iterative methods such as the Lanczos and Jacobi– systems have been developed over the years. Although Davidson [2] algorithms, as well as their variants [3], these algorithms may have slower convergence rates, [4], [5], can be used for this purpose. The key ker- their high degree of parallelism may lead to better time- nels for these methods can be crudely summarized as to-solution on modern hardware [1]. In this paper, we (repeated) sparse matrix–vector multiplications (SpMV) consider the solution of the quantum many-body prob- and orthonormalization of vectors (level-1 BLAS). As lem using the configuration interaction (CI) formulation. alternatives, block versions of these algorithms have We present algorithms and techniques to significantly been developed [6], [7], [8] which improve the arithmetic speed up eigenvalue computations in CI by using a intensity of computations at the cost of a reduced con- block eigensolver and optimizing the key computational vergence rate and increased total number of matrix– kernels involved. vector operations [9]. In block methods, SpMV becomes The quantum many-body problem transcends several a sparse matrix multiple vector multiplication (SpMM) areas of physics and chemistry. The CI method enables and vector operations become level-3 BLAS operations. computing the wave functions associated with discrete Related Work: Due to its importance in scientific com- energy levels of these many-body systems with high puting and machine learning, several optimization tech- accuracy. Since only a small number of low energy states niques have been proposed for SpMV [10], [11], [12], [13], are typically needed to compute the physical observables [14], [15]. Performance of SpMV is ultimately bounded by memory bandwidth [16]. The widening gap between • H. M. Aktulga and M. Afibuzzaman are with Michigan State University, 428 S. Shaw Lane, Room 3115, East Lansing, MI 48824. H. M. Aktulga processor performance and memory bandwidth signifi- also has an affiliate appointment with the Lawrence Berkeley National cantly limits the achievable performance in several im- Laboratory. Corresponding author e-mail: [email protected]. portant applications. On the other hand, in SpMM, one • S. Williams, A. Bulu¸c, M. Shao, C. Yang, and E. G. Ng are with the Computational Research Division, Lawrence Berkeley National Laboratory, can make use of the increased data locality in the vector 1 Cyclotron Rd, MS 50F-1650 Berkeley, CA 94720. block and attain much higher FLOP rates on modern • P. Maris and J. Vary are with Iowa State University, Dept. of Physics and architectures. Gropp et al. was the first to exploit this Astronomy, Ames, IA 50011. idea by using multiple right hand sides for SpMV in a 2 computational fluid dynamics application [17]. SpMM nuclear CI code [23], [24], [25]. We demonstrate through is one of the core operations supported by the auto- numerical experiments that the resulting block eigen- tuned sequential sparse matrix library OSKI [12]. OSKI’s solver can outperform the widely used Lanczos algo- shared memory parallel successor, pOSKI, currently does rithm (based on single vector iterations) with modern not support SpMM [18]. More recently, Liu et al. [1] multicore architectures (Sect. 5.4). We also analyze the investigated strategies to improve the performance of performance of our techniques on an Intel Xeon Phi SpMM 1 using SIMD (AVX/SSE) instructions for Stoke- Knights Corner (KNC) processor to assess the feasibility sian dynamics simulation of biological macromolecules of our implementations for future architectures. on modern multicore CPUs. Rohrig-Z¨ ollner¨ et al. [19] While we focus on nuclear CI computations, the im- discuss performance optimization techniques for the pact of optimizing the performance of key kernels in block Jacobi–Davidson method to compute a few eigen- block iterative solvers is broader. For example, spectral pairs of large-scale sparse matrices, and report reduced clustering, one of the most promising clustering tech- time-to-solution using block methods over single vec- niques, uses eigenvectors associated with the smallest tor counterparts for quantum mechanics problems and eigenvalues of the Laplacian of the data similarity matrix PDEs. Finally, Anzt et al. [20] describe an SpMM imple- to cluster vertices in large symmetric graphs [26], [27]. mentation based on the SELLC matrix format, and show Due to the size of the graphs, it is desirable to exploit that performance improvements in the SpMM kernel the symmetry, and for a k-way clustering problem, k can translate into performance improvements in a block eigenvectors are needed, where typically 10 ≤ k ≤ 100, eigensolver running on GPUs. an ideal range for block eigensolvers. Block methods are Our Contributions: Our work differs from previous also used in solving large-scale sparse singular value efforts substantially, in part due to the immense size decomposition (SVD) problems [28], with most popular of the sparse matrices (with dimensions on the order methods being the subspace iteration and block Lanzcos. of several billions and total number of nonzero matrix SVDs are critical for dimensionality reduction in applica- elements on the order of tens of trillions) involved. We tions like latent semantic indexing [29]. In SVD, singular therefore exploit symmetry to reduce the overall mem- values are obtained by solving the associated symmet- ric eigenproblem that requires subsequent SpMM and ory footprint, and offer an efficient solution to perform T SpMM on a sparse matrix and its transpose (SpMMT ) SpMM computations in each iteration [30]. Thus, our with roughly the same performance [21]. This is achieved techniques can have a positive impact on the adoption through a novel thread parallel SpMM implementation, of block solvers in closely related applications. CSB/OpenMP, which is based on the Compressed Sparse 2 BACKGROUND Blocks (CSB) storage format [22] (Sect.3). We demon- 2.1 Eigenvalue Problem in CI Calculations strate the efficiency of CSB/OpenMP on a series of CI matrices where we obtain 3–4× speedup over the Computational simulations of nuclear systems face mul- commonly used compressed sparse row (CSR) format. tiple hurdles, as the underlying physics involves a To estimate the performance characteristics and better very strong interaction, three-nucleon interactions, and understand the bottlenecks of the SpMM kernel, we complicated collective motion dynamics. The eigenvalue propose an extended Roofline model to account for cache problem arises in nuclear structure calculations because bandwidth limitations (Sect.3). the nuclear wave functions Ψ are solutions of the many- In this paper, we extend our previous work (presented body Schrodinger’s¨ equation expressed as a Hamiltonian in [21]) by considering an end-to-end optimization of a matrix eigenvalue problem, HΨ = EΨ. block eigensolver. As discussed in Sect.4, performance of In the CI approach, both the wave functions Ψ and the tall-skinny matrix operations in block
Recommended publications
  • CUDA 6 and Beyond

    CUDA 6 and Beyond

    MUMPS USERS DAYS JUNE 1ST / 2ND 2017 Programming heterogeneous architectures with libraries: A survey of NVIDIA linear algebra libraries François Courteille |Principal Solutions Architect, NVIDIA |[email protected] ACKNOWLEDGEMENTS Joe Eaton , NVIDIA Ty McKercher , NVIDIA Lung Sheng Chien , NVIDIA Nikolay Markovskiy , NVIDIA Stan Posey , NVIDIA Steve Rennich , NVIDIA Dave Miles , NVIDIA Peng Wang, NVIDIA Questions: [email protected] 2 AGENDA Prolegomena NVIDIA Solutions for Accelerated Linear Algebra Libraries performance on Pascal Rapid software development for heterogeneous architecture 3 PROLEGOMENA 124 NVIDIA Gaming VR AI & HPC Self-Driving Cars GPU Computing 5 ONE ARCHITECTURE BUILT FOR BOTH DATA SCIENCE & COMPUTATIONAL SCIENCE AlexNet Training Performance 70x Pascal [CELLR ANGE] 60x 16nm FinFET 50x 40x CoWoS HBM2 30x 20x [CELLR ANGE] NVLink 10x [CELLR [CELLR ANGE] ANGE] 0x 2013 2014 2015 2016 cuDNN Pascal & Volta NVIDIA DGX-1 NVIDIA DGX SATURNV 65x in 3 Years 6 7 8 8 9 NVLINK TO CPU IBM Power Systems Server S822LC (codename “Minsky”) DDR4 DDR4 2x IBM Power8+ CPUs and 4x P100 GPUs 115GB/s 80 GB/s per GPU bidirectional for peer traffic IB P8+ CPU P8+ CPU IB 80 GB/s per GPU bidirectional to CPU P100 P100 P100 P100 115 GB/s CPU Memory Bandwidth Direct Load/store access to CPU Memory High Speed Copy Engines for bulk data movement 1615 UNIFIED MEMORY ON PASCAL Large datasets, Simple programming, High performance CUDA 8 Enable Large Oversubscribe GPU memory Pascal Data Models Allocate up to system memory size CPU GPU Higher Demand
  • Singular Value Decomposition (SVD)

    Singular Value Decomposition (SVD)

    San José State University Math 253: Mathematical Methods for Data Visualization Lecture 5: Singular Value Decomposition (SVD) Dr. Guangliang Chen Outline • Matrix SVD Singular Value Decomposition (SVD) Introduction We have seen that symmetric matrices are always (orthogonally) diagonalizable. That is, for any symmetric matrix A ∈ Rn×n, there exist an orthogonal matrix Q = [q1 ... qn] and a diagonal matrix Λ = diag(λ1, . , λn), both real and square, such that A = QΛQT . We have pointed out that λi’s are the eigenvalues of A and qi’s the corresponding eigenvectors (which are orthogonal to each other and have unit norm). Thus, such a factorization is called the eigendecomposition of A, also called the spectral decomposition of A. What about general rectangular matrices? Dr. Guangliang Chen | Mathematics & Statistics, San José State University3/22 Singular Value Decomposition (SVD) Existence of the SVD for general matrices Theorem: For any matrix X ∈ Rn×d, there exist two orthogonal matrices U ∈ Rn×n, V ∈ Rd×d and a nonnegative, “diagonal” matrix Σ ∈ Rn×d (of the same size as X) such that T Xn×d = Un×nΣn×dVd×d. Remark. This is called the Singular Value Decomposition (SVD) of X: • The diagonals of Σ are called the singular values of X (often sorted in decreasing order). • The columns of U are called the left singular vectors of X. • The columns of V are called the right singular vectors of X. Dr. Guangliang Chen | Mathematics & Statistics, San José State University4/22 Singular Value Decomposition (SVD) * * b * b (n>d) b b b * b = * * = b b b * (n<d) * b * * b b Dr.
  • Segment-Based Clustering of Hyperspectral Images Using Tree-Based Data Partitioning Structures

    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.
  • Eigen Values and Vectors Matrices and Eigen Vectors

    Eigen Values and Vectors Matrices and Eigen Vectors

    EIGEN VALUES AND VECTORS MATRICES AND EIGEN VECTORS 2 3 1 11 × = [2 1] [3] [ 5 ] 2 3 3 12 3 × = = 4 × [2 1] [2] [ 8 ] [2] • Scale 3 6 2 × = [2] [4] 2 3 6 24 6 × = = 4 × [2 1] [4] [16] [4] 2 EIGEN VECTOR - PROPERTIES • Eigen vectors can only be found for square matrices • Not every square matrix has eigen vectors. • Given an n x n matrix that does have eigenvectors, there are n of them for example, given a 3 x 3 matrix, there are 3 eigenvectors. • Even if we scale the vector by some amount, we still get the same multiple 3 EIGEN VECTOR - PROPERTIES • Even if we scale the vector by some amount, we still get the same multiple • Because all you’re doing is making it longer, not changing its direction. • All the eigenvectors of a matrix are perpendicular or orthogonal. • This means you can express the data in terms of these perpendicular eigenvectors. • Also, when we find eigenvectors we usually normalize them to length one. 4 EIGEN VALUES - PROPERTIES • Eigenvalues are closely related to eigenvectors. • These scale the eigenvectors • eigenvalues and eigenvectors always come in pairs. 2 3 6 24 6 × = = 4 × [2 1] [4] [16] [4] 5 SPECTRAL THEOREM Theorem: If A ∈ ℝm×n is symmetric matrix (meaning AT = A), then, there exist real numbers (the eigenvalues) λ1, …, λn and orthogonal, non-zero real vectors ϕ1, ϕ2, …, ϕn (the eigenvectors) such that for each i = 1,2,…, n : Aϕi = λiϕi 6 EXAMPLE 30 28 A = [28 30] From spectral theorem: Aϕ = λϕ 7 EXAMPLE 30 28 A = [28 30] From spectral theorem: Aϕ = λϕ ⟹ Aϕ − λIϕ = 0 (A − λI)ϕ = 0 30 − λ 28 = 0 ⟹ λ = 58 and
  • 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

    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

    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.
  • A Parallel Lanczos Algorithm for Eigensystem Calculation

    A Parallel Lanczos Algorithm for Eigensystem Calculation

    A Parallel Lanczos Algorithm for Eigensystem Calculation Hans-Peter Kersken / Uwe Küster Eigenvalue problems arise in many fields of physics and engineering science for example in structural engineering from prediction of dynamic stability of structures or vibrations of a fluid in a closed cavity. Their solution consumes a large amount of memory and CPU time if more than very few (1-5) vibrational modes are desired. Both make the problem a natural candidate for parallel processing. Here we shall present some aspects of the solution of the generalized eigenvalue problem on parallel architectures with distributed memory. The research was carried out as a part of the EC founded project ACTIVATE. This project combines various activities to increase the performance vibro-acoustic software. It brings together end users form space, aviation, and automotive industry with software developers and numerical analysts. Introduction The algorithm we shall describe in the next sections is based on the Lanczos algorithm for solving large sparse eigenvalue problems. It became popular during the past two decades because of its superior convergence properties compared to more traditional methods like inverse vector iteration. Some experiments with an new algorithm described in [Bra97] showed that it is competitive with the Lanczos algorithm only if very few eigenpairs are needed. Therefore we decided to base our implementation on the Lanczos algorithm. However, when implementing the Lanczos algorithm one has to pay attention to some subtle algorithmic details. A state of the art algorithm is described in [Gri94]. On parallel architectures further problems arise concerning the robustness of the algorithm. We implemented the algorithm almost entirely by using numerical libraries.
  • Singular Value Decomposition (SVD) 2 1.1 Singular Vectors

    Singular Value Decomposition (SVD) 2 1.1 Singular Vectors

    Contents 1 Singular Value Decomposition (SVD) 2 1.1 Singular Vectors . .3 1.2 Singular Value Decomposition (SVD) . .7 1.3 Best Rank k Approximations . .8 1.4 Power Method for Computing the Singular Value Decomposition . 11 1.5 Applications of Singular Value Decomposition . 16 1.5.1 Principal Component Analysis . 16 1.5.2 Clustering a Mixture of Spherical Gaussians . 16 1.5.3 An Application of SVD to a Discrete Optimization Problem . 22 1.5.4 SVD as a Compression Algorithm . 24 1.5.5 Spectral Decomposition . 24 1.5.6 Singular Vectors and ranking documents . 25 1.6 Bibliographic Notes . 27 1.7 Exercises . 28 1 1 Singular Value Decomposition (SVD) The singular value decomposition of a matrix A is the factorization of A into the product of three matrices A = UDV T where the columns of U and V are orthonormal and the matrix D is diagonal with positive real entries. The SVD is useful in many tasks. Here we mention some examples. First, in many applications, the data matrix A is close to a matrix of low rank and it is useful to find a low rank matrix which is a good approximation to the data matrix . We will show that from the singular value decomposition of A, we can get the matrix B of rank k which best approximates A; in fact we can do this for every k. Also, singular value decomposition is defined for all matrices (rectangular or square) unlike the more commonly used spectral decomposition in Linear Algebra. The reader familiar with eigenvectors and eigenvalues (we do not assume familiarity here) will also realize that we need conditions on the matrix to ensure orthogonality of eigenvectors.
  • A Singularly Valuable Decomposition: the SVD of a Matrix Dan Kalman

    A Singularly Valuable Decomposition: the SVD of a Matrix Dan Kalman

    A Singularly Valuable Decomposition: The SVD of a Matrix Dan Kalman Dan Kalman is an assistant professor at American University in Washington, DC. From 1985 to 1993 he worked as an applied mathematician in the aerospace industry. It was during that period that he first learned about the SVD and its applications. He is very happy to be teaching again and is avidly following all the discussions and presentations about the teaching of linear algebra. Every teacher of linear algebra should be familiar with the matrix singular value deco~??positiolz(or SVD). It has interesting and attractive algebraic properties, and conveys important geometrical and theoretical insights about linear transformations. The close connection between the SVD and the well-known theo1-j~of diagonalization for sylnmetric matrices makes the topic immediately accessible to linear algebra teachers and, indeed, a natural extension of what these teachers already know. At the same time, the SVD has fundamental importance in several different applications of linear algebra. Gilbert Strang was aware of these facts when he introduced the SVD in his now classical text [22, p. 1421, obselving that "it is not nearly as famous as it should be." Golub and Van Loan ascribe a central significance to the SVD in their defini- tive explication of numerical matrix methods [8, p, xivl, stating that "perhaps the most recurring theme in the book is the practical and theoretical value" of the SVD. Additional evidence of the SVD's significance is its central role in a number of re- cent papers in :Matlgenzatics ivlagazine and the Atnericalz Mathematical ilironthly; for example, [2, 3, 17, 231.
  • Learning on Hypergraphs: Spectral Theory and Clustering

    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.
  • Exact Diagonalization of Quantum Lattice Models on Coprocessors

    Exact Diagonalization of Quantum Lattice Models on Coprocessors

    Exact diagonalization of quantum lattice models on coprocessors T. Siro∗, A. Harju Aalto University School of Science, P.O. Box 14100, 00076 Aalto, Finland Abstract We implement the Lanczos algorithm on an Intel Xeon Phi coprocessor and compare its performance to a multi-core Intel Xeon CPU and an NVIDIA graphics processor. The Xeon and the Xeon Phi are parallelized with OpenMP and the graphics processor is programmed with CUDA. The performance is evaluated by measuring the execution time of a single step in the Lanczos algorithm. We study two quantum lattice models with different particle numbers, and conclude that for small systems, the multi-core CPU is the fastest platform, while for large systems, the graphics processor is the clear winner, reaching speedups of up to 7.6 compared to the CPU. The Xeon Phi outperforms the CPU with sufficiently large particle number, reaching a speedup of 2.5. Keywords: Tight binding, Hubbard model, exact diagonalization, GPU, CUDA, MIC, Xeon Phi 1. Introduction hopping amplitude is denoted by t. The tight-binding model describes free electrons hopping around a lattice, and it gives a In recent years, there has been tremendous interest in utiliz- crude approximation of the electronic properties of a solid. The ing coprocessors in scientific computing, including condensed model can be made more realistic by adding interactions, such matter physics[1, 2, 3, 4]. Most of the work has been done as on-site repulsion, which results in the well-known Hubbard on graphics processing units (GPU), resulting in impressive model[9]. In our basis, however, such interaction terms are di- speedups compared to CPUs in problems that exhibit high data- agonal, rendering their effect on the computational complexity parallelism and benefit from the high throughput of the GPU.
  • Fast Approximate Spectral Clustering

    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.