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447
On the equivalence of nonnegative matrix factorization and spectral clustering
 in SIAM International Conference on Data Mining
, 2005
"... Current nonnegative matrix factorization (NMF) deals with X = FG T type. We provide a systematic analysis and extensions of NMF to the symmetric W = HH T, and the weighted W = HSHT. We show that (1) W = HHT is equivalent to Kernel Kmeans clustering and the Laplacianbased spectral clustering. (2) X ..."
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Cited by 159 (20 self)
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Current nonnegative matrix factorization (NMF) deals with X = FG T type. We provide a systematic analysis and extensions of NMF to the symmetric W = HH T, and the weighted W = HSHT. We show that (1) W = HHT is equivalent to Kernel Kmeans clustering and the Laplacianbased spectral clustering. (2
Efficient Parallel Solutions Of Large Sparse SPD Systems On DistributedMemory Multiprocessors
 Advanced Computing Research Institute, Center for Theory and Simulation in Science and Engineering, Cornell
"... . We consider several issues involved in the solution of sparse symmetric positive definite systems by multifrontal method on distributedmemory multiprocessors. First, we present a new algorithm for computing the partial factorization of a frontal matrix on a subset of processors which significantl ..."
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Cited by 16 (2 self)
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. We consider several issues involved in the solution of sparse symmetric positive definite systems by multifrontal method on distributedmemory multiprocessors. First, we present a new algorithm for computing the partial factorization of a frontal matrix on a subset of processors which
Multifrontal Parallel Distributed Symmetric and Unsymmetric Solvers
, 1998
"... We consider the solution of both symmetric and unsymmetric systems of sparse linear equations. A new parallel distributed memory multifrontal approach is described. To handle numerical pivoting efficiently, a parallel asynchronous algorithm with dynamic scheduling of the computing tasks has been dev ..."
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Cited by 187 (30 self)
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We consider the solution of both symmetric and unsymmetric systems of sparse linear equations. A new parallel distributed memory multifrontal approach is described. To handle numerical pivoting efficiently, a parallel asynchronous algorithm with dynamic scheduling of the computing tasks has been
The Design, Implementation, and Evaluation of a Symmetric Banded Linear Solver for DistributedMemory Parallel Computers
 ACM Trans. Math. Softw
, 1998
"... This article describes the design, implementation, and evaluation of a parallel algorithm for the Cholesky factorization of symmetric banded matrices. The algorithm is part of IBM's Parallel Engineering and Scientific Subroutine Library version 1.2 and is compatible with ScaLAPACK's banded ..."
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Cited by 10 (1 self)
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LAPACK's banded solver. Analysis, as well as experiments on an IBM SP2 distributedmemory parallel computer, shows that the algorithm efficiently factors banded matrices with wide bandwidth. For example, a 31node SP2 factors a large matrix more than 16 times faster than a single node would factor it using
WSSMP: A HighPerformance Shared and DistributedMemory Parallel Sparse Symmetric Linear Equation Solver
"... . The Watson Symmetric Sparse Matrix Package, WSSMP, is a highperformance, robust, and easy to use software package for solving large sparse symmetric systems of linear equations. It can be used as a serial package, or in a sharedmemory multiprocessor environment, or as a scalable parallel solver ..."
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. The Watson Symmetric Sparse Matrix Package, WSSMP, is a highperformance, robust, and easy to use software package for solving large sparse symmetric systems of linear equations. It can be used as a serial package, or in a sharedmemory multiprocessor environment, or as a scalable parallel solver
On reduced rank nonnegative matrix factorizations for symmetric matrices
 Special Issue on Positivity in Linear Algebra
, 2004
"... Let V ∈ Rm,n be a nonnegative matrix. The nonnegative matrix factorization (NNMF) problem consists of finding nonnegative matrix factors W ∈ Rm,r and H ∈ Rr,n such that V ≈ WH. Lee and Seung proposed two algorithms which find nonnegative W and H such that �V − WH�F is minimized. After examining the ..."
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Cited by 33 (4 self)
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approximation cannot be a symmetric matrix. Finally, we show that the class of positive semidefinite symmetric nonnegative matrices V generated via a Soules basis admit for every 1 ≤ r ≤ rank(V), a nonnegative factorization WH which coincides with the best approximation in the Frobenius norm to V in Rn
LU FACTORIZATION ALGORITHMS ON DISTRIBUTEDMEMORY MULTIPROCESSOR ARCHITECTURES*
"... Abstract. In this paper, we consider the effect that the datastorage scheme and pivoting scheme have on the efficiency of LU factorization on a distributedmemory multiprocessor. Our presentation will focus on the hypercube architecture, but most of our results are applicable to distributedmemory ..."
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Abstract. In this paper, we consider the effect that the datastorage scheme and pivoting scheme have on the efficiency of LU factorization on a distributedmemory multiprocessor. Our presentation will focus on the hypercube architecture, but most of our results are applicable to distributedmemory
Symmetric Nonnegative Matrix Factorization for Graph Clustering
"... Nonnegative matrix factorization (NMF) provides a lower rank approximation of a nonnegative matrix, and has been successfully used as a clustering method. In this paper, we offer some conceptual understanding for the capabilities and shortcomings of NMF as a clustering method. Then, we propose Symme ..."
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Cited by 21 (4 self)
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Symmetric NMF (SymNMF) as a general framework for graph clustering, which inherits the advantages of NMF by enforcing nonnegativity on the clustering assignment matrix. Unlike NMF, however, SymNMF is based on a similarity measure between data points, and factorizes a symmetric matrix containing pairwise
WSMP: A HighPerformance Shared and DistributedMemory Parallel Sparse Linear Equation Solver
, 2001
"... The Watson Sparse Matrix Package, WSMP, is a highperformance, robust, and easy to use software package for solving large sparse systems of linear equations. It can be used as a serial package, or in a sharedmemory multiprocessor environment, or as a scalable parallel solver in a messagepassing en ..."
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Cited by 6 (1 self)
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passing environment, where each node can either be a uniprocessor or a sharedmemory multiprocessor. A unique aspect of WSMP is that it exploits both SMP and MPP parallelism using Pthreads and MPI, respectively, while mostly shielding the user from the details of the architecture. Sparse symmetric factorization
Highly scalable parallel algorithms for sparse matrix factorization
 IEEE Transactions on Parallel and Distributed Systems
, 1994
"... In this paper, we describe a scalable parallel algorithm for sparse matrix factorization, analyze their performance and scalability, and present experimental results for up to 1024 processors on a Cray T3D parallel computer. Through our analysis and experimental results, we demonstrate that our algo ..."
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Cited by 130 (27 self)
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communication overhead and is more scalable than any previously known parallel formulation of sparse matrix factorization. Although, in this paper, we discuss Cholesky factorization of symmetric positive definite matrices, the algorithms can be adapted for solving sparse linear least squares problems
Results 1  10
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