38908855
Large-Scale SVD and Subspace-Based Methods for Information Retrieval
A theoretical foundation for latent semantic indexing (LSI) is proposed by adapting a model first used in array signal processing to the context of information retrieval using the concept of subspaces. It is shown that this subspace-based model coupled with minimal description length (MDL) principle leads to a statistical test to determine the dimensions of the latent-concept subspaces in LSI. The effect of weighting on the choice of the optimal dimensions of latent-concept subspaces is illustrated. It is also shown that the model imposes a so-called low-rank-plus-shift structure that is approximately satisfied by the cross-product of the term-document matrices. This structure can be exploited to give a more accurate updating scheme for LSI and to correct some of the misconception about the achievable retrieval accuracy in LSI updating. Variants of Lanczos algorithms are illustrated with numerical test results on Cray T3E using document collections generated from World Wide Web.
62731399
NAS parallel benchmark results
The NAS (Numerical Aerodynamic Simulation) parallel benchmarks have been developed at NASA Ames Research Center to study the performance of parallel supercomputer. The eight benchmark problems are specified in a 'pencil and paper' fashion. The performance results of various systems using the NAS parallel benchmarks are presented. These results represent the best results that have been reported to the authors for the specific systems listed. They represent implementation efforts performed by personnel in both the NAS Applied Research Branch of NASA Ames Research Center and in other organizations.<<ETX>>
6270234
Spectral Relaxation for K-means Clustering
The popular K-means clustering partitions a data set by minimizing a sum-of-squares cost function. A coordinate descend method is then used to find local minima. In this paper we show that the minimization can be reformulated as a trace maximization problem associated with the Gram matrix of the data vectors. Furthermore, we show that a relaxed version of the trace maximization problem possesses global optimal solutions which can be obtained by computing a partial eigendecomposition of the Gram matrix, and the cluster assignment for each data vectors can be found by computing a pivoted QR decomposition of the eigenvector matrix. As a by-product we also derive a lower bound for the minimum of the sum-of-squares cost function.
35264400
Parallel Efficiency of the Lanczos Method for Eigenvalue Problems
LBNL-42828 Parallel Efficiency of the Lanczos Method for Eigenvalue Problems ^ Kesheng Wu+and Horst Simon* Abstract Two of the commonly used versions of the Lanczos method for eigenvalues problems are the shift-and-invert Lanczos method and the restarted Lanczos method. In this talk, we will address two questions, is the shift-and-invert Lanczos method a viable option on massively parallel machines and which one is more appropriate for a given eigenvalue problem? Introduction T h i s talk is o n how to compute eigenvalues a n d eigenvectors of large sparse s y m m e t r i c matrices o n massively parallel machines. One of the the most c o m m o n l y used algorithms for this task is the Lanczos m e t h o d w h i c h projects the large eigenvalue p r o b l e m onto a low dimensional K r y l o v subspace [7, 10]. I n most cases, the K r y l o v subspace basis is built t h r o u g h a series of matrix-vector m u l t i p l i c a t i o n s , the whole Lanczos m e t h o d c a n be implemented w i t h a matrix-vector m u l t i p l i c a t i o n routine a n d a few simple vector operations. T h i s a l g o r i t h m is highly efficient on parallel machines a n d it is effective for c o m p u t i n g extreme a n d well separated eigenvalues. To compute the interior or not well-separated eigenvalues, the shift-and-invert Lanczos m e t h o d is one of the most effective methods. Since the eigenvalues of a m a t r i x A are related to the eigenvalues of (A — c r l ) by a simple relation X(A) = a + 1/X((A — c r l ) ) a n d the corresponding eigenvectors of A a n d (A — c r l ) are identical, the shift-and-invert scheme computes the extreme eigenvalues of (A — c r l ) a n d deduce the corresponding eigenvalues of A. W i t h appropriate choice of a, the extreme eigenvalues of (A — al)^ are well separated and can be easily computed. T h e shift-and-invert Lanczos m e t h o d needs to b u i l d a K r y l o v subspace basis of (A — c r l ) w h i c h is done by solving a series of linear systems involving the coefficient m a t r i x (A — a I). T h e linear systems need to be solved accurately a n d the only reliable means to accomplish this is by using a direct m e t h o d [5]. Because it is difficult to implement direct methods o n d i s t r i b u t e d parallel computers, efficient parallel i m p l e m e n t a t i o n has not been widely available u n t i l recently [1, 8]. T h i s talk w i l l present our s t u d y of the parallel efficiency of the shift-and-invert Lanczos m e t h o d using these newly available direct solvers. T h e study w i l l show that the shift-and-invert Lanczos m e t h o d is ' T h i s w o r k was s u p p o r t e d b y t h e D i r e c t o r , Office o f E n e r g y R e s e a r c h , Office o f L a b o r a t o r y P o l i c y a n d Infrastructure M a n a g e m e n t , of the U . S . D e p a r t m e n t of E n e r g y under C o n t r a c t N o . D E - A C 0 3 - 7 6 S F 0 0 0 9 8 . T h i s r e s e a r c h u s e d r e s o u r c e s o f t h e N a t i o n a l E n e r g y R e s e a r c h S c i e n t i f i c C o m p u t i n g C e n t e r , w h i c h is s u p p o r t e d b y t h e Office o f E n e r g y R e s e a r c h o f t h e U . S . D e p a r t m e n t o f E n e r g y . 'Lawrence Berkeley National Laboratory/NERSC, Berkeley, CA Email: {ksni, hdsimon}81bl.gov.
17421545
Recent trends in the marketplace of high performance computing
In this paper we analyze major recent trends and changes in the High Performance Computing (HPC) market place. The introduction of vector computers started the area of 'Supercomputing'. The initial success of vector computers in the seventies was driven by raw performance. Massive parallel systems (MPP) became successful in the early nineties due to their better price/performance ratios, which was enabled by the attack of the 'killer-micros'. The success of microprocessor based on the shared memory concept (referred to as symmetric multiprocessors (SMP)) even for the very high-end systems, was the basis for the emerging cluster concepts in the early 2000s. Within the first half of this decade clusters of PC's and workstations have become the prevalent architecture for many HPC application areas on all ranges of performance. However, the Earth Simulator vector system demonstrated that many scientific applications could benefit greatly from other computer architectures. At the same time there is renewed broad interest in the scientific HPC community for new hardware architectures and new programming paradigms. The IBM BlueGene/L system is one early example of a shifting design focus for large-scale system. The DARPA HPCS program has the declared goal of building a Petaflops computer system by the end of the decade using novel computer architectures.
9150119
Thick-Restart Lanczos Method for Symmetric Eigenvalue Problems
This paper describes a restarted Lanczos algorithm that particularly suitable for implementation on distributed machines. The only communication operation is requires outside of the matrix-vector multiplication is a global sum. For most large eigenvalue problems, the global sum operation takes a small fraction of the total execution time. The majority of the computer is spent in the matrix-vector multiplication. Efficient parallel matrix-vector multiplication routines can be found in many parallel sparse matrix packages such as AZTEC [9], BLOCK-SOLVE [10], PETSc [3], P_SPARSLIB. For this reason, our main emphasis in this paper is to demonstrate the correctness and the effectiveness of the new algorithm.
45719384
A Characterization of Strong Learnability in the Statistical Query Model
In this paper, we consider Kearns' [4] Statistical Query Model of learning. It is well known [3] that the number of statistical queries, needed for "weakly learning" an unknown target concept (i.e. for gaining significant advantage over random guessing) is polynomially related to the so-called Statistical Query dimension of the concept class. In this paper, we provide a similar characterization for "strong learning" where the learners final hypothesis is required to approximate the unknown target concept up to a small rate of misclassification. The quantity that characterizes strong learnability in the Statistical Query model is a surprisingly close relative of (though not identical to) the Statistical Query dimension. For the purpose of proving the main result, we provide other characterizations of strong learnability which are given in terms of covering numbers and related notions. These results might find some interest in their own right. All characterizations are purely information-theoretical and ignore computational issues.
9982978
General Polynomial Time Decomposition Algorithms
We present a general decomposition algorithm that is uniformly applicable to every (suitably normalized) instance of Convex Quadratic Optimization and efficiently approaches the optimal solution. The number of iterations required to be within e of optimality grows linearly with 1/e and quadratically with the number m of variables. The working set selection can be performed in polynomial time. If we restrict our considerations to instances of Convex Quadratic Optimization with at most k 0 equality constraints for some fixed constant k 0 plus some so-called box-constraints (conditions that hold for most variants of SVM-optimization), the working set is found in linear time. Our analysis builds on a generalization of the concept of rate certifying pairs that was introduced by Hush and Scovel. In order to extend their results to arbitrary instances of Convex Quadratic Optimization, we introduce the general notion of a rate certifying q-set. We improve on the results of Hush and Scovel [8] in several ways. First our result holds for Convex Quadratic Optimization whereas the results of Hush and Scovel are specialized to SVM-optimization. Second, we achieve a higher rate of convergence even for the special case of SVM-optimization (despite the generality of our approach). Third, our analysis is technically simpler.
37804897
Directions for Qualitative Reasoning
Sacks & Doyle provide an excellent overview of the fundamental limitations of the SPQR representations for reasoning about the qualitative properties of dynamic systems. We take this opportunity to outline some new directions for qualitative reasoning. In this paper, we provide a rigorous mathematical characterization for the term “qualitative property” in the context of static and dynamic systems. Based on these characterizations, we show that interval representations are well suited for reasoning about the qualitative properties of static systems such as qualitative comparative statics and qualitative stability. Moreover, we also show that symbolic computations help in the derivation of useful global properties of dynamic systems which can be used to guide numerical sampling of differential equations. The integration of symbolic and numeric methods provides a powerful approach for automating the qualitative analysis of differential equations.
8839735
Internal representation and rule development in object-oriented design
This article proposes a cognitive framework describing the software development process in object-oriented design (OOD) as building internal representations and developing rules. Rule development (method construction) is performed in two problem spaces: a rule space and an instance space. Rules are generated, refined, and evaluated in the rule space by using three main cognitive operations: Infer, Derive, and Evoke. Cognitive activities in the instance space are called mental simulations and are used in conjunction with the Infer operation in the rule space. In an empirical study with college students, we induced different representations to the same problem by using problem isomorphs. Initially, subjects built a representation based on the problem description. As rule development proceeded, the initial internal representation and designed objects were refined, or changed if necessary, to correspond to knowledge gained during rule development. Differences in rule development processes among groups created final designs that are radically different in terms of their level of abstraction and potential reusability. The article concludes by discussing the implications of these results for object-oriented design.
5520100
The Theory of Problem Solving
It is now about fifteen years since the first computer programs were written and tested that used the method of heuristic search to solve problems. Dozens of such programs, some designed for specific task domains, others claiming various degrees of generality, have now been described in the literature, and many experiments with their performance have been reported and analysed. It is an appropriate time to ask what has been learned from these experiments about the general theory of problem solving, and to try to summarize the present state of that theory.