Using reconfigurable computing technology to accelerate matrix decomposition and applications

Matrix decomposition plays an increasingly significant role in many scientific and engineering applications. Among numerous techniques, Singular Value Decomposition (SVD) and Eigenvalue Decomposition (EVD) are widely used as factorization tools to perform Principal Component Analysis for dimensionality reduction and pattern recognition in image processing, text mining and wireless communications, while QR Decomposition (QRD) and sparse LU Decomposition (LUD) are employed to solve the dense or sparse linear system of equations in bioinformatics, power system and computer vision. Matrix decompositions are computationally expensive and their sequential implementations often fail to meet the requirements of many time-sensitive applications. The emergence of reconfigurable computing has provided a flexible and low-cost opportunity to pursue high-performance parallel designs, and the use of FPGAs has shown promise in accelerating this class of computation. In this research, we have proposed and implemented several highly parallel FPGA-based architectures to accelerate matrix decompositions and their applications in data mining and signal processing. Specifically, in this dissertation we describe the following contributions: • We propose an efficient FPGA-based double-precision floating-point architecture for EVD, which can efficiently analyze large-scale matrices. • We implement a floating-point Hestenes-Jacobi architecture for SVD, which is capable of analyzing arbitrary sized matrices. • We introduce a novel deeply pipelined reconfigurable architecture for QRD, which can be dynamically configured to perform either Householder transformation or Givens rotation in a manner that takes advantage of the strengths of each. • We design a configurable architecture for sparse LUD that supports both symmetric and asymmetric sparse matrices with arbitrary sparsity patterns. • By further extending the proposed hardware solution for SVD, we parallelize a popular text mining tool-Latent Semantic Indexing with an FPGA-based architecture. • We present a configurable architecture to accelerate Homotopy l1-minimization, in which the modification of the proposed FPGA architecture for sparse LUD is used at its core to parallelize both Cholesky decomposition and rank-1 update. Our experimental results using an FPGA-based acceleration system indicate the efficiency of our proposed novel architectures, with application and dimension-dependent speedups over an optimized software implementation that range from 1.5ÃÂ to 43.6ÃÂ in terms of computation time

Iterative Residual Rescaling: An Analysis and Generalization of LSI

We consider the problem of creating document representations in which inter-document similarity measurements correspond to semantic similarity. We first present a novel subspace-based framework for formalizing this task. Using this framework, we derive a new analysis of Latent Semantic Indexing (LSI), showing a precise relationship between its performance and the uniformity of the underlying distribution of documents over topics. This analysis helps explain the improvements gained by Ando's (2000) Iterative Residual Rescaling (IRR) algorithm: IRR can compensate for distributional non-uniformity. A further benefit of our framework is that it provides a well-motivated, effective method for automatically determining the rescaling factor IRR depends on, leading to further improvements. A series of experiments over various settings and with several evaluation metrics validates our claims.Comment: To appear in the proceedings of SIGIR 2001. 11 page

On the Performance of Latent Semantic Indexing-based Information Retrieval

Conventional vector based Information Retrieval (IR) models, Vector Space Model (VSM) and Generalized Vector Space Model (GVSM), represents documents and queries as vectors in a multidimensional space. This high dimensional data places great demands for computing resources. To overcome these problems, Latent Semantic Indexing (LSI): a variant of VSM, projects the documents into a lower dimensional space, computed via Singular Value Decomposition. It is stated in IR literature that LSI model is 30% more effective than classical VSM models. However statistical significance tests are required to evaluate the reliability of such comparisons. But to the best of our knowledge significance of performance of LSI model is not analyzed so far. Focus of this paper is to address this issue. We discuss the tradeoffs of VSM, GVSM and LSI and empirically evaluate the difference in performance on four testing document collections. Then we analyze the statistical significance of these performance differences

Using SVD on Clusters to Improve Precision of Interdocument Similarity Measure

Recently, LSI (Latent Semantic Indexing) based on SVD (Singular Value Decomposition) is proposed to overcome the problems of polysemy and homonym in traditional lexical matching. However, it is usually criticized as with low discriminative power for representing documents although it has been validated as with good representative quality. In this paper, SVD on clusters is proposed to improve the discriminative power of LSI. The contribution of this paper is three manifolds. Firstly, we make a survey of existing linear algebra methods for LSI, including both SVD based methods and non-SVD based methods. Secondly, we propose SVD on clusters for LSI and theoretically explain that dimension expansion of document vectors and dimension projection using SVD are the two manipulations involved in SVD on clusters. Moreover, we develop updating processes to fold in new documents and terms in a decomposed matrix by SVD on clusters. Thirdly, two corpora, a Chinese corpus and an English corpus, are used to evaluate the performances of the proposed methods. Experiments demonstrate that, to some extent, SVD on clusters can improve the precision of interdocument similarity measure in comparison with other SVD based LSI methods