Person:
Park, Haesun

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    Workshop on Future Direction in Numerical Algorithms and Optimization
    (Georgia Institute of Technology, 2008-01-15) Park, Haesun ; Golub, Gene ; Wu, Weili ; Du, Ding-Zhu
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    Sparse Nonnegative Matrix Factorization for Clustering
    (Georgia Institute of Technology, 2008) Kim, Jingu ; Park, Haesun
    Properties of Nonnegative Matrix Factorization (NMF) as a clustering method are studied by relating its formulation to other methods such as K-means clustering. We show how interpreting the objective function of K-means as that of a lower rank approximation with special constraints allows comparisons between the constraints of NMF and K-means and provides the insight that some constraints can be relaxed from K-means to achieve NMF formulation. By introducing sparsity constraints on the coefficient matrix factor in NMF objective function, we in term can view NMF as a clustering method. We tested sparse NMF as a clustering method, and our experimental results with synthetic and text data shows that sparse NMF does not simply provide an alternative to K-means, but rather gives much better and consistent solutions to the clustering problem. In addition, the consistency of solutions further explains how NMF can be used to determine the unknown number of clusters from data. We also tested with a recently proposed clustering algorithm, Affinity Propagation, and achieved comparable results. A fast alternating nonnegative least squares algorithm was used to obtain NMF and sparse NMF.
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    Toward Faster Nonnegative Matrix Factorization: A New Algorithm and Comparisons
    (Georgia Institute of Technology, 2008) Kim, Jingu ; Park, Haesun
    Nonnegative Matrix Factorization (NMF) is a dimension reduction method that has been widely used for various tasks including text mining, pattern analysis, clustering, and cancer class discovery. The mathematical formulation for NMF appears as a non-convex optimization problem, and various types of algorithms have been devised to solve the problem. The alternating nonnegative least squares (ANLS) framework is a block coordinate descent approach for solving NMF, which was recently shown to be theoretically sound and empirically efficient. In this paper, we present a novel algorithm for NMF based on the ANLS framework. Our new algorithm builds upon the block principal pivoting method for the nonnegativity constrained least squares problem that overcomes some limitations of active set methods. We introduce ideas to efficiently extend the block principal pivoting method within the context of NMF computation. Our algorithm inherits the convergence theory of the ANLS framework and can easily be extended to other constrained NMF formulations. Comparisons of algorithms using datasets that are from real life applications as well as those artificially generated show that the proposed new algorithm outperforms existing ones in computational speed.
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    ALGORITHMS: Collaborative research: development of vector space based methods for protein structure prediction
    (Georgia Institute of Technology, 2007-07-16) Park, Haesun ; Vazirani, Vijay V.
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    Fast Linear Discriminant Analysis using QR Decomposition and Regularization
    (Georgia Institute of Technology, 2007-03-23) Park, Haesun ; Drake, Barry L. ; Lee, Sangmin ; Park, Cheong Hee
    Linear Discriminant Analysis (LDA) is among the most optimal dimension reduction methods for classification, which provides a high degree of class separability for numerous applications from science and engineering. However, problems arise with this classical method when one or both of the scatter matrices is singular. Singular scatter matrices are not unusual in many applications, especially for high-dimensional data. For high-dimensional undersampled and oversampled problems, the classical LDA requires modification in order to solve a wider range of problems. In recent work the generalized singular value decomposition (GSVD) has been shown to mitigate the issue of singular scatter matrices, and a new algorithm, LDA/GSVD, has been shown to be very robust for many applications in machine learning. However, the GSVD inherently has a considerable computational overhead. In this paper, we propose fast algorithms based on the QR decomposition and regularization that solve the LDA/GSVD computational bottleneck. In addition, we present fast algorithms for classical LDA and regularized LDA utilizing the framework based on LDA/GSVD and preprocessing by the Cholesky decomposition. Experimental results are presented that demonstrate substantial speedup in all of classical LDA, regularized LDA, and LDA/GSVD algorithms without any sacrifice in classification performance for a wide range of machine learning applications.
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