UT EE 381K - Literature Survey: Non-Negative Matrix Factorization (10 pages)

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Literature Survey: Non-Negative Matrix Factorization



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Literature Survey: Non-Negative Matrix Factorization

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Pages:
10
School:
University of Texas at Austin
Course:
Ee 381k - Analysis & Design of Communication Networks
Analysis & Design of Communication Networks Documents

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Literature Survey Non Negative Matrix Factorization Joel A Tropp Institute for Computational Engineering and Sciences 1 University Station C0200 The University of Texas at Austin Austin TX 78712 E mail address jtropp ticam utexas edu Abstract This article surveys recent research on Non Negative Matrix Factorization NNMF a relatively new technique for dimensionality reduction It is based on the idea that in many data processing tasks negative numbers are physically meaningless The NNMF technique addresses this problem by placing non negativity constraints on the data model I discuss the applications of NNMF the algorithms and the qualitative results Since many of the algorithms proposed for NNMF seem to lack a firm theoretical foundation this article also surveys techniques for proving that iterative algorithms converge It concludes with a description of additional investigations which are presently underway 1 INTRODUCTION 3 1 Introduction A basic technique in dimensionality reduction is Principal Component Analysis PCA which calculates a set basis vectors that can be used to approximate high dimensional data optimally in the least squares sense The number of basis vectors is much smaller than the number of dimensions so encoding the data as linear combinations of the basis vectors transforms it to a lower dimensional space This reduction can be used to improve the tractability of data analysis algorithms to discover features in the data or to find hidden variables See Jolliffe 1 for a survey One major problem with PCA is that the basis vectors have both positive and negative components and the data are represented as linear combinations of these vectors with positive and negative coefficients The optimality of PCA can be traced back to construction cancelation of the signs In many applications the negative components contradict physical realities For example the pixels in a grayscale image have non negative intensities so an image with negative intensities cannot be



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