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Generalized Nonnegative Matrix Approximations using Bregman Divergences

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Dhillon, I., & Sra, S.(2005). Generalized Nonnegative Matrix Approximations using Bregman Divergences (TR-06-27). Austin, TX, USA: Department of Computer Sciences: University of Texas.

Cite as: http://hdl.handle.net/11858/00-001M-0000-0013-D567-3
Nonnegative matrix approximation (NNMA) is a recent technique for dimensionality reduction anddata analysis that yields a parts based, sparse nonnegativerepresentation of the nonnegative input data. Dueto these advantages, NNMA has found a wide variety of applications, including text analysis, documentclustering, face/image recognition, language modeling, speech processing and many others. Despite thesenumerous applications, the algorithmic development for computing the NNMA factors has been relativelydeficient. This paper makes algorithmic progress by modeling andsolving(using multiplicative updates)new generalized NNMA problems that minimize Bregman divergences between the input matrix and itslow-rank approximation. The multiplicative update formulae in the pioneering work by Lee and Seung [20]arise as a special case of our algorithms. In addition, the paper shows how to use penalty functions forincorporating constraints other than nonnegativity into the problem. Further, some interesting extensions tothe use of “link” functions for modeling non-linear relationships are also discussed.