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120 – Non-Negative Matrix Factorization
Discussion of NMF Papers
Jon R. Kettenring
In statistical practice, two-way tables of numeric data are commonplace, and are often analyzed using dimension reduction methods like the singular value decomposition and its close cousin, principal component analysis . This analysis produces score and loading matrices representing the rows and the columns of the original table and these matrices may be used for both prediction purposes and to gain structural understanding of the data. In some tables, the data entries are necessarily non-negative, and so the matrix factors meant to represent them should arguably also contain only non-negative elements. This thinking, and the desire for parsimony, underlies such techniques as rotating factors in a search for "simple structure." These attempts to transform matrices of mixed sign into non-negative, parsimonious forms are however indirect and at best imperfect. The recent development of non-negative matrix factorization, or NMF, is an attractive alternative. Rather than attempt to transform a loading or score matrix of mixed signs into one with only non-negative elements, it directly seeks matrix factors containing only non-negative elements. The resulting factorization often leads to substantial improvements in interpretability of the factors. The purpose of this session is to introduce NMF to a wider statistical audience through explicating theory, demonstrating software and showing examples.