On the Complexity of Nonnegative Matrix Factorization

Nonnegative matrix factorization (NMF) has become a prominent technique for the analysis of image databases, text databases, and other information retrieval and clustering applications. The problem is most naturally posed as continuous optimization. In this report, we define an exact version of NMF. Then we establish several results about exact NMF: (i) that it is equivalent to a problem in polyhedral combinatorics; (ii) that it is NP-hard; and (iii) that a polynomial-time local search heuristic exists.

[1]  J. Edmonds.  Systems of distinct representatives and linear algebra , 1967 .

[2]  A. Berman.  Rank Factorization of Nonnegative Matrices , 1973 .

[3]  L. B. Thomas.  Rank Factorization of Nonnegative Matrices (A. Berman) , 1974 .

[4]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[5]  Gene H. Golub,et al.  Matrix computations , 1983 .

[6]  Joel E. Cohen,et al.  Nonnegative ranks, decompositions, and factorizations of nonnegative matrices , 1993 .

[7]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[8]  Lenore Blum,et al.  Complexity and Real Computation , 1997, Springer New York.

[9]  H. Sebastian Seung,et al.  Learning the parts of objects by non-negative matrix factorization , 1999, Nature.

[10]  Thomas Hofmann,et al.  Probabilistic Latent Semantic Analysis , 1999, UAI.

[11]  Sven Bergmann,et al.  Iterative signature algorithm for the analysis of large-scale gene expression data. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Russell Greiner,et al.  Using Rank-One Biclusters to Classify Microarray Data , 2007 .

[13]  Golub Gene H. Et.Al.  Matrix Computations, 3rd Edition , 2007 .

[14]  Hyunsoo Kim,et al.  Sparse Non-negative Matrix Factorizations via Alternating Non-negativity-constrained Least Squares , 2006 .

[15]  Christos Boutsidis,et al.  SVD based initialization: A head start for nonnegative matrix factorization , 2008, Pattern Recognit..

[16]  Ali Ghodsi,et al.  Nonnegative matrix factorization via rank-one downdate , 2008, ICML '08.

[17]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.