Computing symmetric nonnegative rank factorizations

An algorithm is described for the nonnegative rank factorization (NRF) of some completely positive (CP) matrices whose rank is equal to their CP-rank. The algorithm can compute the symmetric NRF of any nonnegative symmetric rank-r matrix that contains a diagonal principal submatrix of that rank and size with leading cost O(rm2) operations in the dense case. The algorithm is based on geometric considerations and is easy to implement. The implications for matrix graphs are also discussed.

[1]  Stephen L. Campbell,et al.  Computing nonnegative rank factorizations , 1981 .

[2]  Matthew M. Lin,et al.  NONNEGATIVE RANK FACTORIZATION VIA RANK REDUCTION , 2008 .

[3]  L. J. Gray,et al.  Nonnegative factorization of positive semidefinite nonnegative matrices , 1980 .

[4]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[5]  Nicolas Gillis,et al.  Using underapproximations for sparse nonnegative matrix factorization , 2009, Pattern Recognit..

[6]  H. W. Turnbull,et al.  Lectures on Matrices , 1934 .

[7]  Samuel Burer,et al.  On the copositive representation of binary and continuous nonconvex quadratic programs , 2009, Math. Program..

[8]  A. Berman,et al.  Inverses of nonnegative matrices , 1974 .

[9]  Ronald C. Henry,et al.  Current factor analysis receptor models are ill-posed , 1987 .

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

[11]  A. Berman,et al.  Completely Positive Matrices , 2003 .

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

[13]  G. Golub,et al.  Rank Modifications of Semi-definite Matrices with Applications to Secant Updates , 1996 .

[14]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[15]  Nicolas Gillis,et al.  Dimensionality reduction, classification, and spectral mixture analysis using nonnegative underapproximation , 2010, Defense + Commercial Sensing.

[16]  Gene H. Golub,et al.  A Rank-One Reduction Formula and Its Applications to Matrix Factorizations , 1995, SIAM Rev..

[17]  Naomi Shaked-Monderer.  A note on upper bounds on the cp-rank , 2009 .

[18]  S. Vanduffel,et al.  On the Parameterization of the Creditrisk-Plus Model for Estimating Credit Portfolio Risk , 2008 .

[19]  N. Higham.  Analysis of the Cholesky Decomposition of a Semi-definite Matrix , 1990 .

[20]  M. Kaykobad.  On Nonnegative Factorization of Matrices , 1987 .

[21]  I. Duff,et al.  Direct Methods for Sparse Matrices , 1987 .

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

[23]  R. Plemmons,et al.  On reduced rank nonnegative matrix factorization for symmetric nonnegative matrices , 2004 .

[24]  P. Paatero,et al.  Understanding and controlling rotations in factor analytic models , 2002 .

[25]  Bo Dong,et al.  Nonnegative rank factorization—a heuristic approach via rank reduction , 2014, Numerical Algorithms.

[26]  Andrzej Cichocki,et al.  Nonnegative Matrix and Tensor Factorization T , 2007 .

[27]  Stephen A. Vavasis,et al.  On the Complexity of Nonnegative Matrix Factorization , 2007, SIAM J. Optim..

[28]  Uriel G. Rothblum,et al.  A note on the computation of the CP-rank , 2006 .

[29]  T. Markham,et al.  Factorizations of completely positive matrices , 1971, Mathematical Proceedings of the Cambridge Philosophical Society.

[30]  C. Ding,et al.  On the Equivalence of Nonnegative Matrix Factorization and K-means - Spectral Clustering , 2005 .

[31]  A reverse Hadamard inequality , 1996 .

[32]  Nicolas Gillis,et al.  Dimensionality reduction, classification, and spectral mixture analysis using non-negative underapproximation , 2011 .

[33]  R. Henry.  Receptor Model Applied to Patterns in Space (RMAPS) , 1997 .

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

[35]  Mirjam Dür,et al.  The difference between 5×5 doubly nonnegative and completely positive matrices , 2009 .

[36]  James Curry,et al.  Non-negative matrix factorization: Ill-posedness and a geometric algorithm , 2009, Pattern Recognit..