Uniqueness of Nonnegative Matrix Factorizations by Rigidity Theory

Nonnegative matrix factorizations are often encountered in data mining applications where they are used to explain datasets by a small number of parts. For many of these applications it is desirable that there exists a unique nonnegative matrix factorization up to trivial modifications given by scalings and permutations. This means that model parameters are uniquely identifiable from the data. Rigidity theory of bar and joint frameworks is a field that studies uniqueness of point configurations given some of the pairwise distances. The goal of this paper is to use ideas from rigidity theory to study uniqueness of nonnegative matrix factorizations in the case when nonnegative rank of a matrix is equal to its rank. We characterize infinitesimally rigid nonnegative factorizations, prove that a nonnegative factorization is infinitesimally rigid if and only if it is locally rigid and a certain matrix achieves its maximal possible Kruskal rank, and show that locally rigid nonnegative factorizations can be extended to globally rigid nonnegative factorizations. We also explore connections between rigidity of nonnegative factorizations and boundaries of the set of matrices of fixed nonnegative rank. Finally we extend these results from nonnegative factorizations to completely positive factorizations.

[1]  Chong-Yung Chi,et al.  A Convex Analysis-Based Minimum-Volume Enclosing Simplex Algorithm for Hyperspectral Unmixing , 2009, IEEE Transactions on Signal Processing.

[2]  M. Yannakakis.  Expressing combinatorial optimization problems by linear programs , 1991, Symposium on the Theory of Computing.

[3]  Karthik Devarajan,et al.  Nonnegative Matrix Factorization: An Analytical and Interpretive Tool in Computational Biology , 2008, PLoS Comput. Biol..

[4]  Yaroslav Shitov.  Nonnegative rank depends on the field , 2021, Math. Program..

[5]  Nicolas Gillis,et al.  Sparse and unique nonnegative matrix factorization through data preprocessing , 2012, J. Mach. Learn. Res..

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

[7]  Wing-Kin Ma,et al.  Nonnegative Matrix Factorization for Signal and Data Analytics: Identifiability, Algorithms, and Applications , 2018, IEEE Signal Processing Magazine.

[8]  V. P. Pauca,et al.  Nonnegative matrix factorization for spectral data analysis , 2006 .

[9]  P. Smaragdis,et al.  Non-negative matrix factorization for polyphonic music transcription , 2003, 2003 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (IEEE Cat. No.03TH8684).

[10]  Chong-Yung Chi,et al.  Nonnegative Least-Correlated Component Analysis for Separation of Dependent Sources by Volume Maximization , 2010, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[11]  Michael Joswig,et al.  polymake: a Framework for Analyzing Convex Polytopes , 2000 .

[12]  B. Roth,et al.  The rigidity of graphs, II , 1979 .

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

[14]  David Mond,et al.  Stochastic factorizations, sandwiched simplices and the topology of the space of explanations , 2003, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[15]  Chris H. Q. Ding,et al.  Orthogonal nonnegative matrix t-factorizations for clustering , 2006, KDD '06.

[16]  Xin Liu,et al.  Document clustering based on non-negative matrix factorization , 2003, SIGIR.

[17]  Xiao Fu,et al.  On Identifiability of Nonnegative Matrix Factorization , 2017, IEEE Signal Processing Letters.

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

[19]  Nicolas Gillis,et al.  Heuristics for exact nonnegative matrix factorization , 2014, J. Glob. Optim..

[20]  Ankur Moitra,et al.  An Almost Optimal Algorithm for Computing Nonnegative Rank , 2013, SIAM J. Comput..

[21]  Mark D. Plumbley,et al.  Theorems on Positive Data: On the Uniqueness of NMF , 2008, Comput. Intell. Neurosci..

[22]  Wei-Chiang Li,et al.  Identifiability of the Simplex Volume Minimization Criterion for Blind Hyperspectral Unmixing: The No-Pure-Pixel Case , 2014, IEEE Transactions on Geoscience and Remote Sensing.

[23]  Victoria Stodden,et al.  When Does Non-Negative Matrix Factorization Give a Correct Decomposition into Parts? , 2003, NIPS.

[24]  Bernd Sturmfels,et al.  FIXED POINTS OF THE EM ALGORITHM AND NONNEGATIVE RANK BOUNDARIES , 2013, 1312.5634.

[25]  Nikos D. Sidiropoulos,et al.  Blind Separation of Quasi-Stationary Sources: Exploiting Convex Geometry in Covariance Domain , 2015, IEEE Transactions on Signal Processing.

[26]  Nicolas Gillis,et al.  On the Geometric Interpretation of the Nonnegative Rank , 2010, 1009.0880.

[27]  Toshihisa Tanaka,et al.  First results on uniqueness of sparse non-negative matrix factorization , 2005, 2005 13th European Signal Processing Conference.

[28]  Amit Singer,et al.  Uniqueness of Low-Rank Matrix Completion by Rigidity Theory , 2009, SIAM J. Matrix Anal. Appl..

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

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