Geometrical Formulation of the Nonnegative Matrix Factorization

Nonnegative matrix factorization (NMF) has many applications as a tool for dimension reduction. In this paper, we reformulate the NMF from an information geometrical viewpoint. We show that a conventional optimization criterion is not geometrically natural, thus we propose to use more natural criterion. By this formulation, we can apply a geometrical algorithm based on the Pythagorean theorem. We also show the algorithm can improve the existing algorithm through numerical experiments.

[1]  Shun-ichi Amari,et al.  Information Geometry and Its Applications , 2016 .

[2]  Donna K. Harman,et al.  Overview of the First Text REtrieval Conference (TREC-1) , 1992, TREC.

[3]  Kazuho Watanabe,et al.  Variational Bayesian Mixture Model on a Subspace of Exponential Family Distributions , 2009, IEEE Transactions on Neural Networks.

[4]  David M. Blei,et al.  Probabilistic topic models , 2012, Commun. ACM.

[5]  Shun-ichi Amari,et al.  Differential-geometrical methods in statistics , 1985 .

[6]  Franz Pernkopf,et al.  Model-Based Multiple Pitch Tracking Using Factorial HMMs: Model Adaptation and Inference , 2013, IEEE Transactions on Audio, Speech, and Language Processing.

[7]  S. Akaho.  The e-PCA and m-PCA: dimension reduction of parameters by information geometry , 2004, 2004 IEEE International Joint Conference on Neural Networks (IEEE Cat. No.04CH37541).

[8]  Sanjoy Dasgupta,et al.  A Generalization of Principal Components Analysis to the Exponential Family , 2001, NIPS.

[9]  F. Maxwell Harper,et al.  The MovieLens Datasets: History and Context , 2016, TIIS.

[10]  Seungjin Choi,et al.  Nonnegative features of spectro-temporal sounds for classification , 2005, Pattern Recognit. Lett..

[11]  H. Sebastian Seung,et al.  Algorithms for Non-negative Matrix Factorization , 2000, NIPS.

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

[13]  Thomas Hofmann,et al.  Probabilistic Latent Semantic Indexing , 1999, SIGIR Forum.

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

[15]  Nancy Bertin,et al.  Nonnegative Matrix Factorization with the Itakura-Saito Divergence: With Application to Music Analysis , 2009, Neural Computation.

[16]  Shotaro Akaho,et al.  Progressive evolution of whole‐rock composition during metamorphism revealed by multivariate statistical analyses , 2018 .

[17]  Hideitsu Hino,et al.  Nonparametric e-Mixture Estimation , 2016, Neural Computation.

[18]  Hideitsu Hino,et al.  Non-parametric e-mixture of Density Functions , 2016, ICONIP.

[19]  Inderjit S. Dhillon,et al.  Generalized Nonnegative Matrix Approximations with Bregman Divergences , 2005, NIPS.

[20]  Donna Harman,et al.  The First Text REtrieval Conference (TREC-1) , 1993 .