On reduced rank nonnegative matrix factorization for symmetric nonnegative matrices

By M. Catral, Lixing Han, M. Neumann, Robert J. Plemmons. 2004

68 citations recorded by the source corpus.

Abstract Let V∈ R m,n be a nonnegative matrix. The nonnegative matrix factorization (NNMF) problem consists of finding nonnegative matrix factors W∈ R m,r and H ∈ R r , n such that V ≈ WH . Lee and Seung proposed two algorithms, one of which finds nonnegative W and H such that ∥ V − WH ∥ F is minimized. After examining the case in which r =1 about which a complete characterization of the solution is possible, we consider the case in which m = n and V is symmetric. We focus on questions concerning when the best approximate factorization results in the product WH being symmetric and on cases in which the best approximation cannot be a symmetric matrix . Finally, we show that the class of positive semidefinite symmetric nonnegative matrices V generated via a Soules basis admit for every 1⩽ r ⩽rank( V ), a nonnegative factorization WH which coincides with the best approximation in the Frobenius norm to V in R n,n of rank not exceeding r . An example of applications in which NNMF factorizations for nonnegative symmetric matrices V arise is video and other media summarization technology where V is obtained from a distance matrix.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Electrical and Electronic Engineering

View Paper: https://www.semanticscholar.org/paper/037caf89ab8afe8abbdfc458fee58c5b2c39f077

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In Computational Mechanics

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Citations in the corpus

Algorithms and applications for approximate nonnegative matrix factorization (2007, background): “A theoretical analysis of NMF of symmetric matrices can be found in Catral et al. (2004).”
Graph Regularized Nonnegative Matrix Factorization for Data Representation (2011, methodology): “Our proof essentially follows the idea in the proof of Lee and Seung's [27] paper for the original NMF. Recent studies [8], [30] show that Lee and Seung's [27] multiplicative algorithm cannot guarantee the convergence to a stationary point.”
Nonnegative matrix factorization and I-divergence alternating minimization☆ (2004, background): “∗The authors have been supported in part by the European Community’s Human Potential Programme under contract HPRN-CT-2000-00100, DYNSTOCH.”
Non-negative tensor factorization with applications to statistics and computer vision (2005, background): “…then the rank-k approximation described in the previous section will not necessarily find a supersymmetric decomposition, i.e., where u j 1 = ... = u j n , but a rank-1 fit to a super-symmetric tensor is guaranteed to have a symmetric form (cf. Catral et al. (2004),Kofidis and Regalia (2002)).”
On the Convergence of Multiplicative Update Algorithms for Nonnegative Matrix Factorization (2007, background): “However, Lin [5] stated that due to possible numerical inaccuracy, a mathematical example is desired before drawing conclusions.”
SVD based initialization: A head start for nonnegative matrix factorization (2008, unknown): “[34].”