Four algorithms to solve symmetric multi-type non-negative matrix tri-factorization problem

Authors

R. Hribar,
T. Hrga,
G. Papa, G. Petelin, J. Povh, N. Pržulj, V. Vukašinović

Publication

Journal of Global Optimization, 2021, :30

Abstract

In this paper we consider the symmetric multi-type non-negative matrix tri-factorisation problem (SNMTF), where one tries to simultaneously factorise several symmetric non-negative matrices. This can be considered as a generalisation of the classical non-negative matrix tri-factorisation problem and involves non-convex objective function which is a multivariate polynomial of degree six and has convex feasibility set. It has a special importance in data science since it serves as a mathematical model for fusion of different data sources in data clustering.
We develop four methods to solve SNMTF. They are based on four theoretical approaches known from the literature: the fixed point method (FPM), block-coordinate descent with projected gradient (BCD), gradient method with exact line search (GM-ELS) and adaptive moment estimation method (ADAM). For each of these method we provide software implementation: for the former two methods we use Matlab and for the latter Python with TensorFlow library.
We test these methods on three data-sets: the synthetic one generated by us, while the others represent real-life similarities between different objects.
Extensive numerical results show that if we have enough computation time then all four methods perform satisfactorily and the ADAM gives the most often the best mean square error (MSE). However, if computation time is limited, then the FPM gives the best MSE since it shows the fastest convergence at the beginning.
All data-sets and codes are publicly available on our GitLab profile.

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