Convex analysis of Nonnegative Matrix Factorization

This report focuses on the literature review of convex analysis of non-negative matrix factorization (NMF). NMF is an ill-posed problem with non-unique solutions and is a NP-hard problem and is difficult to solve. NMF has wide applicability in the domains of Information Retrieval, Spectral Clustering, Data Compression, and many other domains, thereby making the problem important. The existence of many sub-optimal solutions makes it important to design algorithms/techniques so that good solutions can be found. We talk about some variations of NMF, for example, Convex-NMF, Semi-NMF, that are solvable using iterative techniques and have interesting properties, and how they can be applied to some problems with missing data. We also talk about how NMF can be cast as a conic program over the cone of completely-positive(CP) matrices. This relation helps us prove the existence of a non-trivial NMF for every non-negative matrix. We also summarize different ways in which NMF can be relaxed to a convex program. We also briefly summarize how matrix under-approximation constraints can be used to give sparser factorizations than what the general NMF delivers which is practically useful in many cases. List of papers we summarize in this report: • Introduction to nonnegative matrix factorization, Nicholas Gillis, 2017 • Convex and semi-nonnegative matrix factorizations, Ding, Chris HQ and Li, Tao and Jordan, Michael I, 2010 • Convex nonnegative matrix factorization with missing data, Hamon, Ronan and Emiya, Valentin and Févotte, Cédric, 2016 • Convex Algorithms for Nonnegative Matrix Factorization, Krishnamurthy, V. and d’Aspremont, A., 2012 • Using underapproximations for sparse nonnegative matrix factorization, Gillis, Nicolas and Glineur, François, 2010 • Non-Negative Matrix Factorization, Convexity and Isometry, Nikolaos Vasiloglou, Alexander G. Gray, David V. Anderson, 2008