BNCA: Full-rank factorisable subgraphs based unique structure-constrained matrix factorisation

Abstract

A Structure-Constrained Matrix Factorisation (SCMF) problem is considered where data and structural constraints on one of the matrix factors specific to an application are known. A simple two-step iterative optimisation algorithm can produce unique solutions provided both matrix factors are full-ranked and constraints matrix satisfies certain additional rank conditions. Constraints matrix is apriori known and hence, it can be tested for these rank conditions. Graph theoretical approaches can be used to decompose a graph representing incompatible constraints matrix into compatible subgraphs. However, there is no method available in relevant literature to compute the rank of second matrix factor as it is apriori unknown. Previously, it has been argued that the second matrix factor will naturally be full-ranked, but we show that this is not necessarily true. We develop theoretical bounds on rank of the second matrix factor in terms of ranks of constraints matrix, data matrix and their dimensions. With this new result, uniqueness of a solution can be guaranteed solely based on available constraints and data. Furthermore, we propose Beaded Network Component Analysis algorithm that introduces necessary corrections to the available graph decomposition and mixing algorithms to obtain unique solutions by computing a convex-combination of full-rank factors of subgraphs. The key contributions in this paper are theoretical bounds on rank of a matrix factor and unique solutions to a general SCMF problem.