User Assisted Exploration and Sampling of the Solution Set of Non-negative Matrix Factorizations

The non-negative matrix factorization provides a valuable tool for the analysis of positive data by representing it as an additive linear superposition of a small number of non-negative basis elements. This property allows the base elements to be interpreted in the same domain as the input data. The problem though lies in the ambiguity of equally valid solutions from which only one is obtained. Its selection depends on the initialization of the applied factorization algorithm or further constraints. We propose a new approach which is based on sampling the set of valid factorizations, given one initial solution. First we derive a parameterization of the set of valid solutions by means of a strong membership oracle. This function returns true if a parameter tuple represents a valid solution and false otherwise. Furthermore, we present an algorithm that explores and samples parts of the non-convex solution set. To assist the otherwise automatic process and to alleviate the drawbacks of sampling a nonconvex space, we provide a graphical user interface that puts the user in the loop. From an initial set of samples the user is allowed to select elements that serve as the starting point for subsequent samplings. With this browser-like tool a steering of the sampling of the NMF can be performed without further knowledge on the underlying algorithm and without the need to express possibly hard to formulate constraints. An evaluation of the sampling procedure reveals promising results for a factorization of data in up to 4 basis elements.