Ghostbusters: A parts-based NMF algorithm

Abstract

An exact nonnegative matrix decomposition algorithm is proposed. This is achieved by 1) Taking a nonlinear approximation of a sparse real-valued dataset at a given tolerance-to-error constraint, c; Choosing an arbitrary lectic ordering on the rows or column entries; And, then systematically applying a closure operator, so that all closures are selected. Assuming a nonnegative hierarchical closure structure (a Galois lattice) ensures the data has a unique ordered overcomplete dictionary representation. Parts-based constraints on these closures can then be used to specify and supervise the form of the solution. We illustrate that this approach outperforms NMF on two standard NMF datasets: it exhibits the properties described above; It is correct and exact.