Nonnegative Matrix Factorization Using Nonnegative Polynomial Approximations

Nonnegative matrix factorization is a key tool in many data analysis applications such as feature extraction, compression, and noise filtering. Many existing algorithms impose additional constraints to take into account prior knowledge and to improve the physical interpretation. This letter proposes a novel algorithm for nonnegative matrix factorization, in which the factors are modeled by nonnegative polynomials. Using a parametric representation of finite-interval nonnegative polynomials, we obtain an optimization problem without external nonnegativity constraints, which can be solved using conventional quasi-Newton or nonlinear least-squares methods. The polynomial model guarantees smooth solutions and may realize a noise reduction. A dedicated orthogonal compression enables a significant reduction of the matrix dimensions, without sacrificing accuracy. The overall approach scales well to large matrices. The approach is illustrated with applications in hyperspectral imaging and chemical shift brain imaging.