Nonnegative matrix factorization with matrix exponentiation

Nonnegative matrix factorization (NMF) has been successfully applied to different domains as a technique able to find part-based linear representations for nonnegative data. However, when extra constraints are incorporated into NMF, simple gradient descent optimization can be inefficient for high-dimensional problems, due to the overhead to enforce the nonnegativity constraints. We describe an alternative formulation based on matrix exponentiation, where the nonnegativity constraints are enforced implicitly, and a direct gradient descent algorithm can have better efficiency. In numerical experiments, such a reformulation leads to significant improvement in running time.