Controlling orthogonality constraints for better NMF clustering
In this paper we study a variation of a Non-negative Matrix Factorization (NMF) called the Orthogonal NMF(ONMF). This special type of NMF was proposed in order to increase the quality of clustering results of standard NMF by imposing orthogonality on clustering indicator matrix and/or the matrix of basis vectors. We develop an extension of ONMF which we call Weighted ONMF and propose a novel approach for imposing orthogonality on the matrix of basis vectors obtained via NMF using Gram-Schmidt process.
We prove existence of solutions for a class of systems of subelliptic PDEs arising from Mean Field Game systems with H\"ormander diffusion. These results are motivated by the feedback synthesis Mean Field Game solutions and the Nash equilibria of a large class of $N$-player differential games.