This paper provides a theoretical explanation on the clustering aspect of nonnegative matrix factorization (NMF). We prove that even without imposing orthogonality nor sparsity constraint on the basis and or coefficient matrix, NMF still can give clustering results, thus providing a theoretical support for many works, e.g., [1] and [2], that show the superiority of the standard NMF as a clustering method.
@cite_10 provides the theoretical analysis on the equivalences between orthogonal NMF to @math -means clustering for both rectangular data matrices and symmetric matrices. However as their proofs utilize the zero gradient conditions, the hidden assumptions (setting the Lagrange multipliers to zeros) are not revealed there. Actually it can be easily shown that their approach is the KKT conditions applied to the unconstrained version of eq. . Thus there is no guarantee that minimizing eq. by using the zero gradient conditions leads to the stationary point located on the nonnegative orthant as required by the objective.
@cite_10
2043545458
Let B i be deterministic real symmetric m  m matrices, and  i be independent random scalars with zero mean and of order of one (e.g., @math ). We are interested to know under what conditions typical norm of the random matrix @math is of order of 1. An evident necessary condition is @math , which, essentially, translates to @math ; a natural conjecture is that the latter condition is sufficient as well. In the paper, we prove a relaxed version of this conjecture, specifically, that under the above condition the typical norm of S N is @math : @math for all  > 0 We outline some applications of this result, primarily in investigating the quality of semidefinite relaxations of a general quadratic optimization problem with orthogonality constraints @math , where F is quadratic in X = (X 1,... ,X k ). We show that when F is convex in every one of X j , a natural semidefinite relaxation of the problem is tight within a factor slowly growing with the size m of the matrices @math .