A Convergent Algorithm for Bi-orthogonal Nonnegative Matrix Tri-Factorization

Abstract. We extend our previous work on a convergent algorithm for uni-orthogonal nonnegative matrix factorization (UNMF) to the case where the data matrix is decomposed into three factors with two of them are constrained orthogonally and the third one is used to absorb the approximation error. Due to the way the factorization is performed, we name it as bi-orthogonal nonnegative matrix tri-factorization, i.e., BNMtF. This factorization was first introduced by Ding et al. [9] with intent to further improve clustering capability of their version of UNMF. However, as shown in this paper, not only their BNMtF algorithm does not have convergent property but also it does not minimize the objective function it intends to minimize. We tackle this problem by utilizing a technique presented in our previous work and prove that our algorithm converges to a stationary point inside the solution space. As a practical demonstration, the proposed algorithm is utilized for clustering a text corpus; however, contrary to the claim in the original work, both BNMtF algorithms (the original one by Ding et al. [9] and our proposed algorithm) perform poorly compared to the standard NMF algorithm by Lee & Seung [24] and our UNMF algorithm based on multiplicative update rules. This implies that the additional complexity introduced by BNMtF which was originally intended to improve UNMF clustering capability probably is not necessary.