Transductive Nonnegative Matrix Tri-Factorization

Nonnegative matrix factorization (NMF) decomposes a nonnegative matrix into the product of two lower-rank nonnegative matrices. Since NMF learns parts-based representation, it has been widely used as a feature learning component in many fields. However, standard NMF algorithms ignore the training labels as well as unlabeled data in the test domain. In this paper, we propose a transductive nonnegative matrix tri-factorization method (T-NMTF) to simultaneously exploit the label information of training examples and the statistical structure of features in the test domain. Different from standard NMF, nonnegative matrix tri-factorization (NMTF) decomposes a nonnegative matrix into the product of three lower-rank nonnegative matrices, and thus provides a flexible framework to transduce discriminative information of training examples to test examples. In particular, the proposed T-NMTF projects both training examples and test examples into a unified subspace, and expects the coefficients of training examples close to their label vectors. Since training examples and test examples are assumed to identically distributed, it is reasonable to expect the learned coefficients of test examples approximate their label vectors well. To estimate the T-NMTF parameters, we develop an efficient multiplicative update rule and prove its convergence. In addition, we propose a manifold regularized T-NMTF (MT-NMTF) algorithm that exploits the local geometry structure of the dataset to boost discriminant power. Experimental results on face recognition demonstrate the effectiveness of T-NMTF and MT-NMTF.