Abstract
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.
Highlights
Nonnegative matrix factorization(NMF) [1] decomposes a nonnegative matrix into the product of two lower-rank nonnegative matrices
The semi-supervised CNMF, NMF-α and transductive nonnegative matrix tri-factorization method (T-nonnegative matrix tri-factorization (NMTF)) can deliver better results than the unsupervised NMF, NMTF and K-means methods, and the proposed T-NMTF method is superior to other methods in the experiment. These results show the superiority of our proposed T-NMTF, and the reason is probably that it can simultaneously exploit the label information of training examples and the statistical structure of features in the test domain, it can get a better representation of the original data
These results show that the performance of both T-NMTF and manifold regularized T-NMTF (MT-NMTF) is significantly superior to other methods, and this superiority increases with the increase in the percentage of labeled examples in the training set
Summary
Nonnegative matrix factorization(NMF) [1] decomposes a nonnegative matrix into the product of two lower-rank nonnegative matrices. We propose a new transductive nonnegative matrix tri-factorization method (T-NMTF) to simultaneously utilize the labels of training examples and exploit the statistical structure of test examples in the framework of NMTF. Several transductive NMF are studied recently, Cho and Saul [30] proposed to use a pre-trained SVM classifier to learn the discriminative support vectors and apply them in the decomposition. Zhang et al [51] proposed a joint label prediction based Robust Semi-Supervised Adaptive Concept Factorization (RS2ACF) framework recently, which utilized class information of labeled data and propagated it to unlabeled data by jointly learning an explicit label indicator for unlabeled data. In [55], to acquire a more accurate prediction in classification, the triple matrix recovery-based robust auto-weighted label propagation framework (ALP-TMR) was proposed by introducing a TMR mechanism to remove noise or mixed signs from the estimated soft labels and improve the robustness to noise and outliers in the steps of assigning weights and predicting the labels simultaneously
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