Soft-constrained nonnegative matrix factorization via normalization

Semi-supervised clustering aims at boosting the clustering performance on unlabeled samples by using labels from a few labeled samples. Constrained NMF (CNMF) is one of the most significant semi-supervised clustering methods, and it factorizes the whole dataset by NMF and constrains those labeled samples from the same class to have identical encodings. In this paper, we propose a novel soft-constrained NMF (SCNMF) method by softening the hard constraint in CNMF. Particularly, SCNMF factorizes the whole dataset into two lower-dimensional factor matrices by using multiplicative update rule (MUR). To utilize the labels of labeled samples, SCNMF iteratively normalizes both factor matrices after updating them with MURs to make encodings of labeled samples close to their label vectors. It is therefore reasonable to believe that encodings of unlabeled samples are also close to their corresponding label vectors. Such strategy significantly boosts the clustering performance even when the labeled samples are rather limited, e.g., each class owns only a single labeled sample. Since the normalization procedure never increases the computational complexity of MUR, SCNMF is quite efficient and effective in practices. Experimental results on face image datasets illustrate both efficiency and effectiveness of SCNMF compared with both NMF and CNMF.