Abstract
Non-negative matrix factorization (NMF) approximates a non-negative data matrix with the product of two low-rank non-negative matrices by minimizing the cost of such approximation. However, traditional NMF models cannot be generalized in the cases when the dataset contains outliers and limited knowledge from domain experts. In this paper, we propose a robust semi-supervised NMF model (RSS-NMF) to overcome the aforementioned deficiency. RSS-NMF utilizes the L 2 /L 1 -norm to encourage approximation and makes the model insensitive to outliers by prohibiting them from dominating the cost function. To incorporate the discriminative information, RSS-NMF utilizes the structured normalization method when learns a diagonal matrix to normalize the coefficients such that they get close to the label indicators of the given labeled examples. Although the multiplicative update rule (MUR) can be adopted to minimize RSS-NMF, it converges slowly. In this paper, we adopt a fast gradient descent algorithm (FGD) to optimize RSS-NMF and prove its convergence to a stationary point. FGD uses a Newton method to search the optimal step length and thus, FGD converges faster than MUR. The experimental results show the promise of RSS-NMF comparing with the representative clustering models on several face image datasets.
Highlights
Semi-supervised clustering is a longstanding problem in machine learning filed and has extremely widespread applications, ranging from document processing [36], segmentation [37], behavioral analysis [38], to face recognition [39]
To optimize RSS-negative matrix factorization (NMF), we present a fast gradient descent algorithm (FGD) for much faster convergence
robust semi-supervised NMF model (RSS-NMF) is robust to outliers by using the L2/L1-norm and incorporates few labeled examples by utilizing the structured normalization
Summary
Semi-supervised clustering is a longstanding problem in machine learning filed and has extremely widespread applications, ranging from document processing [36], segmentation [37], behavioral analysis [38], to face recognition [39]. It softens the hard constraint in CNMF by introducing a diagonal matrix with positive diagonal elements to normalize the decomposition These semi-supervised NMF models incorporate prior information only, but are not robust to noises and outliers. This model utilizes the L2/L1-norm [6] to measure the cost of NMF approximation and make the model insensitive to outliers by prohibiting them from dominating the cost function. This paper is organized as follows: Section II briefly reviews the related works; Section III proposes the robust semi-supervised NMF model (RSS-NMF) and proposes a multiplicative update rule algorithm (MUR) for optimizing RSS-NMF, and proposes a fast gradient decent (FGD) to accelerate MUR; Section IV empirically evaluates RSS-NMF by showing its efficiency and effectiveness; Section V concludes this paper
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