Incremental Graph Regulated Nonnegative Matrix Factorization for Face Recognition

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 928051, 10 pages http://dx.doi.org/10.1155/2014/928051 Research Article Incremental Graph Regulated Nonnegative Matrix Factorization for Face Recognition Zhe-Zhou Yu, Yu-Hao Liu, Bin Li, Shu-Chao Pang, and Cheng-Cheng Jia College of Computer Science and Technology, Jilin University, Changchun 130012, China Correspondence should be addressed to Zhe-Zhou Yu; Received 25 March 2014; Revised 4 May 2014; Accepted 4 May 2014; Published 19 May 2014 Academic Editor: Jen-Tzung Chien Copyright © 2014 Zhe-Zhou Yu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In a real world application, we seldom get all images at one time. Considering this case, if a company hired an employee, all his images information needs to be recorded into the system; if we rerun the face recognition algorithm, it will be time consuming. To address this problem, In this paper, firstly, we proposed a novel subspace incremental method called incremental graph regularized nonnegative matrix factorization (IGNMF) algorithm which imposes manifold into incremental nonnegative matrix factorization algorithm (INMF); thus, our new algorithm is able to preserve the geometric structure in the data under incremental study framework; secondly, considering we always get many face images belonging to one person or many different people as a batch, we improved our IGNMF algorithms to Batch-IGNMF algorithms (B-IGNMF), which implements incremental study in batches. Experiments show that (1) the recognition rate of our IGNMF and B-IGNMF algorithms is close to GNMF algorithm while it runs faster than GNMF. (2) The running times of our IGNMF and B-IGNMF algorithms are close to INMF while the recognition rate outperforms INMF. (3) Comparing with other popular NMF-based face recognition incremental algorithms, our IGNMF and B-IGNMF also outperform then both the recognition rate and the running time. 1. Introduction Nonnegative matrix factorization (NMF) is a widely used method for low-rank approximation of a nonnegative matrix (matrix with only nonnegative entries), where nonnegative constraints are imposed on factor matrices in the decomposition. There are large bodies of past work on NMF [1]. Lee and Seung [2, 3] proposed NMF for learning parts of faces, and in their work the reconstruction error function 𝐹(𝑈, 𝑉) is introduced: 𝐹(𝑈, 𝑉) = ‖𝑋 − 𝑈𝑉‖2𝐹 , where 𝑋 denotes the data matrix, 𝑈 can be considered as the basic matrix, and 𝑉 can be considered as the coefficient matrix; all elements of 𝑋, 𝑈, and 𝑉 are nonnegative. Sparse coding is a famous parts-based representation method, by minimizing a 𝐿 1 regularization-related objective function of NMF-based algorithms, sparse constraints can be achieved. Hoyer [4] proposed a method by keeping 𝐿 2 norm unchanged in each iteration, but 𝐿 1 norm set to achieve desired sparseness. Li et al. [5] proposed another sparse representation method which focused on sparse NMF algorithm with KullbackLeibler based cost function. Discrimination method was also introduced into NMF algorithm. Wang et al. [6] proposed a Fisher nonnegative matrix factorization which introduces Fisher constraint (discrimination) method into NMF algorithm. later, Nikitidis et al. [7] introduced subclass discriminant into NMF algorithm, by separating each class into several subclasses; this method was able to ensure that the underlying data distribution in each subclass is unimodal. Because the convergence of canonical NMF algorithms is slow, gradient descent based methods are introduced to NMF to improve its speed of convergence. Guan et. al [8] applied Nesterov’s optimal gradient method to alternatively optimize one factor with another. By introducing fast gradient descent method into search of the optimal step size for gradient descent based NMF algorithm, Guan et al. [9] introduced nonnegative patch alignment framework (NPAF) and nonnegative discriminative locality alignment (NDLA). Canonical NMF algorithm aims to minimize the Euclidean distance or the Kullback-Leibler distance between the data matrix 𝑋 and its reconstruction matrix 𝑈 × 𝑉. By introducing Manhattan distance into NMF algorithm, Guan et al. [10] introduced Manhattan nonnegative matrix 2 factorization (MahNMF), then rank-one residual iteration method and Nesterov’s smoothing method are both introduced to optimizing MahNMF. The geometrical information of the original data space is an important information for face recognition. Zhang et al. [11] proposed a novel topology preserving nonnegative matrix factorization (TPNMF) which considers the gradient distance instead of the Euclidean distance or the Kullback-Leibler distance. By constructing the intrinsic and penalty graphs, Liu et al. [12] proposed the projective nonnegative graph embedding (PNGE). Cai et al. [13] proposed graph regularized nonnegative matrix factorization (GNMF) algorithm for face recognition. GNMF imposes manifold into NMF, which makes it able to preserve geometric structure of data after it maps the original image into low dimension space. NMF-based algorithms are notoriously slow to converge. In practice, if new image comes, algorithm needs to rerun which is time consuming. To address this problem, incremental process used for online study has attracted a lot of attention. During an incremental study process of NMF, once a factorization, such as 𝑈 and 𝑉 which mentioned above, is obtained for a group of face images, the representation of 𝑈 should be updated with a small computational cost, but the canonical NMF and many other algorithms are too expensive to compute 𝑈 and 𝑉 every time when new face image comes. Bucak and Gunsel [14] introduced an Incremental nonnegative matrix factorization algorithm (INMF), which imposes NMF into incremental study. This makes it possible to represent data content online and reduce dimension significantly. Chen et al. [15] proposed another incremental nonnegative matrix factorization (INMF) for face representation and recognition. In order to distinguish Chen’s algorithms [15] from Bucak and Gunsel’s [14], we called Chen’s algorithms CINMF. There are two main differences between CINMF and INMF: first, INMF is an unsupervised method while CINMF is a supervised method; second, INMF can only deal with one new image while CINMF can only deal with a batch of images which belong to a new class. Wang and Lu [16] introduced an incremental orthogonal projective nonnegative matrix factorization algorithm (IOPNMF) which introduced orthogonal projective constraint into incremental NMF. Lefèvre et al. [17] introduced incremental study into Itakura-Saito NMF algorithm; the Istakura-Saito divergence was defined as 𝑑IS (𝑦, 𝑥) = ∑𝑖 (𝑦𝑖 /𝑥𝑖 − log 𝑦𝑖 /𝑥𝑖 − 1). The proposed algorithm is used in the field of au (...truncated)