Elsevier

Signal Processing

Volume 124, July 2016, Pages 103-114
Signal Processing

Local coordinate based graph-regularized NMF for image representation

https://doi.org/10.1016/j.sigpro.2015.09.038Get rights and content

Abstract

Non-negative matrix factorization (NMF) has been a powerful data representation tool which has been widely applied in pattern recognition and computer vision due to its simplicity and effectiveness. However, existing NMF methods suffer from one or both of the following deficiencies: (1) they cannot theoretically guarantee the decomposition results to be sparse, and (2) they completely neglect geometric structure of data, especially when some examples are heavily corrupted. In this paper, we propose a local coordinate based graph regularized NMF method (LCGNMF) to simultaneously overcome both deficiencies. In particular, LCGNMF enforces the learned coefficients to be sparse by incorporating the local coordinate constraint over both factors meanwhile preserving the geometric structure of the data by incorporating graph regularization. To enhance the robustness of NMF, LCGNMF removes the effect of the outliers via the maximum correntropy criterion (MCC). LCGNMF is difficult because the MCC induced objective function is neither quadratic nor convex. We therefore developed a multiplicative update rule to solve LCGNMF and theoretically proved its convergence. Experiments of image clustering on several popular image datasets verify the effectiveness of LCGNMF compared to the representative methods in quantities.

Introduction

Data representation learns the intrinsic structure of the data and reduces data redundancy to facilitate subsequent data analysis. It has played an important role in pattern recognition [1], [2], computer vision [3], [4], and biological tasks [5], [6], [7] due to its efficacy and efficiency. Recently, non-negative matrix factorization (NMF [8], [9]) has been proven to be a powerful data representation method. It represents data matrix as the product of two lower-dimensional factors, i.e., the bases and coefficients of examples on these bases. Since NMF learns sparse representation, it has been successfully applied in computer vision.

Since NMF preserves the non-negativity property of practical data and learns intuitive interpretation consistent with human brain [5], [6], [7], it has been widely used in computer vision [10] and data mining[11]. However, NMF neglects the geometric structure proven beneficial for various vision tasks. Cai et al. [12] proposed graph regularized NMF (GNMF) which can preserve the geometric structure of dataset in the lower-dimensional space. Guan et al. [13] proposed a manifold regularized discriminative NMF (MD-NMF) by preserving both the neighborhood relationships and marginal maximization among examples. Shen and Si [14] developed the multiple manifold NMF method (MM-NMF) to model the intrinsic geometrical structure of data on multiple manifolds. Guan et al. [15] proposed a non-negative patch alignment framework (NPAF) to unify NMF, GNMF, MD-NMF, MM-NMF, and other related methods. However, both NMF and NPAF cannot guarantee any of decomposition results to be sparse in theory.

Many NMF methods have been developed by imposing sparseness constraints over the factors to overcome this deficiency. For instance, Hoyer et al. [16] proposed the sparse NMF method (SNMF) which explicitly incorporates sparseness constraints over both factors via the L1-norm regularization. Li et al. also proposed the local NMF (LNMF [1]) which imposes localization constraint over basis to learn spatially localized, parts-based representation of visual patterns. Besides, Yuan et al. [17] developed the projective NMF (PNMF) to learn sparse representation by implicitly enforcing orthogonal constraint over the basis. However, PNMF cannot effectively learn the basis, and thus fails to reveal the grouping memberships of examples. Chen et al. [18] proposed the non-negative local coordinate factorization (NLCF) to induce sparse coefficients via the local coordinate constraint. But it often easily induces trivial basis. To overcome this deficiency, Liu et al. [19] developed the local coordinate concept factorization method (LCF) to learn sparse coefficients. Since LCF implicitly requires that the learned basis vector be close to several original data points, each data point can be approximated by a linear combination of as few basis vectors as possible. However, since these methods assume that data noises follow either Gaussian or Poisson distribution, they often fail in situation where some examples are heavily corrupted.

To address this issue, Zhang et al. [20] and Shen et al. [21] proposed the sparse robust NMF (SR-NMF) method to decompose the original matrix into sparse and low-rank components. The sparse component captures the outliers, and meanwhile the low-rank component models the intrinsic structure of the data. Kong et al. [22], [23] proposed the L2,1-NMF which penalizes the reconstruction with the L2,1-norm. It has been claimed to be robust to the outliers. Besides, Du et al. [24] proposed the CIM-NMF method which employs the correntropy induced metric (CIM) to remove the effect of the outliers. It possesses the ability to handle the non-Gaussian noises. However, both traditional NMF and robust NMF methods cannot guarantee the decomposition results of NMF to be sparse in theory and do not consider the geometric structure of the datasets. However, they still obtain unsatisfactory results in clustering tasks because they neither guarantee the learned factors to be sparse, nor preserve the geometric structure, both of which have been proven beneficial for clustering tasks.

In this paper, we propose a local coordinate based NMF method with the signed graph regularization (LCGNMF) to overcome the above deficiencies. Particularly, LCGNMF enforces the learned coefficients to be sparse by using local coordinate constraint over both factors meanwhile preserving the geometric structure of the data by incorporating graph regularization. To further boost the robustness of NMF, LCGNMF removes the effect of the outliers via the maximum correntropy criterion (MCC). It is well-known that the MCC induced loss function is non-quadratic and NMF׳s objective function is non-convex, and thus it is difficult to optimize LCGNMF. In this paper, we developed a multiplicative update rule to optimize LCGNMF, and theoretically proved its convergence. Experiments of image clustering on several popular image datasets including Yale [25], Extended Yale B [26], UMIST [27] and ORL [28] datasets verify the effectiveness of LCGNMF compared to the representative methods in terms of average accuracy and normalized mutual information.

The rest of this paper is organized as follows: Section 2 briefly reviews related works on NMF and its variants. Section 3 proposes LCGNMF and optimizes it via the multiplicative update rule (MUR). Section 4 conducts experiments to evaluate the effectiveness of LCGNMF, and Section 5 concludes this paper.

Section snippets

Related works

This section briefly reviews most related works with LCGNMF, including non-negative matrix factorization (NMF [8], [9]), its robust variants [22], [23], [24] and non-negative local coordinate factorization (NLCF [18]) .

Local coordinate based graph-regularized NMF

To address the above issues, this section introduces a local coordinate based graph-regularized NMF method (LCGNMF) via maximum correntropy criterion. The maximum correntropy criterion is robust to the non-Gaussian large outliers because it is dominated by slightly corrupted examples. Particularly, LCGNMF enforces the learned coefficients to be sparse by using local coordinate constraint over both factors meanwhile preserving the geometric structure of the data by incorporating graph

Experiments

This section verifies the effectiveness by comparing the clustering performance of LCGNMF with the representative methods including NMF [8], [9], L2,1-NMF [22], PNMF [17], CIM-NMF [24] and Kmeans on four popular datasets including Yale [25], Extended Yale B [26], UMIST [27], and ORL [28]. The image instances of these datasets are shown in Fig. 5. In clustering tasks, we adopt the raw pixels to learn the coefficients of examples and we choose K-means to cluster the coefficients. The number of

Conclusion

This paper proposes a local coordinate based graph-regularized NMF (LCGNMF) to induce the sparse coefficients and consider the geometric structure of data space under the real noise datasets. Benefiting from these effective strategies, LCGNMF enhances the representation ability of NMF. Besides, LCGNMF utilizes the correntropy induced metric as the loss function to remove the effect of the outliers. To optimize LCGNMF, we developed a multiplicative update rule and proved its convergence.

Acknowledgment

The research was supported in part by grants from 973 project 2013C B329006, China NSFC under Grant 61173156, RGC under the contracts CERG 622613, 16212714, HKUST6/CRF/12R, and M-HKUST609/13, as well as the grant from Huawei-HKUST joint lab.

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