Abstract
Matrix factorization based methods have widely been used in data representation. Among them, Non-negative Matrix Factorization (NMF) is a promising technique owing to its psychological and physiological interpretation of spontaneously occurring data. On one hand, although traditional Laplacian regularization can enhance the performance of NMF, it still suffers from the problem of its weak extrapolating ability. On the other hand, standard NMF disregards the discriminative information hidden in the data and cannot guarantee the sparsity of the factor matrices. In this paper, a novel algorithm called ℓ 2 , 1 norm and Hessian Regularized Non-negative Matrix Factorization with Discriminability (ℓ 2 , 1 HNMFD), is developed to overcome the aforementioned problems. In ℓ 2 , 1 HNMFD, Hessian regularization is introduced in the framework of NMF to capture the intrinsic manifold structure of the data. ℓ 2 , 1 norm constraints and approximation orthogonal constraints are added to assure the group sparsity of encoding matrix and characterize the discriminative information of the data simultaneously. To solve the objective function, an efficient optimization scheme is developed to settle it. Our experimental results on five benchmark data sets have demonstrated that ℓ 2 , 1 HNMFD can learn better data representation and provide better clustering results.
Highlights
In many real-world applications, the input data is usually high-dimensional
Proposed Neighborhood-Preserving Non-negative Matrix Factorization, which imposed an additional constraint on NMF that each item be able to be represented as a linear combination of its neighbors
As far as we can see, the objective function of2,1 HNMFD is not convex in both U and V, so we cannot result in a closed-form solution
Summary
In many real-world applications, the input data is usually high-dimensional. On one hand, this is a serious challenge for storage and computation. Lots of recent work has focused on preserving the intrinsic geometry of the data space by adding different constraints to the objective function of NMF. Cai et al [7] proposed graph regularized NMF (GNMF) by constructing a nearest neighbor graph while preserving the local geometrical information of the data space. Proposed Neighborhood-Preserving Non-negative Matrix Factorization, which imposed an additional constraint on NMF that each item be able to be represented as a linear combination of its neighbors. Proposed robust structured NMF a semi-supervised NMF learning algorithm, which learns a robust discriminative data representation by pursuing the block-diagonal structure and the2,p norm loss function. Non-negative Matrix Factorization with Discriminability (`2,1 HNMFD), is developed in this paper, which is designed to include local geometrical structure preservation, row sparsity and to exploit discriminative information at the same time.
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