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Non-negative Matrix Factorization (NMF) is an effective way to solve the redundancy of non-negative high-dimensional data. Most of the traditional probability-based NMF methods use Gaussian distribution to model the differences between the matrices before and after decomposition. However, the Gaussian distribution is strongly affected by outliers, and it may not fit all datasets accurately when there are no outliers in the data. In this article, we propose a novel Bayesian NMF with the Student's t-distribution, i.e., TNMF. specifically, in order to reduce the impact of outliers on the algorithm, we use the Student's t-distribution to fit the data points instead of the Gaussian distribution. In addition, it is possible to adjust the Degree of Freedom (DF) to make the Student's t-distribution more flexible than the Gaussian distribution to fit data points when there are no outliers. Next, we combine the Automatic Relevance Determination (ARD) prior in our algorithm to simplify the model and allow for better performance of the algorithm. Finally, the article used 10 datasets to design two kinds of experiments, outlier removal and data clustering. The outlier removal results of this proposed algorithm are significantly better than the other methods, and it performs better in clustering compared to the other methods in the majority of cases. [ABSTRACT FROM AUTHOR]

Non-negative matrix factorization (NMF), as a good data dimensionality reduction method, is widely used in the field of image recognition. Incremental non-negative matrix factorization (INMF) as an improvement solves the problem of inefficiency caused by repeated running of data samples during online learning. However, in the traditional incremental non-negative matrix factorization algorithm, the newly added training samples do not contain label information. Some samples may be marked in both the initial sample and the new training sample in the real application scenario. In order to make full use of the label information carried by the dataset, in this paper, we propose a semi-supervised non-negative matrix factorization model for batch incremental data, incremental semi-supervised graph learning NMF with block diagonal (ISGDNMF). The model is divided into three cases according to the degree of label-carrying for the new batch data: all label-carrying, no label-carrying, and partial label-carrying. The label information is also used to add a diagonal structure to the coefficient matrix, which makes it possible to have stronger discriminatory ability and to distinguish different classes of images more easily. And graph regularization information is added in order to maintain the spatial-geometric structure of the data. Experiments on six image datasets show that this algorithm has superior performance relative to the other seven NMF-based algorithms. [ABSTRACT FROM AUTHOR]

We propose a novel NMF algorithm, named Total Variation constrained Graph-regularized Convex Non-negative Matrix Factorization (TV-GCNMF), to incorporate total variation and graph Laplacian with convex NMF. In this model, the feature details of the data are preserved by a diffusion coefficient based on the gradient information. The graph regularization and convex constraints reveal the intrinsic geometry and structure information of the features; thereby, obtaining sparse and parts-based representations. Furthermore, we give the multiplicative update rules and prove convergence of the proposed algorithm. The results of clustering experiments on multiple datasets, under various noise conditions, show the effectiveness and robustness of the proposed method compared to state-of-the-art clustering methods and other related work. [ABSTRACT FROM AUTHOR]

Non-negative matrix factorization (NMF) has been widely applied in information retrieval and computer vision. However, its performance has been restricted due to its limited tolerance to data noise, as well as its inflexibility in setting regularization parameters. In this paper, we propose a novel sparse matrix factorization method for data representation to solve these problems, termed Adaptive Total-Variation Constrained based Non-Negative Matrix Factorization on Manifold (ATV-NMF). The proposed ATV can adaptively choose the anisotropic smoothing scheme based on the gradient information of data to denoise or preserve feature details by incorporating adaptive total variation into the factorization process. Notably, the manifold graph regularization is also incorporated into NMF, which can discover intrinsic geometrical structure of data to enhance the discriminability. Experimental results demonstrate that the proposed method is very effective for data clustering in comparison to the state-of-the-art algorithms on several standard benchmarks. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new supervised non-negative matrix factorization algorithm, named Robust Dual-graph Discriminative Non-negative Matrix Factorization (RDGDNMF). This model not only considers the differentiability between data samples, but also explores the intrinsic geometric structure information of data and feature space simultaneously, so it can obtain a more effective low-dimensional representation. Considering that the real data usually contains noise and outliers, we adopt ℓ 2 , 1 2 norm to minimize the loss function of Non-negative Matrix Factorization (NMF) to improve the robustness of the algorithm. In addition, we also give the multiplication update rules and prove the convergence of the algorithm. Then, we use this model and K-Nearest Neighbor (KNN) to reduce dimension and classify data. By comparing the classification results with other advanced methods on multiple datasets, the effectiveness and robustness of the proposed method are demonstrated. • We propose a novel NMF-based model to get low-dimensional representation of data. • It solves the problems that discriminative NMF ignores inherent local geometric structure and data contains noise and outliers. • We give the multiplication update rules and prove the convergence of the algorithm. • The results of data classification using multiple classifiers are better than other methods. • Our method is most suitable for face data sets. [ABSTRACT FROM AUTHOR]

Non-Negative Matrix Factorization (NMF) has become a commonly used method for data representation. Orthogonal NMF improves the clustering performance by adding orthogonal constraints to the decomposed matrices. The existing orthogonal NMF methods typically use Euclidean distance to measure the difference between before and after factorization for convenience and simplicity. However, limitations of the Euclidean distance can lead to inflexibilities. In addition, failure to consider orthogonality of the decomposed features and sparsity of the data representation can also lead to degraded performance of the algorithm. In order to overcome the above shortcomings, we propose a novel β -divergence-based NMF with biorthogonal regularization (BO- β NMF). Our BO- β NMF method uses generalized β -divergence instead of Euclidean distance to measure the similarity between matrices, and selects an appropriate β for each type of data to obtain a more flexible way of measuring similarity. In addition, we also incorporate biorthogonal constraints into the minimized objective function, which ensures both orthogonality of the decomposed features and sparsity of the data representation. Furthermore, we use trace rather than Euclidean distance to measure the orthogonality of the decomposed matrices, which reduces execution time. Finally, clustering experiments on image datasets show that the overall clustering effect of BO- β NMF is better than state-of-the-art methods. [ABSTRACT FROM AUTHOR]