Nonnegative matrix factorization (NMF) has been shown to be a strong data representation technique, with applications in text mining, pattern recognition, image processing, clustering and other fields. In this paper, we propose a hypergraph regularized Lp smooth nonnegative matrix factorization (HGSNMF), by incorporating hypergraph regularization and Lp smoothing constraint terms into the standard NMF. The hypergraph regularization term can capture the intrinsic geometry structure of the high dimension space data more comprehensively than simple graph, the Lp smoothing constraint may yield a smooth and more accurate solution to the optimization problem. The updating rules are given using multiplicative update techniques, and the convergence of HGSNMF is theoretically investigated. The experimental results on four different data sets show that the proposed method has better clustering effect than the related state-of-the-art methods in the vast majority of cases.

Integration of multi-view datasets which are comprised of heterogeneous sources or different representations is challenging to understand the subtle and complex relationship in data. Such data integration methods attempt to combine efficiently the complementary information of multiple data types to construct a comprehensive view of underlying data. Nonnegative matrix factorization (NMF), an approach that can be used for signal compression and noise reduction, has aroused widespread attention in the last two decades. The Kullback–Leibler divergence (or relative entropy) information distance can be used to measure the loss function of NMF. In this article, we propose a fast and robust framework (RSNMF) based on symmetric nonnegative matrix factorization (SNMF) and similarity network fusion (SNF) for clustering human microbiome data including functional, metabolic and phylogenetic profiles. Many existing methods typically utilize all the information provided by each view to create a consensus representation, which often suffers a lot from noise in data and cannot provide a precise representation of the latent data structures. In contrast, RSNMF combines the strength of SNMF and the advantage of SNF to form a robust clustering indicator matrix thus can reduce the noise influence. We conduct experiments on one synthetic and two real dataset (microbiome data, text data) and the results show that the proposed RSNMF has better performance over the baseline and the state-of-art methods, which demonstrates the potential application of RSNMF for microbiome data analysis.

In this paper, some mixed type bounds on the spectral radius $\rho(A\circ B)$ for the Hadamard product of two nonnegative matrices ($A$ and $B$) and the minimum eigenvalue $\tau(C\star D)$ of the Fan product of two $M$-matrices ($C$ and $D$) are researched. These bounds complement some corresponding results on the simple type bounds. In addition, a new lower bound on the minimum eigenvalue of the Fan product of several $M$-matrices is also presented: $$ \tau(A_{1}\star A_{2}\cdots\star A_{m})\geq \min_{1\leq i\leq n}\{\prod^{m}_{k=1}A_{k}(i,i)-\prod^{m}_{k=1}[A_{k}(i,i)^{P_{k}}-\tau(A_{k}^{(P_{k})})]^\frac{1}{P_{k}}\}, $$ where $A_{1},\ldots, A_{k}$ are $n\times n$ $M$-matrices and $P_{1},\ldots, P_{k}>0$ satisfy $\sum^{m}_{k=1}\frac{1}{P_{k}}\geq 1$. Some special cases of the above result and numerical examples show that this new bound improves some existing results.

In this paper, we use a machine learning method, known as Nonnegative Matrix Factorization, to extract from a given image some new information obtained from a number of color space representations. Such kind of information, termed ``metacolor'', can be considered as a particular color space representation of the image, which can be used to obtain a binary segmentation of the investigated image by distinguishing foreground and background pixels. Some numerical experiments prove that the proposed method shows improved results when compared with common simpler thresholding algorithms.