Nonnegative matrix factorization (NMF) is the problem of decomposing a given nonnegative n × m matrix M into a product of a nonnegative n × d matrix W and a nonnegative d × m matrix H. Restricted NMF requires in addition that the column spaces of M and W coincide. Finding the minimal inner dimension d is known to be NP-hard, both for NMF and restricted NMF. We show that restricted NMF is closel...

We investigate the homology representation of symmetric group on rank-selected subposets subword order. show that module for words bounded length, over an alphabet size $n,$ decomposes into a sum tensor powers $S_n$-irreducible $S_{(n-1,1)}$ indexed by partition $(n-1,1),$ recovering, as special case, theorem Bj\"orner and Stanley length at most $k.$ For arbitrary ranks we is integer combinatio...

With every graph (or digraph) one can associate several different matrices. Here we shall concentrate mainly on the adjacency matrix of (undirected) graphs, and also discuss briefly the Laplacian. We shall show that spectral properties (the eigenvalues and eigenvectors) of these matrices provide useful information about the structure of the graph. It turns out that for regular graphs, the infor...

Nonnegative matrix factorization is a powerful technique to realize dimension reduction and pattern recognition through single-layer data representation learning. Deep learning, however, with its carefully designed hierarchical structure, is able to combine hidden features to form more representative features for pattern recognition. In this paper, we proposed sparse deep nonnegative matrix fac...

In sparse nonnegative component analysis (sparse NMF) a given dataset is decomposed into a mixing matrix and a feature data set, which are both nonnegative and fulfill certain sparsity constraints. In this paper, we extend the sparse NMF algorithm to allow for varying sparsity in each feature and discuss the uniqueness of an involved projection step. Furthermore, the eligibility of the extended...

Nonnegative Matrix Factorization (NMF) has been widely used in computer vision and pattern recognition. It aims to find two nonnegative matrices whose product can well approximate the nonnegative data matrix, which naturally leads to parts-based and non-subtractive representation. In this paper, we present a neighborhood preserving nonnegative matrix factorization (NPNMF) for dimensionality red...

In this paper, we introduce and provide a short overview of nonnegative matrix factorization (NMF). Several aspects of NMF are discussed, namely, the application in hyperspectral imaging, geometry and uniqueness of NMF solutions, complexity, algorithms, and its link with extended formulations of polyhedra. In order to put NMF into perspective, the more general problem class of constrained low-r...

Nonnegative Matrix Factorization (NMF) techniques has aroused considerable interests from the field of artificial intelligence in recent years because of its good interpretability and computational efficiency. However, in many real world applications, the data features usually evolve over time smoothly. In this case, it would be very expensive in both computation and storage to rerun the whole ...