Feature extraction plays a crucial role in dimensionality reduction in machine learning applications. Nonnegative Matrix Factorization (NMF) has emerged as a powerful technique for dimensionality reduction; however, its equal treatment of all features may limit accuracy. To address this challenge, this paper introduces Graph-Regularized entropy-Weighted Nonnegative Matrix Factorization (GEWNMF) for enhanced feature representation. The proposed method improves feature extraction through two key innovations: optimizable feature weights and Graph regularization. GEWNMF uses optimizable weights to prioritize the extraction of crucial features that best describe the underlying data structure. These weights, determined using entropy measures, ensure a diverse selection of features, thereby enhancing the fidelity of the data representation. This adaptive weighting not only improves interpretability but also strengthens the model against noisy or outlier-prone datasets. Furthermore, GEWNMF integrates robust Graph regularization techniques to preserve local data relationships. By constructing an adjacency Graph that captures these relationships, the method enhances its ability to discern meaningful patterns amid noise and variability. This regularization not only stabilizes the method but also ensures that nearby data points appropriately influence feature extraction. Thus, GEWNMF produces representations that capture both global trends and local nuances, making it applicable across various domains. Extensive experiments on four widely used datasets validate the efficacy of GEWNMF compared to existing methods, demonstrating its superior performance in capturing meaningful data patterns and enhancing interpretability.

A useful tool for investigation various problems in mathematical chemistry and  computational physics is Graph entropy.  In this paper, we introduce a new version of Graph entropy  and  then we determine it for some classes of Graphs.

Graph representation of data can better define relationships among data components and thus provide better and richer analysis. So far, movies have been represented in Graphs many times using different features for clustering, genre prediction, and even for use in recommender systems. In constructing movie Graphs, little attention has been paid to their textual features such as subtitles, while they contain the entire content of the movie and there is a lot of hidden information in them. So, in this paper, we propose a method called MoGaL to construct movie Graph using LDA on subtitles. In this method, each node is a movie and each edge represents the novel relationship discovered by MoGaL among two associated movies. First, we extracted the important topics of the movies using LDA on their subtitles. Then, we visualized the relationship between the movies in a Graph, using the cosine similarity. Finally, we evaluated the proposed method with respect to measures genre homophily and genre entropy. MoGaL succeeded to outperforms the baseline method significantly in these measures. Accordingly, our empirical results indicate that movie subtitles could be considered a rich source of informative information for various movie analysis tasks.

The independence Graph Ind(G) of a Graph G is the Graph with vertices as maximum independent sets of G and two vertices are adjacent, if and only if the corresponding maximum independent sets are disjoint. In this work, we find the independence Graph of Cartesian product of d copies of complete Graphs Kq, which is known as the Hamming Graph H(d, q). Greenwell and Lovasz [7] found that the independence number of direct product of d copies of Kq as qd−1. We prove that the independence number of Hamming Graph H(d, q), which is cartesian product of d copies of Kq, is also qd−1. As an application of our findings, we find answers for rook problem in higher dimensional square chess board.