On approximating the non-negative rank: Applications to unsupervised image reduction

Unsupervised Learning (UL) methods are a class of machine learning which aims to disentangle the representations and reduce the dimensionality among the data without any prede\ufb01ned labels. Among all UL methods, the Non-negative Matrix Factorization (NMF) factorizes the data into two subspaces of non-negative components. Moreover, the NMF enforces the non-negativity, sparsity, and part-based analysis, thus the representations can be interpreted and explained for the Explainable Arti\ufb01cial Intelligence (XAI) applications. However, one of the main issues when using the NMF is to impose the factorization rank r to identify the dimensionality of the subspaces, where the rank is usually unknown in advance and known as the non-negative rank that is used as a prior to carrying out the factorization. Accordingly, we propose a novel method for the non-negative rank r approximation to help solving this problem, and we generalize our method among different image scales. Where, the results on different image data sets con\ufb01rm the validity of our approach