Analysis of deep non-smooth symmetric nonnegative matrix factorization on hierarchical clustering

Deep matrix factorization (deep MF) is an increasingly popular unsupervised data-mining technique that operates as a deep decomposition rooted in traditional nonnegative matrix factorization (NMF). Compared with standard NMF, deep MF has shown excellent performance in the extraction of hierarchical information from complex datasets. For cases in which the data matrices corresponding to the dataset are symmetric—such as the adjacency matrix of an undirected graph in network analysis—this paper proposes a deep MF variant called deep non-smooth nonnegative symmetric matrix factorization (DNSSNMF). The aim of this work is to enhance the extraction of complex hierarchical structures in high-dimensional datasets and achieve the clustering of structures inherent in graphical representations by improving the goodness-of-fit of the factor matrix product. Accordingly, we successfully applied DNSSNMF to post-traumatic-stress-disorder (PTSD) datasets and synthetic datasets to extract several hierarchical communities. In particular, we extracted non-disjoint communities in the partial correlation network of psychiatric symptoms in PTSD, revealing correlations between different symptoms and leading to meaningful clinical interpretations. The results of our numerical experiments indicated promising applications of DNSSNMF in fields including network analysis and medicine.