{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:55:00Z","timestamp":1760144100366,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,3,6]],"date-time":"2024-03-06T00:00:00Z","timestamp":1709683200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12161020","12061025","Qian-Jiao-He KY Zi [2021]298","QKHJC-ZK[2023]YB245","QKHJC [2020]1Z002"],"award-info":[{"award-number":["12161020","12061025","Qian-Jiao-He KY Zi [2021]298","QKHJC-ZK[2023]YB245","QKHJC [2020]1Z002"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation of Educational Commission of Guizhou Province","award":["12161020","12061025","Qian-Jiao-He KY Zi [2021]298","QKHJC-ZK[2023]YB245","QKHJC [2020]1Z002"],"award-info":[{"award-number":["12161020","12061025","Qian-Jiao-He KY Zi [2021]298","QKHJC-ZK[2023]YB245","QKHJC [2020]1Z002"]}]},{"name":"Guizhou Provincial Basis Research Program (Natural Science)","award":["12161020","12061025","Qian-Jiao-He KY Zi [2021]298","QKHJC-ZK[2023]YB245","QKHJC [2020]1Z002"],"award-info":[{"award-number":["12161020","12061025","Qian-Jiao-He KY Zi [2021]298","QKHJC-ZK[2023]YB245","QKHJC [2020]1Z002"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Nowadays, deep representations have gained significant attention due to their outstanding performance in a wide range of tasks. However, the interpretability of deep representations in specific applications poses a significant challenge. For instances where the generated quantity matrices exhibit symmetry, this paper introduces a variant of deep matrix factorization (deep MF) called deep Lp smooth symmetric non-negative matrix factorization (DSSNMF), which aims to improve the extraction of clustering structures inherent in complex hierarchical and graphical representations in high-dimensional datasets by improving the sparsity of the factor matrices. We successfully applied DSSNMF to synthetic datasets as well as datasets related to post-traumatic stress disorder (PTSD) to extract several hierarchical communities. Specifically, we identified non-disjoint communities within the partial correlation networks of PTSD psychiatric symptoms, resulting in highly meaningful clinical interpretations. Numerical experiments demonstrate the promising applications of DSSNMF in fields like network analysis and medicine.<\/jats:p>","DOI":"10.3390\/sym16030312","type":"journal-article","created":{"date-parts":[[2024,3,6]],"date-time":"2024-03-06T05:48:01Z","timestamp":1709704081000},"page":"312","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Hierarchical Object Part Learning Using Deep Lp Smooth Symmetric Non-Negative Matrix Factorization"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-0614-5442","authenticated-orcid":false,"given":"Shunli","family":"Li","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, China"},{"name":"College of Mathematics and Information Science, Guiyang University, Guiyang 550005, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chunli","family":"Song","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Linzhang","family":"Lu","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, China"},{"name":"School of Mathematical Sciences, Xiamen University, Xiamen 361005, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhen","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,6]]},"reference":[{"key":"ref_1","unstructured":"Lee, D., and Seung, H.S. 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