{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T14:32:52Z","timestamp":1774621972229,"version":"3.50.1"},"reference-count":36,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2016,5,26]],"date-time":"2016-05-26T00:00:00Z","timestamp":1464220800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Postdoctoral Sustentation Fund of Heilongjiang Province of China","award":["LBH-Z14051"],"award-info":[{"award-number":["LBH-Z14051"]}]},{"name":"Natural Science Fund of Heilongjiang Province of China","award":["F2015033"],"award-info":[{"award-number":["F2015033"]}]},{"DOI":"10.13039\/501100001809","name":"Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61573114"],"award-info":[{"award-number":["61573114"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Recently manifold learning has received extensive interest in the community of pattern recognition. Despite their appealing properties, most manifold learning algorithms are not robust in practical applications. In this paper, we address this problem in the context of the Hessian locally linear embedding (HLLE) algorithm and propose a more robust method, called RHLLE, which aims to be robust against both outliers and noise in the data. Specifically, we first propose a fast outlier detection method for high-dimensional datasets. Then, we employ a local smoothing method to reduce noise. Furthermore, we reformulate the original HLLE algorithm by using the truncation function from differentiable manifolds. In the reformulated framework, we explicitly introduce a weighted global functional to further reduce the undesirable effect of outliers and noise on the embedding result. Experiments on synthetic as well as real datasets demonstrate the effectiveness of our proposed algorithm.<\/jats:p>","DOI":"10.3390\/a9020036","type":"journal-article","created":{"date-parts":[[2016,5,26]],"date-time":"2016-05-26T12:13:20Z","timestamp":1464264800000},"page":"36","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Robust Hessian Locally Linear Embedding Techniques for High-Dimensional Data"],"prefix":"10.3390","volume":"9","author":[{"given":"Xianglei","family":"Xing","sequence":"first","affiliation":[{"name":"College of Automation, Harbin Engineering University, Harbin 150001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sidan","family":"Du","sequence":"additional","affiliation":[{"name":"School of Electronic Science and Engineering, Nanjing University, Nanjing 210046, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kejun","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Automation, Harbin Engineering University, Harbin 150001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2016,5,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2319","DOI":"10.1126\/science.290.5500.2319","article-title":"A global geometric framework for nonlinear dimensionality reduction","volume":"290","author":"Tenenbaum","year":"2000","journal-title":"Science"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2323","DOI":"10.1126\/science.290.5500.2323","article-title":"Nonlinear dimensionality reduction by locally linear embedding","volume":"290","author":"Roweis","year":"2000","journal-title":"Science"},{"key":"ref_3","first-page":"585","article-title":"Laplacian eigenmaps and spectral techniques for embedding and clustering","volume":"Volume 14","author":"Dietterich","year":"2001","journal-title":"Advances in Neural Information Processing Systems 14 (NIPS)"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"406","DOI":"10.1007\/s11741-004-0051-1","article-title":"Principal manifolds and nonlinear dimensionality reduction via tangent space alignment","volume":"8","author":"Zhang","year":"2004","journal-title":"J. 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