{"url":"/method/large-scale-spectral-clustering","slug":"large-scale-spectral-clustering","name":"Large-scale spectral clustering","full_name":"Large-scale spectral clustering","full_name_withheld":false,"description_markdown":"# [Spectral Clustering](https://paperswithcode.com/method/spectral-clustering)\r\n\r\nSpectral clustering aims to partition the data points into $k$ clusters using the spectrum of the graph Laplacians \r\nGiven a dataset $X$ with $N$ data points, spectral clustering algorithm first constructs similarity matrix ${W}$, where ${w_{ij}}$ indicates the similarity between data points $x_i$ and $x_j$ via a similarity measure metric.\r\n\r\nLet $L=D-W$, where $L$ is called graph Laplacian and ${D}$ is a diagonal matrix with $d_{ii} = \\sum_ {j=1}^n w_{ij}$.\r\nThe objective function of spectral clustering can be formulated based on the graph Laplacian as follow:\r\n\\begin{equation}\r\n  \\label{eq:SC_obj}\r\n  {\\max_{{U}}  \\operatorname{tr}\\left({U}^{T} {L} {U}\\right)}, \\\\ {\\text { s.t. } \\quad {U}^{T} {{U}={I}}},\r\n\\end{equation}\r\nwhere $\\operatorname{tr(\\cdot)}$ denotes the trace norm of a matrix.\r\nThe rows of matrix ${U}$ are the low dimensional embedding of the original data points.\r\nGenerally, spectral clustering computes ${U}$ as the bottom $k$ eigenvectors of ${L}$, and finally applies $k$-means on ${U}$ to obtain the clustering results.\r\n\r\n\r\n# Large-scale Spectral Clustering\r\n\r\nTo capture the relationship between all data points in $X$, an $N\\times N$ similarity matrix is needed to be constructed in conventional spectral clustering, which costs $O(N^2d)$ time and $O(N^2)$ memory and is not feasible for large-scale clustering tasks.\r\nInstead of a full similarity matrix, many accelerated spectral clustering methods are using a similarity sub-matrix to represent each data points by the cross-similarity between data points and a set of representative data points (i.e., landmarks) via some similarity measures, as\r\n\\begin{equation}\r\n    \\label{eq: cross-similarity}\r\n    B = \\Phi(X,R),\r\n\\end{equation}\r\nwhere $R = \\{r_1,r_2,\\dots, r_p \\}$ ($p \\ll N$) is a set of landmarks with the same dimension to $X$, $\\Phi(\\cdot)$ indicate a similarity measure metric, and $B\\in \\mathbb{R}^{N\\times p}$ is the similarity sub-matrix to represent the $X \\in \\mathbb{R}^{N\\times d}$ with respect to the $R\\in \\mathbb{R}^{p\\times d}$.\r\n\r\nFor large-scale spectral clustering using such similarity matrix,\r\na symmetric similarity matrix $W$ can be designed as \r\n\\begin{equation}\r\n  \\label{eq: WusedB }\r\n  W=\\left[\\begin{array}{ll}\r\n      \\mathbf{0} & B         ; \\\\\r\n      B^{T}      & \\mathbf{0}\r\n    \\end{array}\\right].\r\n\\end{equation}\r\nThe size of matrix $W$ is $(N+p)\\times (N+p)$. \r\nTaking the advantage of the bipartite structure, some fast eigen-decomposition methods can then  be used to obtain the spectral embedding.\r\nFinally, $k$-means is conducted on the embedding to obtain clustering results.\r\n\r\nThe clustering result is directly related to the quality of $B$ that consists of the similarities between data points and landmarks.\r\nThus, the performance of landmark selection is crucial to the clustering result.","description_state":"present","introduced_year":null,"introduced_by":{"title":"Divide-and-conquer based Large-Scale Spectral Clustering","paper":"/paper/divide-and-conquer-based-large-scale-spectral","first_author":null,"n_authors":0,"url_abs":null,"archive_paper_url":"https://paperswithcode.com/paper/divide-and-conquer-based-large-scale-spectral"},"source":{"url":"https://www.researchgate.net/publication/351270623_Divide-and-conquer_based_Large-Scale_Spectral_Clustering","title":"Divide-and-conquer based Large-Scale Spectral Clustering","url_on_a_paper_host":true},"code_snippet_url":null,"code_snippet_url_on_a_code_host":false,"categories":[{"area":"General","area_id":"general","collection":"Clustering","url":"/methods/category/clustering","pwc_aliases":[]}],"n_papers_tagged":4,"archive_num_papers":5,"papers_newest_first":[{"paper":null,"title":"Clustering and classification of low-dimensional data in explicit feature map domain: intraoperative pixel-wise diagnosis of adenocarcinoma of a colon in a liver","date":"2022-03-07","arxiv_id":"2203.03636","n_code_links":0,"syntology":null},{"paper":"/paper/lsec-large-scale-spectral-ensemble-clustering","title":"LSEC: Large-scale spectral ensemble clustering","date":"2021-06-18","arxiv_id":"2106.09852","n_code_links":1,"syntology":null},{"paper":"/paper/ultra-scalable-spectral-clustering-and","title":"Ultra-Scalable Spectral Clustering and Ensemble Clustering","date":"2019-03-04","arxiv_id":"1903.01057","n_code_links":0,"syntology":null},{"paper":"/paper/large-scale-spectral-clustering-using","title":"Large-scale spectral clustering using diffusion coordinates on landmark-based bipartite graphs","date":"2018-06-01","arxiv_id":null,"n_code_links":0,"syntology":null}],"papers_shown":4,"tasks":[{"task":"/task/clustering","name":"Clustering","papers":4},{"task":"/task/imagedocument-clustering","name":"Image/Document Clustering","papers":2},{"task":"/task/semantic-segmentation","name":"Semantic Segmentation","papers":2},{"task":"/task/incremental-learning","name":"Incremental Learning","papers":1}],"tasks_shown":4,"n_tasks":4,"usage_by_year":[{"year":"2018","papers":1},{"year":"2019","papers":1},{"year":"2021","papers":1},{"year":"2022","papers":1}],"row_source":"methods_table","archive":{"source":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","archive_url":"https://paperswithcode.com/method/large-scale-spectral-clustering"},"syntology_read_at":"2026-09-24T18:15:14+00:00"}