Spectral clustering: Difference between revisions

Revision as of 14:58, 28 November 2023

Revision as of 02:32, 3 December 2023

Line 54:

If the similarity matrix <math>A</math> has not already been explicitly constructed, the efficiency of spectral clustering may be improved if the solution to the corresponding eigenvalue problem is performed in a [[Matrix-free methods|matrix-free fashion]] (without explicitly manipulating or even computing the similarity matrix), as in the [[Lanczos algorithm]].

For large-sized graphs, the second eigenvalue of the (normalized) graph [[Laplacian matrix]] is often [[ill-conditioned]], leading to slow convergence of iterative eigenvalue solvers. [[Preconditioner#Preconditioning for eigenvalue problems|Preconditioning]] is a key technology accelerating the convergence, e.g., in the matrix-free [[LOBPCG]] method. Spectral clustering has been successfully applied on large graphs by first identifying their [[community structure]], and then clustering communities.<ref>{{cite journal|last1=Zare|first1=Habil |first2=P. |last2=Shooshtari |first3=A. |last3=Gupta |first4=R. |last4=Brinkman|title=Data reduction for spectral clustering to analyze high throughput flow cytometry data|journal=BMC Bioinformatics|date=2010|doi=10.1186/1471-2105-11-403|volume=11|pages=403 |pmid=20667133 |pmc=2923634 |doi-access=free }}</ref>

Spectral clustering is closely related to [[nonlinear dimensionality reduction]], and dimension reduction techniques such as locally-linear embedding can be used to reduce errors from noise or outliers.<ref>{{Citation

Line 75:

Moreover, a normalized Laplacian has exactly the same eigenvectors as the normalized adjacency matrix, but with the order of the eigenvalues reversed. Thus, instead of computing the eigenvectors corresponding to the smallest eigenvalues of the normalized Laplacian, one can equivalently compute the eigenvectors corresponding to the largest eigenvalues of the normalized adjacency matrix, without even talking about the Laplacian matrix.

Naive constructions of the graph [[adjacency matrix]], e.g., using the RBF kernel, make it dense, thus requiring <math>n^2</math> memory and <math>n^2</math> AO to determine each of the <math>n^2</math> entries of the matrix. Nystrom method<ref>{{Cite journal|last=Fowlkes|first=C|date=2004|title=Spectral grouping using the Nystrom method.|url=https://escholarship.org/uc/item/29z29233|journal=IEEE Transactions on Pattern Analysis and Machine Intelligence|volume=26|issue=2|pages=214–25|doi=10.1109/TPAMI.2004.1262185|pmid=15376896|s2cid=2384316}}</ref> can be used to approximate the similarity matrix, but the approximate matrix is not elementwise positive,<ref>{{Cite journal|firstfirst1=S. |last1=Wang |first2=A. |last2=Gittens |first3=M.W. |last3=Mahoney|year=2019|title=Scalable Kernel K-Means Clustering with Nystrom Approximation: Relative-Error Bounds|journal=Journal of Machine Learning Research|volume=20|pages=1–49|arxiv=1706.02803}}</ref> i.e. cannot be interpreted as a distance-based similarity.

Algorithms to construct the graph adjacency matrix as a [[sparse matrix]] are typically based on a [[nearest neighbor search]], which estimate or sample a neighborhood of a given data point for nearest neighbors, and compute non-zero entries of the adjacency matrix by comparing only pairs of the neighbors. The number of the selected nearest neighbors thus determines the number of non-zero entries, and is often fixed so that the memory footprint of the <math>n</math>-by-<math>n</math> graph adjacency matrix is only <math>O(n)</math>, only <math>O(n)</math> sequential arithmetic operations are needed to compute the <math>O(n)</math> non-zero entries, and the calculations can be trivially run in parallel.