Abstract
Spectral clustering methods allow datasets to be partitioned into clusters by mapping the input datapoints into the space spanned by the eigenvectors of the Laplacian matrix. In this article, we make use of the incomplete Cholesky decomposition (ICD) to construct an approximation of the graph Laplacian and reduce the size of the related eigenvalue problem from N to m, with m ≪ N . In particular, we introduce a new stopping criterion based on normalized mutual information between consecutive partitions, which terminates the ICD when the change in the cluster assignments is below a given threshold. Compared with existing ICD-based spectral clustering approaches, the proposed method allows the reduction of the number m of selected pivots (i.e., to obtain a sparser model) and at the same time, to maintain high clustering quality. The method scales linearly with respect to the number of input datapoints N and has low memory requirements, because only matrices of size N × m and m × m are calculated (in contrast to standard spectral clustering, where the construction of the full N × N similarity matrix is needed). Furthermore, we show that the number of clusters can be reliably selected based on the gap heuristics computed using just a small matrix R of size m × m instead of the entire graph Laplacian. The effectiveness of the proposed algorithm is tested on several datasets.
Highlights
In this paper, we deal with the data clustering problem
We introduce a spectral clustering algorithm that exploits the incomplete Cholesky decomposition to reduce the size of the eigenvalue problem
The incomplete Cholesky decomposition (ICD) [21] allows the reduciton of the computational time required by the Cholesky decomposition by computing a low rank approximation of accuracy τ of the matrix A in O(m2 N ), such that
Summary
Clustering refers to a technique for partitioning unlabeled data into natural groups, where data points that are related to each other are grouped together and points that are dissimilar are assigned to different groups [1] In this context, spectral clustering [2,3,4,5] has been shown to be among the most successful methods in many application domains, due mainly to its ability to discover nonlinear clustering boundaries. The algorithm is based on computing the eigendecomposition of a matrix derived from the data called Laplacian. We introduce a spectral clustering algorithm that exploits the incomplete Cholesky decomposition to reduce the size of the eigenvalue problem.
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