Spectral clustering is a tool for extracting meaningful information from data by grouping similar objectsDtogether [1]. The method uses the eigenvector of an adjacency matrix for embedding the data into a space that captures the underlying group structure [2]. High-dimensional signals, magnetic resonance images, and hyperspectral images can be costly to acquire; even simple direct comparisons could be infeasible among such data sets. Our work shows that the meaningful organization extracted from spectral clustering is preserved under the perturbation from making compressed, incomplete and inaccurate measurements. Using bounds on the perturbation of eigenvectors, we establish error bounds of the spectral embedding when matrix completion and compressed sensing measurements are used. Given some error Nǫ in the entries of an affinity matrix A RN N, we show that the space spanned by the first k eigenvector are all within O(Nǫ) of the span of the unperturbed eigenvectors. We prove that the perturbed spectral coordinates are within O(Nǫ)of a unitary transform of the unperturbed coordinates and can give k-means cluster assignments within O(Nǫ) of the unperturbed case. This analysis holds true when the error perturbation in the entries of an affinity matrix |A(i,j) A (i,j)| ǫ is caused from making compressed arXiv:1011.0997v1

Different from Laplacian or normal matrix, the properties of the adjacency eigenspace received much less attention. Recent work showed that nodes projected into the adjacency eigenspace exhibit an orthogonal line pattern and nodes from the same community locate along the same line. In this paper, we conduct theoretical studies based on graph perturbation to demonstrate why this line orthogonality property holds in the adjacency eigenspace and why it generally disappears in the Laplacian and normal eigenspaces. Using the orthogonality property in the adjacency eigenspace, we present a graph partition algorithm, AdjCluster, which first projects node coordinates to the unit sphere and then applies the classic k-means to find clusters. Empirical evaluations on synthetic data and real-world social networks validate our theoretical findings and show the effectiveness of our graph partition algorithm.

Compressive spectral clustering combines the distance preserving measurements of compressed sensing with the power of spectral clustering. Our analysis provides rigorous bounds on how small errors in the affinity matrix can affect the spectral coordinates and clusterability. This work generalizes the current perturbation results of two-class spectral clustering to incorporate multiclass clustering using k eigenvectors.