Compressive Spectral Clustering - Error Analysis

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. One of the most common and powerful techniques for extracting meaningful information from a data set is spectral clustering. Spectral clustering uses local information to embed the data into a space which captures the global group structure. Standard learning techniques require an appropriate transformation to higher dimension where dimensionality reduction is done before clustering. Compressed sensing provides a mathematically rigorous way to obtain optimal dimensionality reduction for exact reconstruction. Hyperspectral images and MRIs are examples of high dimensional signals where the true underlying data may only have a few degrees of freedom or be sparse in some unknown basis. We show that the meaningful organization extracted from spectral clustering is preserved under the perturbation from making compressed sensing measurements.