Approximate Spectral Clustering Using Topology Preserving Methods and Local Scaling

Abstract

Spectral clustering is the type of unsupervised learning that separates data based on their connectivity instead of convexity. However, its computational demands increase cubically with the number of points n. This triggered a stream of studies to ease these demands. An effective solution is to provide an approximated graph $G^{*} = \left( {V^{*} ,E^{*} } \right)$ for the input data with a reduced set of vertices and edges. Recent similarity measures used to construct the approximated graph $G^{*} = \left( {V^{*} ,E^{*} } \right)$ have some deficiencies such as: (1) weights on edges highly depend on the cluster density, and (2) larger memory footprint compared to conventional similarity measures. In this work, we employed topology preserving methods (e.g., neural gas) to obtain $G^{*} = \left( {V^{*} ,E^{*} } \right)$ due to their ability to preserve input data topology. Then we used a conventional similarity measure to assign weights on the graph. The experiments reveal that graphs obtained through topology preserving methods and passed to a locally scaled similarity measure, produce performances comparable to the recent measures with a significantly smaller memory footprint.