  This paper focuses on obtaining clustering information about a distribution
from its i.i.d. samples. We develop theoretical results to understand and use
clustering information contained in the eigenvectors of data adjacency matrices
based on a radial kernel function with a sufficiently fast tail decay. In
particular, we provide population analyses to gain insights into which
eigenvectors should be used and when the clustering information for the
distribution can be recovered from the sample. We learn that a fixed number of
top eigenvectors might at the same time contain redundant clustering
information and miss relevant clustering information. We use this insight to
design the data spectroscopic clustering (DaSpec) algorithm that utilizes
properly selected eigenvectors to determine the number of clusters
automatically and to group the data accordingly. Our findings extend the
intuitions underlying existing spectral techniques such as spectral clustering
and Kernel Principal Components Analysis, and provide new understanding into
their usability and modes of failure. Simulation studies and experiments on
real-world data are conducted to show the potential of our algorithm. In
particular, DaSpec is found to handle unbalanced groups and recover clusters of
different shapes better than the competing methods.
