  Graph-Laplacians and their spectral embeddings play an important role in
multiple areas of machine learning. This paper is focused on graph-Laplacian
dimension reduction for the spectral clustering of data as a primary
application. Spectral embedding provides a low-dimensional parametrization of
the data manifold which makes the subsequent task (e.g., clustering) much
easier. However, despite reducing the dimensionality of data, the overall
computational cost may still be prohibitive for large data sets due to two
factors. First, computing the partial eigendecomposition of the graph-Laplacian
typically requires a large Krylov subspace. Second, after the spectral
embedding is complete, one still has to operate with the same number of data
points. For example, clustering of the embedded data is typically performed
with various relaxations of k-means which computational cost scales poorly with
respect to the size of data set. In this work, we switch the focus from the
entire data set to a subset of graph vertices (target subset). We develop two
novel algorithms for such low-dimensional representation of the original graph
that preserves important global distances between the nodes of the target
subset. In particular, it allows to ensure that target subset clustering is
consistent with the spectral clustering of the full data set if one would
perform such. That is achieved by a properly parametrized reduced-order model
(ROM) of the graph-Laplacian that approximates accurately the diffusion
transfer function of the original graph for inputs and outputs restricted to
the target subset. Working with a small target subset reduces greatly the
required dimension of Krylov subspace and allows to exploit the conventional
algorithms (like approximations of k-means) in the regimes when they are most
robust and efficient.
