Out-of-sample extension of graph adjacency spectral embedding

TL;DR

This paper develops and compares two out-of-sample extension methods for adjacency spectral embedding in graph analysis, demonstrating their effectiveness under a latent position model and establishing asymptotic normality for one approach.

Contribution

It introduces two novel out-of-sample extension methods for adjacency spectral embedding and proves their consistency and asymptotic properties under the random dot product graph model.

Findings

Both methods estimate true latent positions with the same error rate.

A central limit theorem is established for the least-squares extension.

Extensions are effective for graphs generated by the random dot product model.

Abstract

Many popular dimensionality reduction procedures have out-of-sample extensions, which allow a practitioner to apply a learned embedding to observations not seen in the initial training sample. In this work, we consider the problem of obtaining an out-of-sample extension for the adjacency spectral embedding, a procedure for embedding the vertices of a graph into Euclidean space. We present two different approaches to this problem, one based on a least-squares objective and the other based on a maximum-likelihood formulation. We show that if the graph of interest is drawn according to a certain latent position model called a random dot product graph, then both of these out-of-sample extensions estimate the true latent position of the out-of-sample vertex with the same error rate. Further, we prove a central limit theorem for the least-squares-based extension, showing that the estimate is asymptotically normal about the truth in the large-graph limit.