Limit theorems for out-of-sample extensions of the adjacency and Laplacian spectral embeddings

TL;DR

This paper establishes theoretical limit theorems for out-of-sample extensions of adjacency and Laplacian spectral embeddings in graph analysis, providing insights into their statistical properties and trade-offs.

Contribution

It introduces new limit theorems for out-of-sample spectral embedding extensions under the random dot product graph model, including CLTs and concentration inequalities.

Findings

Out-of-sample extension obeys a CLT for latent positions.

Concentration inequality established for adjacency spectral embedding.

Framework for analyzing accuracy versus computational cost.

Abstract

Graph embeddings, a class of dimensionality reduction techniques designed for relational data, have proven useful in exploring and modeling network structure. Most dimensionality reduction methods allow out-of-sample extensions, by which an embedding can be applied to observations not present in the training set. Applied to graphs, the out-of-sample extension problem concerns how to compute the embedding of a vertex that is added to the graph after an embedding has already been computed. In this paper, we consider the out-of-sample extension problem for two graph embedding procedures: the adjacency spectral embedding and the Laplacian spectral embedding. In both cases, we prove that when the underlying graph is generated according to a latent space model called the random dot product graph, which includes the popular stochastic block model as a special case, an out-of-sample extension based on a least-squares objective obeys a central limit theorem about the true latent position of the out-of-sample vertex. In addition, we prove a concentration inequality for the out-of-sample extension of the adjacency spectral embedding based on a maximum-likelihood objective. Our results also yield a convenient framework in which to analyze trade-offs between estimation accuracy and computational expense, which we explore briefly.