On the Effect of Misspecifying the Embedding Dimension in Low-rank Network Models
Roddy Taing, Keith Levin
TL;DR
The paper addresses how adjacency spectral embedding (ASE) behaves when the embedding dimension is misspecified in latent-space network models, notably the random dot product graph (RDPG) and its binary and weighted variants. It develops a signal-plus-noise framework $\mathbf{A}=\mathbf{P}+\mathbf{E}$ with $\mathbf{P}=\rho_n\mathbf{X}\mathbf{X}^\top$ and proves a key delocalization result: all non-signal eigenvectors of $\mathbf{A}$ are delocalized under mild assumptions. The authors quantify misspecification effects: when the chosen dimension $d=r+k$ is too small ($k<0$), they obtain a lower bound on estimation error of order $\sqrt{\rho_n|k|}$, while for over-specification ($k>0$) they establish an upper bound showing consistency with a slower rate $\|\hat{\mathbf{X}}_{1:r+k}-\rho_n^{1/2}\mathbf{X}_{1:r+k}\|_{2,\infty} \lesssim \phi_n + \sqrt{k}\,r^2(\log n)^{5+6\gamma}/n^{1/4}$. The results extend to weighted and binary networks, with experiments demonstrating the predicted rates and showing consistency persists under over-specification. The work provides guidance for dimension selection in network embeddings and contributes a delocalization-toolkit for random-matrix-based spectral analysis in low-rank perturbations.
Abstract
As network data has become ubiquitous in the sciences, there has been growing interest in network models whose structure is driven by latent node-level variables in a (typically low-dimensional) latent geometric space. These "latent positions" are often estimated via embeddings, whereby the nodes of a network are mapped to points in Euclidean space so that "similar" nodes are mapped to nearby points. Under certain model assumptions, these embeddings are consistent estimates of the latent positions, but most such results require that the embedding dimension be chosen correctly, typically equal to the dimension of the latent space. Methods for estimating this correct embedding dimension have been studied extensive in recent years, but there has been little work to date characterizing the behavior of embeddings when this embedding dimension is misspecified. In this work, we provide theoretical descriptions of the effects of misspecifying the embedding dimension of the adjacency spectral embedding under the random dot product graph, a class of latent space network models that includes a number of widely-used network models as special cases, including the stochastic blockmodel. We consider both the case in which the dimension is chosen too small, where we prove estimation error lower-bounds, and the case where the dimension is chosen too large, where we show that consistency still holds, albeit at a slower rate than when the embedding dimension is chosen correctly.A range of synthetic data experiments support our theoretical results. Our main technical result, which may be of independent interest, is a generalization of earlier work in random matrix theory, showing that all non-signal eigenvectors of a low-rank matrix subject to additive noise are delocalized.