Search for: All records

Abstract Varimax factor rotations, while popular among practitioners in psychology and statistics since being introduced by Kaiser (1958), have historically been viewed with skepticism and suspicion by some theoreticians and mathematical statisticians. Now, work by Rohe & Zeng (2023) provides new, fundamental insight: varimax rotations provably perform statistical estimation in certain classes of latent variable models when paired with spectral-based matrix truncations for dimensionality reduction. We build on this new-found understanding of varimax rotations by developing further connections to network analysis and spectral methods rooted in entrywise matrix perturbation analysis. Concretely, this paper establishes the asymptotic multivariate normality of vectors in varimax-transformed Euclidean point clouds that represent low-dimensional node embeddings in certain latent space random graph models. We address related concepts including network sparsity, data denoising and the role of matrix rank in latent variable parameterizations. Collectively, these findings, at the confluence of classical and contemporary multivariate analysis, reinforce methodology and inference procedures grounded in matrix factorization-based techniques. Numerical examples illustrate our findings and supplement our discussion. 

Abstract We pursue the problem of modelling and analysing latent space dynamics in collections of networks. Towards this end, we pose and study latent space generative models for signed networks that are amenable to inference via spectral methods. Permitting signs, rather than restricting to unsigned networks, enables richer latent space structure and permissible dynamic mechanisms that can be provably inferred via low rank truncations of observed adjacency matrices. Our treatment of and ability to recover latent space dynamics holds across different levels of granularity, namely, at the overall graph level, for communities of nodes, and even at the individual node level. We provide synthetic and real data examples to illustrate the effectiveness of methodologies and to corroborate accompanying theory. The contributions set forth in this paper complement an emerging statistical paradigm for random graph inference encompassing random dot product graphs and generalizations thereof. 

This note revisits the classical orthogonal Procrustes problem and investigates the norm-dependent geometric behavior underlying Procrustes alignment for subspaces. It presents generic, deterministic bounds quantifying the performance of a specified Procrustes-based choice of subspace alignment. Numerical examples illustrate the theoretical observations and offer additional, empirical findings which are discussed in detail. This note complements recent advances in statistics involving Procrustean matrix perturbation decompositions and eigenvector estimation.