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On varimax asymptotics in network models and spectral methods for dimensionality reduction

Joshua Cape

TL;DR

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 and addresses related concepts including network sparsity, data denoising, and the role of matrix rank in latent variable parameterizations.

Abstract

Varimax factor rotations, while popular among practitioners in psychology and statistics since being introduced by H. Kaiser, have historically been viewed with skepticism and suspicion by some theoreticians and mathematical statisticians. Now, work by K. Rohe and M. Zeng 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 newfound 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.

On varimax asymptotics in network models and spectral methods for dimensionality reduction

TL;DR

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 and addresses related concepts including network sparsity, data denoising, and the role of matrix rank in latent variable parameterizations.

Abstract

Varimax factor rotations, while popular among practitioners in psychology and statistics since being introduced by H. Kaiser, have historically been viewed with skepticism and suspicion by some theoreticians and mathematical statisticians. Now, work by K. Rohe and M. Zeng 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 newfound 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.
Paper Structure (21 sections, 3 theorems, 48 equations, 2 figures, 1 table)

This paper contains 21 sections, 3 theorems, 48 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Contributions of this paper
  4. Preliminaries
  5. Main results
  6. Setup
  7. Undirected stochastic blockmodels
  8. Directed stochastic blockmodels
  9. Degree-corrected stochastic blockmodels
  10. Coda to main results
  11. Numerical examples
  12. Four-block undirected stochastic blockmodel graphs
  13. Two-block directed stochastic blockmodel graphs
  14. Degree-corrected stochastic blockmodels with affinity network structure
  15. Asymptotic variance and covariance expressions
  16. ...and 6 more sections

Key Result

theorem 1

Assume the undirected stochastic blockmodel setting of sec:main_SBM with $n \rho = \omega(\log^{c} n)$ for some sufficiently large constant $c > 1$ where either $\rho = 1$ or $\rho \rightarrow 0$ with limit $\rho_{\infty}$. It suffices to set $c=20$. There exists a sequence of signed permutation mat converges in distribution to a multivariate Gaussian random vector with mean zero and covariance ma

Figures (2)

  • Figure 1: Varimax-rotated left and right embeddings for two directed graphs. Embedding vectors are uncentered. Latent vectors are scaled in agreement with \ref{['sec:main_DCSBM']}. Dotted lines denote circles with radii given by the reciprocals of the square-roots of the block membership probabilities. Solid black circles denote the sample mean vectors computed for each block. Additional details are provided in \ref{['sec:numerics_directed']}.
  • Figure 2: Varimax-rotated degree-corrected stochastic blockmodel embedding for one graph. Dotted circles have radius equal to the magnitude of each block's non-zero theoretical coordinate centroid, while solid black circles denote empirical coordinate centroids. Additional details are provided in \ref{['sec:numerics_degree-corrected']}.

Theorems & Definitions (6)

  • theorem 1
  • theorem 2
  • theorem 3
  • proof : of Theorem 1
  • proof : of Theorem 2
  • proof : outline of Theorem 3