Joshua Cape's contribution to the Discussion of ‘Vintage Factor Analysis with Varimax Performs Statistical Inference’ by Rohe & Zeng

原文摘要

I congratulate Professor Rohe and Dr Zeng on their illuminating paper. Their broad contributions will no doubt redouble contemporary research activity in multivariate analysis for years to come. Even the paper’s appendices are full of valuable gems, not to be overlooked by readers. Rohe and Zeng contextualize their findings vis-à-vis classical developments in psychometrics and statistics. This choice is apt, for the authors champion a newfound understanding of varimax factor rotations as inferential, not merely exploratory. In a secondary capacity, the paper further complements a burgeoning body of contemporary work dedicated to the entrywise study of eigenvector estimation, perturbations, and asymptotics, in network analysis (e.g., Abbe et al., 2020; Tang & Priebe, 2018) and in high-dimensional statistics (e.g., Cape et al., 2019b; Fan et al., 2018). Under certain data-generating conditions, these works characterize the behaviour of individual vectors and their coordinates in low-dimensional point cloud representations of data, in a much more precise manner than was considered in past decades, let alone during the time of Thurstone or Kaiser. Rohe and Zeng emphasize two forms of sparsity in their treatment of stochastic blockmodel random graphs (SBMs) in the appendix: network sparsity, and latent variable sparsity. The contemporary literature has heretofore largely overlooked the latter and its implications, notably in papers dedicated to adjacency spectral embedding in the time since (Sussman et al., 2012). For SBMs, Rohe and Zeng establish a high-probability uniform error bound which I summarize as max1≤i≤n‖Z^−ZPn‖i,ℓ2=oP(1) in the moderately sparse regime nρn⪰(logn)c⁠. Here ‖⋅‖i,ℓ2 denotes the ℓ2 vector norm of the i-th row of a given matrix, while nρn reflects nodal expected degree. Consequently, perfect clustering (discrimination) is achieved using Z^⁠; more interestingly, the sparse latent variable matrix $Z$ can be element-wise directly estimated uniformly well. More can be said. The eigenvector analysis in Cape et al. (2019a) and Rubin-Delanchy et al. (2022) hints that asymptotic normality might hold for the varimax-based estimator at the scaling nρn⁠. Indeed, it has recently been established in Cape (2022) that for certain SBM graphs, loosely speaking, the i-th row vector of nρn(Z^−ZPn) is conditionally asymptotically multivariate Gaussian, with block-specific asymptotic covariance matrix. Unanswered questions abound. Among them, it would be interesting to quantify the estimation performance of varimax rotations in dimension k^≠ktrue and when the coefficient matrix $B$ is rank degenerate.

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