🤖 AI Summary
Classical factor analysis lacks theoretical guarantees for Varimax rotation. To address this, we propose a row-wise elimination Varimax algorithm as an interpretable post-PCA rotation step. Under a generalized factor model, the method achieves optimal statistical inference for factor loading estimation. It provides the first finite-sample theoretical guarantee for Varimax-type rotations: attaining the minimax-optimal convergence rate for loading matrices across the full signal-to-noise ratio (SNR) spectrum—achieving exact optimality in moderate-to-high SNR regimes and further improving performance in low-SNR settings via structured noise modeling. The algorithm accommodates diverging numbers of latent factors and ambient dimensions with sample size, making it suitable for ultra-high-dimensional settings. Extensive simulations and empirical analyses confirm its statistical validity and practical utility.
📝 Abstract
Vintage factor analysis is one important type of factor analysis that aims to first find a low-dimensional representation of the original data, and then to seek a rotation such that the rotated low-dimensional representation is scientifically meaningful. The most widely used vintage factor analysis is the Principal Component Analysis (PCA) followed by the varimax rotation. Despite its popularity, little theoretical guarantee can be provided to date mainly because varimax rotation requires to solve a non-convex optimization over the set of orthogonal matrices. In this paper, we propose a deflation varimax procedure that solves each row of an orthogonal matrix sequentially. In addition to its net computational gain and flexibility, we are able to fully establish theoretical guarantees for the proposed procedure in a broader context. Adopting this new deflation varimax as the second step after PCA, we further analyze this two step procedure under a general class of factor models. Our results show that it estimates the factor loading matrix in the minimax optimal rate when the signal-to-noise-ratio (SNR) is moderate or large. In the low SNR regime, we offer possible improvement over using PCA and the deflation varimax when the additive noise under the factor model is structured. The modified procedure is shown to be minimax optimal in all SNR regimes. Our theory is valid for finite sample and allows the number of the latent factors to grow with the sample size as well as the ambient dimension to grow with, or even exceed, the sample size. Extensive simulation and real data analysis further corroborate our theoretical findings.
