Ying Zhou and Xinyi Zhang's contribution to the Discussion of ‘Vintage Factor Analysis with Varimax Performs Statistical Inference’ by Rohe & Zeng
本文讨论了Rohe和Zeng关于半参数因子模型中Varimax旋转的理论支持，提出了关于经典因子模型、载荷矩阵、独立成分分析以及样本协方差矩阵可逆性等问题的疑问。
We congratulate Rohe and Zeng for their inspiring work on providing theoretical support for Varimax rotation in factor analysis. A key contribution of the article is the demonstration that if the factors in the semi-parametric factor model follow heavy-tailed distribution, then performing principal components analysis (PCA) on the observed data matrix along with Varimax rotation applied to the principal components does produce interpretable explanatory variables. We have the following comments and questions: The article mainly discussed the semi-parametric model, which seems to exclude the classical factor model in the form AT=LF+ε, where observation matrix A∈Rn×d, loading matrix L∈Rd×k, factor matrix F∈Rk×n, error term matrix ε∈Rd×n. One question is whether there is a similar result applicable to the classical factor model. If not, what are the obstacles? And what properties of semi-parametric factor model facilitate the identification? The main theorem in the article concerns the factor matrix F (A in their notation), while in many applications, people are interested in the loading matrix L. It appears there is no loading matrix in semi-parametric factor model. Perhaps BYT in Definition 1 is somewhat related to loading matrix. This lead to the question that, if there is a parallel result for Y. As one of the modern factor models, the Independent Components Analysis is an unsupervised learning algorithm and can be applied for feature extraction. Would it be possible to integrate class information with the Varimax rotation for extracting features that belong to well-separated classes? It would be interesting to see if the Vintage Factor Analysis can be used in a supervised fashion. If we understand it correctly, derivation of the population results for PCA with latent variable models and Varimax uses Σ^Z−1/2 to show how U can be recovered from Z. In theory, dimension d can be of the same order as n. However, in this case, the sample covariance matrix Σ^Z may not be invertible and an alternative estimation of ΣZ−1/2 is needed. Can similar results be established? The factors Z are allowed to be correlated, will the corresponding theory be a direct generalisation from the independent setting? The authors replied later, in writing, as follows.