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.

翻译：经典因子分析是因子分析的一种重要类型，其目标首先寻找原始数据的低维表示，进而寻求一种旋转使得旋转后的低维表示具有科学意义。最广泛使用的经典因子分析是主成分分析（PCA）后接方差最大化旋转。尽管该方法流行，但迄今鲜有理论保证，主要因为方差最大化旋转需要在正交矩阵集合上求解非凸优化问题。本文提出一种去膨胀方差最大化程序，可顺序求解正交矩阵的每一行。除了其净计算收益和灵活性外，我们能够在更广泛的背景下为该程序建立完整的理论保证。采用这种新的去膨胀方差最大化作为PCA后的第二步，我们在一般因子模型类别下进一步分析这一两步程序。结果表明，当中高信噪比（SNR）时，该程序能以极小极大最优速率估计因子载荷矩阵。在低SNR情况下，当因子模型下的加性噪声具有结构时，我们提出了对使用PCA和去膨胀方差最大化方法的可能改进。改进后的程序被证明在所有SNR情况下都是极小极大最优的。我们的理论适用于有限样本，允许潜在因子数量随样本量增长，且允许环境维度随样本量增长甚至超过样本量。大量的模拟和真实数据分析进一步证实了我们的理论发现。