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个重要类型，其目标首先在于寻找原始数据的低维表示，随后寻求一种旋转使得旋转后的低维表示具有科学意义。应用最广泛的经典因子分析是主成分分析（PCA）配合方差最大旋转。尽管该方法极为流行，但迄今为止鲜有理论保证，这主要是因为方差最大旋转需要在正交矩阵集合上求解一个非凸优化问题。本文提出一种紧缩方差最大旋转程序，可顺序求解正交矩阵的每一行。除了其净计算收益与灵活性之外，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。采用这种新的紧缩方差最大旋转作为PCA之后的第二步，我们进一步在一般因子模型类别下分析这一两步流程。我们的结果表明，当中高信噪比时，该流程能以极小极大最优速率估计因子载荷矩阵。在低信噪比情况下，当因子模型下的加性噪声具有特定结构时，我们提出了对PCA结合紧缩方差最大旋转方法的可能改进。改进后的流程被证明在所有信噪比区间内均达到极小极大最优。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。大量的模拟与真实数据分析进一步证实了我们的理论发现。