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. Perhaps 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 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 broad context.
Adopting this new varimax approach 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 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
procedure when the additive noise under the factor model is structured. The
modified procedure is shown to be 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.

本文介绍了古董因子分析的一个重要类型 —— 主成分分析（PCA）和 varimax 旋转后的方差极大化旋转。通过将 PCA 与新的 deflation varimax 方法相结合，并使用一般类别的因子模型进行分析，我们发现它在中等或高信噪比条件下以最佳速率估计因子载荷矩阵，在低信噪比条件下当因子模型中的加性噪声具有结构时，改进的方法优于使用 PCA 和 deflation 程序。在有限样本条件下，理论有效且允许潜在因子的数量随样本量的增长而增长，同时允许环境维度随着样本量的增长甚至超过样本量。我们的理论结果得到了广泛的模拟和实际数据分析的验证。