Optimal vintage factor analysis with deflation varimax
Xin Bing Dian Jin Yuqian Zhang
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.