  Transfer learning has aroused great interest in the statistical community. In
this article, we focus on knowledge transfer for unsupervised learning tasks in
contrast to the supervised learning tasks in the literature. Given the
transferable source populations, we propose a two-step transfer learning
algorithm to extract useful information from multiple source principal
component analysis (PCA) studies, thereby enhancing estimation accuracy for the
target PCA task. In the first step, we integrate the shared subspace
information across multiple studies by a proposed method named as Grassmannian
barycenter, instead of directly performing PCA on the pooled dataset. The
proposed Grassmannian barycenter method enjoys robustness and computational
advantages in more general cases. Then the resulting estimator for the shared
subspace from the first step is further utilized to estimate the target private
subspace in the second step. Our theoretical analysis credits the gain of
knowledge transfer between PCA studies to the enlarged eigenvalue gap, which is
different from the existing supervised transfer learning tasks where sparsity
plays the central role. In addition, we prove that the bilinear forms of the
empirical spectral projectors have asymptotic normality under weaker eigenvalue
gap conditions after knowledge transfer. When the set of informativesources is
unknown, we endow our algorithm with the capability of useful dataset selection
by solving a rectified optimization problem on the Grassmann manifold, which in
turn leads to a computationally friendly rectified Grassmannian K-means
procedure. In the end, extensive numerical simulation results and a real data
case concerning activity recognition are reported to support our theoretical
claims and to illustrate the empirical usefulness of the proposed transfer
learning methods.
