We propose a low-rank approach to learning a Mahalanobis metric from data.
Inspired by the recent geometric mean metric learning (GMML) algorithm, we
propose a low-rank variant of the algorithm. This allows to jointly learn a
low-dimensional subspace where the data reside and the Mahalanobis metric that
appropriately fits the data. Our results show that we compete effectively with
GMML at lower ranks.

用低秩方法从数据中学习马氏度量，该方法受到几何平均度量学习算法的启发，可以联合学习低维子空间和适合数据的马氏度量，结果显示在更低的秩下可以有效地与几何平均度量学习算法竞争。

低秩几何平均度量学习

Low-rank geometric mean metric learning

We present a novel framework for domain adaptation, whereby both geometric
and statistical differences between a labeled source domain and unlabeled
target domain can be integrated by exploiting the curved Riemannian geometry of
statistical manifolds. Our approach is based on formulating transfer from
source to target as a problem of geometric mean metric learning on manifolds.
Specifically, we exploit the curved Riemannian manifold geometry of symmetric
positive definite (SPD) covariance matrices. We exploit a simple but important
observation that as the space of covariance matrices is both a Riemannian space
as well as a homogeneous space, the shortest path geodesic between two
covariances on the manifold can be computed analytically. Statistics on the SPD
matrix manifold, such as the geometric mean of two matrices can be reduced to
solving the well-known Riccati equation. We show how the Ricatti-based solution
can be constrained to not only reduce the statistical differences between the
source and target domains, such as aligning second order covariances and
minimizing the maximum mean discrepancy, but also the underlying geometry of
the source and target domains using diffusions on the underlying source and
target manifolds. A key strength of our proposed approach is that it enables
integrating multiple sources of variation between source and target in a
unified way, by reducing the combined objective function to a nested set of
Ricatti equations where the solution can be represented by a cascaded series of
geometric mean computations. In addition to showing the theoretical optimality
of our solution, we present detailed experiments using standard transfer
learning testbeds from computer vision comparing our proposed algorithms to
past work in domain adaptation, showing improved results over a large variety
of previous methods.

通过利用统计流形的曲率黎曼几何，我们提出了一种新的域自适应框架，该框架可以整合标记源域和未标记目标域之间的几何和统计差异，从而实现源到目标的转移。