Symmetric Positive Definite (SPD) matrices have been widely used for data
representation in many visual recognition tasks. The success mainly attributes
to learning discriminative SPD matrices with encoding the Riemannian geometry
of the underlying SPD manifold. In this paper, we propose a geometry-aware SPD
similarity learning (SPDSL) framework to learn discriminative SPD features by
directly pursuing manifold-manifold transformation matrix of column full-rank.
Specifically, by exploiting the Riemannian geometry of the manifold of
fixed-rank Positive Semidefinite (PSD) matrices, we present a new solution to
reduce optimizing over the space of column full-rank transformation matrices to
optimizing on the PSD manifold which has a well-established Riemannian
structure. Under this solution, we exploit a new supervised SPD similarity
learning technique to learn the transformation by regressing the similarities
of selected SPD data pairs to their ground-truth similarities on the target SPD
manifold. To optimize the proposed objective function, we further derive an
algorithm on the PSD manifold. Evaluations on three visual classification tasks
show the advantages of the proposed approach over the existing SPD-based
discriminant learning methods.

本文提出了一种利用黎曼几何学习固定秩半正定矩阵流形的几何感知 SPD 相似性学习框架，通过在 PSD 流形中优化来学习具有判别性的 SPD 特征，优于现有的基于 SPD 的判别学习方法。