Data encoded as symmetric positive definite (SPD) matrices frequently arise in many areas of computer vision and machine learning. While these matrices form an open subset of the Euclidean space of symmetric matrices, viewing them through the lens of non-Euclidean Riemannian geometry often turns out to be better suited in capturing several desirable data properties. However, formulating classical machine learning algorithms within such a geometry is often non-trivial and computationally expensive. Inspired by the great success of dictionary learning and sparse coding for vector-valued data, our goal in this paper is to represent data in the form of SPD matrices as sparse conic combinations of SPD atoms from a learned dictionary via a Riemannian geometric approach. To that end, we formulate a novel Riemannian optimization objective for dictionary learning and sparse coding in which the representation loss is characterized via the affine invariant Riemannian metric. We also present a computationally simple algorithm for optimizing our model. Experiments on several computer vision datasets demonstrate superior classification and retrieval performance using our approach when compared to sparse coding via alternative non-Riemannian formulations.

本研究提出一种新的Riemannian几何方法来通过学习字典中的SPD原子的稀疏锥组合，将数据表示为SPD矩阵。通过比较与其他非Riemannian公式的稀疏编码的分类和检索性能，我们的实验表明了这种方法的卓越性能。

正定矩阵的黎曼字典学习和稀疏编码