Dimensionality Reduction on SPD Manifolds: The Emergence of Geometry-Aware Methods
This addresses a computational bottleneck for researchers and practitioners using SPD matrices in visual recognition, offering a more efficient approach with improved performance.
The paper tackles the high computational cost of processing high-dimensional Symmetric Positive Definite (SPD) matrices in visual recognition by introducing dimensionality reduction algorithms that project onto a lower-dimensional SPD manifold, resulting in significant accuracy gains over state-of-the-art methods on classification tasks.
Representing images and videos with Symmetric Positive Definite (SPD) matrices, and considering the Riemannian geometry of the resulting space, has been shown to yield high discriminative power in many visual recognition tasks. Unfortunately, computation on the Riemannian manifold of SPD matrices -especially of high-dimensional ones- comes at a high cost that limits the applicability of existing techniques. In this paper, we introduce algorithms able to handle high-dimensional SPD matrices by constructing a lower-dimensional SPD manifold. To this end, we propose to model the mapping from the high-dimensional SPD manifold to the low-dimensional one with an orthonormal projection. This lets us formulate dimensionality reduction as the problem of finding a projection that yields a low-dimensional manifold either with maximum discriminative power in the supervised scenario, or with maximum variance of the data in the unsupervised one. We show that learning can be expressed as an optimization problem on a Grassmann manifold and discuss fast solutions for special cases. Our evaluation on several classification tasks evidences that our approach leads to a significant accuracy gain over state-of-the-art methods.