Positive Definite Matrices: Data Representation and Applications to Computer Vision

Numerous applications in computer vision and machine learning rely on representations of data that are compact, discriminative, and robust while satisfying several desirable invariances. One such recently successful representation is offered by symmetric positive definite (SPD) matrices. However, the modeling power of SPD matrices comes at a price: rather than a flat Euclidean view, SPD matrices are more naturally viewed through curved geometry (Riemannian or otherwise) which often complicates matters. We focus on models and algorithms that rely on the geometry of SPD matrices, and make our discussion concrete by casting it in terms of covariance descriptors for images. We summarize various commonly used distance metrics on SPD matrices, before highlighting formulations and algorithms for solving sparse coding and dictionary learning problems involving SPD data. Through empirical results, we showcase the benefits of mathematical models that exploit the curved geometry of SPD data across a diverse set of computer vision applications.