Recently, covariance descriptors have received much attention as powerful representations of set of points. In this research, we present a new metric learning algorithm for covariance descriptors based on the Dykstra algorithm, in which the current solution is projected onto a half-space at each iteration, and runs at O(n^3) time. We empirically demonstrate that randomizing the order of half-spaces in our Dykstra-based algorithm significantly accelerates the convergence to the optimal solution. Furthermore, we show that our approach yields promising experimental results on pattern recognition tasks.
To the best of our knowledge, (2015) @cite_22 were the first to introduce the supervised metric learning approach for covariance descriptors. They vectorized the matrix logarithms of the covariance descriptors to apply existing metric learning methods to the vectorizations of matrices. The dimensionality of the vectorizations is @math when the size of the covariance matrices are @math . Thus, the size of the Mahalanobis matrix is @math , which is computationally prohibitive when @math is large.
@cite_22
1928707779
Over the past few years, symmetric positive definite (SPD) matrices have been receiving considerable attention from computer vision community. Though various distance measures have been proposed in the past for comparing SPD matrices, the two most widely-used measures are affine-invariant distance and log-Euclidean distance. This is because these two measures are true geodesic distances induced by Riemannian geometry. In this work, we focus on the log-Euclidean Riemannian geometry and propose a data-driven approach for learning Riemannian metrics geodesic distances for SPD matrices. We show that the geodesic distance learned using the proposed approach performs better than various existing distance measures when evaluated on face matching and clustering tasks.