Kernel Learning Method on Riemannian Manifold with Geodesic Distance Preservation

Riemannian manifold recently has widely exploited in pattern recognition, data analysis, and machine learning. In this paper, a novel kernel learning method, based on preserving geodesic distance on Riemannian manifold, is proposed. In our approach, the features of data first are extracted by an covariance descriptor, and represented as covariance matrices (SPD matrices). Then an logarithm operation is used to transfer them to column vectors. Different from general doing, we introduce a parameterized Mahalanobis distance to define the distance between two column vectors, aiming to make it equal to the geodetic distance between two corresponding data on Riemannian manifold. Furthermore, an initialization kernel matrix is obtained from the learned Mahalanobis distance matrix by a linear transformation, and with it the Bregman optimization algorithm is applied to find the optimal kernel matrix, in which the distance in kernel space is equal to the geodetic distance on Riemannian manifold. We implement experiments on texture data sets to demonstrate the benefits of the proposed method.