Curvature flow learning: algorithm and analysis

In order to describe the nonlinear distribution of the image dataset, researchers propose a manifold assumption, called manifold learning (MAL). The geometry information based on a manifold is measured by the Riemannian metric, such as the geodesic distance. Thus, owing to mining the intrinsic geometric structure of the dataset, we need to learn the real Riemannian metric of the embedded manifold. By the Taylor expansion equation of the Riemannian metric, it clearly indicates that the Riemannian metric is relative to the Riemannian curvature. Based on it, we propose a new algorithm to learn the Riemannian metric by adding the curvature information into metric learning. By optimizing the objective function, we obtain a set of iterative equations. We call this model curvature flow. By employing this curvature flow, we obtain a Mahalanobis metric that approaches the Riemannian metric infinitely and can well uncover the intrinsic structure of the embedded manifold. In theory, we analyze several properties of our proposed method, e.g., the boundedness of the metric and the convergence of curvature flow. To show the effectiveness of our proposed method, we compare our algorithm with several traditional MAL algorithms on three real world datasets. The corresponding results indicate that our proposed method outperforms the other algorithms.