Geodesic Distance Function Learning via Heat Flow on Vector Fields
This addresses the challenge of preserving distances in machine learning for non-Euclidean data, though it appears incremental as it builds on existing manifold learning concepts.
The paper tackles the problem of learning distance functions on non-Euclidean manifolds by proposing a method that learns the gradient field directly on the manifold without embedding, using heat flow on vector fields, and demonstrates effectiveness on synthetic and real data.
Learning a distance function or metric on a given data manifold is of great importance in machine learning and pattern recognition. Many of the previous works first embed the manifold to Euclidean space and then learn the distance function. However, such a scheme might not faithfully preserve the distance function if the original manifold is not Euclidean. Note that the distance function on a manifold can always be well-defined. In this paper, we propose to learn the distance function directly on the manifold without embedding. We first provide a theoretical characterization of the distance function by its gradient field. Based on our theoretical analysis, we propose to first learn the gradient field of the distance function and then learn the distance function itself. Specifically, we set the gradient field of a local distance function as an initial vector field. Then we transport it to the whole manifold via heat flow on vector fields. Finally, the geodesic distance function can be obtained by requiring its gradient field to be close to the normalized vector field. Experimental results on both synthetic and real data demonstrate the effectiveness of our proposed algorithm.