TY - GEN

T1 - Geodesic distance function learning via heat flow on vector fields

AU - Lin, Binbin

AU - Yangt, Ji

AU - He, Xiaofei

AU - Ye, Jieping

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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. 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.

AB - 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. 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.

UR - http://www.scopus.com/inward/record.url?scp=84919924102&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84919924102&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84919924102

T3 - 31st International Conference on Machine Learning, ICML 2014

SP - 1346

EP - 1354

BT - 31st International Conference on Machine Learning, ICML 2014

PB - International Machine Learning Society (IMLS)

T2 - 31st International Conference on Machine Learning, ICML 2014

Y2 - 21 June 2014 through 26 June 2014

ER -