Geodesic Distance Function Learning via Heat Flow on Vector Fields

Binbin Lin, Ji Yang, Xiaofei He, Jieping Ye
Proceedings of the 31st International Conference on Machine Learning, PMLR 32(2):145-153, 2014.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v32-linb14, title = {Geodesic Distance Function Learning via Heat Flow on Vector Fields}, author = {Lin, Binbin and Yang, Ji and He, Xiaofei and Ye, Jieping}, booktitle = {Proceedings of the 31st International Conference on Machine Learning}, pages = {145--153}, year = {2014}, editor = {Xing, Eric P. and Jebara, Tony}, volume = {32}, number = {2}, series = {Proceedings of Machine Learning Research}, address = {Bejing, China}, month = {22--24 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v32/linb14.pdf}, url = {https://proceedings.mlr.press/v32/linb14.html}, abstract = {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.} }
Endnote
%0 Conference Paper %T Geodesic Distance Function Learning via Heat Flow on Vector Fields %A Binbin Lin %A Ji Yang %A Xiaofei He %A Jieping Ye %B Proceedings of the 31st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2014 %E Eric P. Xing %E Tony Jebara %F pmlr-v32-linb14 %I PMLR %P 145--153 %U https://proceedings.mlr.press/v32/linb14.html %V 32 %N 2 %X 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.
RIS
TY - CPAPER TI - Geodesic Distance Function Learning via Heat Flow on Vector Fields AU - Binbin Lin AU - Ji Yang AU - Xiaofei He AU - Jieping Ye BT - Proceedings of the 31st International Conference on Machine Learning DA - 2014/06/18 ED - Eric P. Xing ED - Tony Jebara ID - pmlr-v32-linb14 PB - PMLR DP - Proceedings of Machine Learning Research VL - 32 IS - 2 SP - 145 EP - 153 L1 - http://proceedings.mlr.press/v32/linb14.pdf UR - https://proceedings.mlr.press/v32/linb14.html 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. ER -
APA
Lin, B., Yang, J., He, X. & Ye, J.. (2014). Geodesic Distance Function Learning via Heat Flow on Vector Fields. Proceedings of the 31st International Conference on Machine Learning, in Proceedings of Machine Learning Research 32(2):145-153 Available from https://proceedings.mlr.press/v32/linb14.html.

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