Multi-Manifold Modeling in Non-Euclidean spaces

Xu Wang, Konstantinos Slavakis, Gilad Lerman
Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics, PMLR 38:1023-1032, 2015.

Abstract

This paper advocates a novel framework for segmenting a dataset on a Riemannian manifold M into clusters lying around low-dimensional submanifolds of M. Important examples of M, for which the proposed algorithm is computationally efficient, include the sphere, the set of positive definite matrices, and the Grassmannian. The proposed algorithm constructs a data-affinity matrix by thoroughly exploiting the intrinsic geometry and then applies spectral clustering. Local geometry is encoded by sparse coding and directional information of local tangent spaces and geodesics, which is important in resolving intersecting clusters and establishing the theoretical guarantees for a simplified variant of the algorithm. To avoid complication, these guarantees assume that the underlying submanifolds are geodesic. Extensive validation on synthetic and real data demonstrates the resiliency of the proposed method against deviations from the theoretical (geodesic) model as well as its superior performance over state-of-the-art techniques.

Cite this Paper


BibTeX
@InProceedings{pmlr-v38-wang15b, title = {{Multi-Manifold Modeling in Non-Euclidean spaces}}, author = {Xu Wang and Konstantinos Slavakis and Gilad Lerman}, booktitle = {Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics}, pages = {1023--1032}, year = {2015}, editor = {Guy Lebanon and S. V. N. Vishwanathan}, volume = {38}, series = {Proceedings of Machine Learning Research}, address = {San Diego, California, USA}, month = {09--12 May}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v38/wang15b.pdf}, url = { http://proceedings.mlr.press/v38/wang15b.html }, abstract = {This paper advocates a novel framework for segmenting a dataset on a Riemannian manifold M into clusters lying around low-dimensional submanifolds of M. Important examples of M, for which the proposed algorithm is computationally efficient, include the sphere, the set of positive definite matrices, and the Grassmannian. The proposed algorithm constructs a data-affinity matrix by thoroughly exploiting the intrinsic geometry and then applies spectral clustering. Local geometry is encoded by sparse coding and directional information of local tangent spaces and geodesics, which is important in resolving intersecting clusters and establishing the theoretical guarantees for a simplified variant of the algorithm. To avoid complication, these guarantees assume that the underlying submanifolds are geodesic. Extensive validation on synthetic and real data demonstrates the resiliency of the proposed method against deviations from the theoretical (geodesic) model as well as its superior performance over state-of-the-art techniques.} }
Endnote
%0 Conference Paper %T Multi-Manifold Modeling in Non-Euclidean spaces %A Xu Wang %A Konstantinos Slavakis %A Gilad Lerman %B Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2015 %E Guy Lebanon %E S. V. N. Vishwanathan %F pmlr-v38-wang15b %I PMLR %P 1023--1032 %U http://proceedings.mlr.press/v38/wang15b.html %V 38 %X This paper advocates a novel framework for segmenting a dataset on a Riemannian manifold M into clusters lying around low-dimensional submanifolds of M. Important examples of M, for which the proposed algorithm is computationally efficient, include the sphere, the set of positive definite matrices, and the Grassmannian. The proposed algorithm constructs a data-affinity matrix by thoroughly exploiting the intrinsic geometry and then applies spectral clustering. Local geometry is encoded by sparse coding and directional information of local tangent spaces and geodesics, which is important in resolving intersecting clusters and establishing the theoretical guarantees for a simplified variant of the algorithm. To avoid complication, these guarantees assume that the underlying submanifolds are geodesic. Extensive validation on synthetic and real data demonstrates the resiliency of the proposed method against deviations from the theoretical (geodesic) model as well as its superior performance over state-of-the-art techniques.
RIS
TY - CPAPER TI - Multi-Manifold Modeling in Non-Euclidean spaces AU - Xu Wang AU - Konstantinos Slavakis AU - Gilad Lerman BT - Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics DA - 2015/02/21 ED - Guy Lebanon ED - S. V. N. Vishwanathan ID - pmlr-v38-wang15b PB - PMLR DP - Proceedings of Machine Learning Research VL - 38 SP - 1023 EP - 1032 L1 - http://proceedings.mlr.press/v38/wang15b.pdf UR - http://proceedings.mlr.press/v38/wang15b.html AB - This paper advocates a novel framework for segmenting a dataset on a Riemannian manifold M into clusters lying around low-dimensional submanifolds of M. Important examples of M, for which the proposed algorithm is computationally efficient, include the sphere, the set of positive definite matrices, and the Grassmannian. The proposed algorithm constructs a data-affinity matrix by thoroughly exploiting the intrinsic geometry and then applies spectral clustering. Local geometry is encoded by sparse coding and directional information of local tangent spaces and geodesics, which is important in resolving intersecting clusters and establishing the theoretical guarantees for a simplified variant of the algorithm. To avoid complication, these guarantees assume that the underlying submanifolds are geodesic. Extensive validation on synthetic and real data demonstrates the resiliency of the proposed method against deviations from the theoretical (geodesic) model as well as its superior performance over state-of-the-art techniques. ER -
APA
Wang, X., Slavakis, K. & Lerman, G.. (2015). Multi-Manifold Modeling in Non-Euclidean spaces. Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 38:1023-1032 Available from http://proceedings.mlr.press/v38/wang15b.html .

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