On A Nonlinear Generalization of Sparse Coding and Dictionary Learning

Jeffrey Ho, Yuchen Xie, Baba Vemuri
Proceedings of the 30th International Conference on Machine Learning, PMLR 28(3):1480-1488, 2013.

Abstract

Existing dictionary learning algorithms are based on the assumption that the data are vectors in an Euclidean vector space, and the dictionary is learned from the training data using the vector space structure and its Euclidean metric. However, in many applications, features and data often originated from a Riemannian manifold that does not support a global linear (vector space) structure. Furthermore, the extrinsic viewpoint of existing dictionary learning algorithms becomes inappropriate for modeling and incorporating the intrinsic geometry of the manifold that is potentially important and critical to the application. This paper proposes a novel framework for sparse coding and dictionary learning for data on a Riemannian manifold, and it shows that the existing sparse coding and dictionary learning methods can be considered as special (Euclidean) cases of the more general framework proposed here. We show that both the dictionary and sparse coding can be effectively computed for several important classes of Riemannian manifolds, and we validate the proposed method using two well-known classification problems in computer vision and medical imaging analysis.

Cite this Paper

BibTeX
@InProceedings{pmlr-v28-ho13a, title = {On A Nonlinear Generalization of Sparse Coding and Dictionary Learning}, author = {Ho, Jeffrey and Xie, Yuchen and Vemuri, Baba}, booktitle = {Proceedings of the 30th International Conference on Machine Learning}, pages = {1480--1488}, year = {2013}, editor = {Dasgupta, Sanjoy and McAllester, David}, volume = {28}, number = {3}, series = {Proceedings of Machine Learning Research}, address = {Atlanta, Georgia, USA}, month = {17--19 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v28/ho13a.pdf}, url = {https://proceedings.mlr.press/v28/ho13a.html}, abstract = {Existing dictionary learning algorithms are based on the assumption that the data are vectors in an Euclidean vector space, and the dictionary is learned from the training data using the vector space structure and its Euclidean metric. However, in many applications, features and data often originated from a Riemannian manifold that does not support a global linear (vector space) structure. Furthermore, the extrinsic viewpoint of existing dictionary learning algorithms becomes inappropriate for modeling and incorporating the intrinsic geometry of the manifold that is potentially important and critical to the application. This paper proposes a novel framework for sparse coding and dictionary learning for data on a Riemannian manifold, and it shows that the existing sparse coding and dictionary learning methods can be considered as special (Euclidean) cases of the more general framework proposed here. We show that both the dictionary and sparse coding can be effectively computed for several important classes of Riemannian manifolds, and we validate the proposed method using two well-known classification problems in computer vision and medical imaging analysis.} }
Endnote
%0 Conference Paper %T On A Nonlinear Generalization of Sparse Coding and Dictionary Learning %A Jeffrey Ho %A Yuchen Xie %A Baba Vemuri %B Proceedings of the 30th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2013 %E Sanjoy Dasgupta %E David McAllester %F pmlr-v28-ho13a %I PMLR %P 1480--1488 %U https://proceedings.mlr.press/v28/ho13a.html %V 28 %N 3 %X Existing dictionary learning algorithms are based on the assumption that the data are vectors in an Euclidean vector space, and the dictionary is learned from the training data using the vector space structure and its Euclidean metric. However, in many applications, features and data often originated from a Riemannian manifold that does not support a global linear (vector space) structure. Furthermore, the extrinsic viewpoint of existing dictionary learning algorithms becomes inappropriate for modeling and incorporating the intrinsic geometry of the manifold that is potentially important and critical to the application. This paper proposes a novel framework for sparse coding and dictionary learning for data on a Riemannian manifold, and it shows that the existing sparse coding and dictionary learning methods can be considered as special (Euclidean) cases of the more general framework proposed here. We show that both the dictionary and sparse coding can be effectively computed for several important classes of Riemannian manifolds, and we validate the proposed method using two well-known classification problems in computer vision and medical imaging analysis.
RIS
TY - CPAPER TI - On A Nonlinear Generalization of Sparse Coding and Dictionary Learning AU - Jeffrey Ho AU - Yuchen Xie AU - Baba Vemuri BT - Proceedings of the 30th International Conference on Machine Learning DA - 2013/05/26 ED - Sanjoy Dasgupta ED - David McAllester ID - pmlr-v28-ho13a PB - PMLR DP - Proceedings of Machine Learning Research VL - 28 IS - 3 SP - 1480 EP - 1488 L1 - http://proceedings.mlr.press/v28/ho13a.pdf UR - https://proceedings.mlr.press/v28/ho13a.html AB - Existing dictionary learning algorithms are based on the assumption that the data are vectors in an Euclidean vector space, and the dictionary is learned from the training data using the vector space structure and its Euclidean metric. However, in many applications, features and data often originated from a Riemannian manifold that does not support a global linear (vector space) structure. Furthermore, the extrinsic viewpoint of existing dictionary learning algorithms becomes inappropriate for modeling and incorporating the intrinsic geometry of the manifold that is potentially important and critical to the application. This paper proposes a novel framework for sparse coding and dictionary learning for data on a Riemannian manifold, and it shows that the existing sparse coding and dictionary learning methods can be considered as special (Euclidean) cases of the more general framework proposed here. We show that both the dictionary and sparse coding can be effectively computed for several important classes of Riemannian manifolds, and we validate the proposed method using two well-known classification problems in computer vision and medical imaging analysis. ER -
APA
Ho, J., Xie, Y. & Vemuri, B.. (2013). On A Nonlinear Generalization of Sparse Coding and Dictionary Learning. Proceedings of the 30th International Conference on Machine Learning, in Proceedings of Machine Learning Research 28(3):1480-1488 Available from https://proceedings.mlr.press/v28/ho13a.html.

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