Sketching the Support of a Probability Measure

Joachim Giesen, Soeren Laue, Lars Kuehne
Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics, PMLR 33:257-265, 2014.

Abstract

We want to sketch the support of a probability measure on Euclidean space from samples that have been drawn from the measure. This problem is closely related to certain manifold learning problems, where one assumes that the sample points are drawn from a manifold that is embedded in Euclidean space. Here we propose to sketch the support of the probability measure (that does not need to be a manifold) by some gradient flow complex, or more precisely by its Hasse diagram. The gradient flow is defined with respect to the distance function to the sample points. We prove that a gradient flow complex (that can be computed) is homotopy equivalent to the support of the measure for sufficiently dense samplings, and demonstrate the feasibility of our approach on real world data sets.

Cite this Paper

BibTeX
@InProceedings{pmlr-v33-giesen14, title = {{Sketching the Support of a Probability Measure}}, author = {Giesen, Joachim and Laue, Soeren and Kuehne, Lars}, booktitle = {Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics}, pages = {257--265}, year = {2014}, editor = {Kaski, Samuel and Corander, Jukka}, volume = {33}, series = {Proceedings of Machine Learning Research}, address = {Reykjavik, Iceland}, month = {22--25 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v33/giesen14.pdf}, url = {https://proceedings.mlr.press/v33/giesen14.html}, abstract = {We want to sketch the support of a probability measure on Euclidean space from samples that have been drawn from the measure. This problem is closely related to certain manifold learning problems, where one assumes that the sample points are drawn from a manifold that is embedded in Euclidean space. Here we propose to sketch the support of the probability measure (that does not need to be a manifold) by some gradient flow complex, or more precisely by its Hasse diagram. The gradient flow is defined with respect to the distance function to the sample points. We prove that a gradient flow complex (that can be computed) is homotopy equivalent to the support of the measure for sufficiently dense samplings, and demonstrate the feasibility of our approach on real world data sets.} }
Endnote
%0 Conference Paper %T Sketching the Support of a Probability Measure %A Joachim Giesen %A Soeren Laue %A Lars Kuehne %B Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2014 %E Samuel Kaski %E Jukka Corander %F pmlr-v33-giesen14 %I PMLR %P 257--265 %U https://proceedings.mlr.press/v33/giesen14.html %V 33 %X We want to sketch the support of a probability measure on Euclidean space from samples that have been drawn from the measure. This problem is closely related to certain manifold learning problems, where one assumes that the sample points are drawn from a manifold that is embedded in Euclidean space. Here we propose to sketch the support of the probability measure (that does not need to be a manifold) by some gradient flow complex, or more precisely by its Hasse diagram. The gradient flow is defined with respect to the distance function to the sample points. We prove that a gradient flow complex (that can be computed) is homotopy equivalent to the support of the measure for sufficiently dense samplings, and demonstrate the feasibility of our approach on real world data sets.
RIS
TY - CPAPER TI - Sketching the Support of a Probability Measure AU - Joachim Giesen AU - Soeren Laue AU - Lars Kuehne BT - Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics DA - 2014/04/02 ED - Samuel Kaski ED - Jukka Corander ID - pmlr-v33-giesen14 PB - PMLR DP - Proceedings of Machine Learning Research VL - 33 SP - 257 EP - 265 L1 - http://proceedings.mlr.press/v33/giesen14.pdf UR - https://proceedings.mlr.press/v33/giesen14.html AB - We want to sketch the support of a probability measure on Euclidean space from samples that have been drawn from the measure. This problem is closely related to certain manifold learning problems, where one assumes that the sample points are drawn from a manifold that is embedded in Euclidean space. Here we propose to sketch the support of the probability measure (that does not need to be a manifold) by some gradient flow complex, or more precisely by its Hasse diagram. The gradient flow is defined with respect to the distance function to the sample points. We prove that a gradient flow complex (that can be computed) is homotopy equivalent to the support of the measure for sufficiently dense samplings, and demonstrate the feasibility of our approach on real world data sets. ER -
APA
Giesen, J., Laue, S. & Kuehne, L.. (2014). Sketching the Support of a Probability Measure. Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 33:257-265 Available from https://proceedings.mlr.press/v33/giesen14.html.

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