Metrics for Probabilistic Geometry

Alessandra Tosi UPC, Søren Hauberg Technical University of Denmar, Alfredo Vellido UPC, Neil Lawrence
Proceedings of the 30th Conference on Uncertainty in Artificial Intelligence, PMLR R12:549-557, 2014.

Abstract

We investigate the geometrical structure of probabilistic generative dimensionality reduction models using the tools of Riemannian geometry. We explicitly define a distribution over the natu- ral metric given by the models. We provide the necessary algorithms to compute expected metric tensors where the distribution over mappings is given by a Gaussian process. We treat the corre- sponding latent variable model as a Riemannian manifold and we use the expectation of the met- ric under the Gaussian process prior to define in- terpolating paths and measure distance between latent points. We show how distances that respect the expected metric lead to more appropriate gen- eration of new data.

Cite this Paper

BibTeX
@InProceedings{pmlr-vR12-upc14a, title = {Metrics for Probabilistic Geometry}, author = {UPC, Alessandra Tosi and Denmar, S{\o}ren Hauberg Technical University of and UPC, Alfredo Vellido and Lawrence, Neil}, booktitle = {Proceedings of the 30th Conference on Uncertainty in Artificial Intelligence}, pages = {549--557}, year = {2014}, editor = {Zhang, Nevin L. and Tian, Jin}, volume = {R12}, series = {Proceedings of Machine Learning Research}, month = {23--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/r12/main/assets/upc14a/upc14a.pdf}, url = {https://proceedings.mlr.press/r12/upc14a.html}, abstract = {We investigate the geometrical structure of probabilistic generative dimensionality reduction models using the tools of Riemannian geometry. We explicitly define a distribution over the natu- ral metric given by the models. We provide the necessary algorithms to compute expected metric tensors where the distribution over mappings is given by a Gaussian process. We treat the corre- sponding latent variable model as a Riemannian manifold and we use the expectation of the met- ric under the Gaussian process prior to define in- terpolating paths and measure distance between latent points. We show how distances that respect the expected metric lead to more appropriate gen- eration of new data.}, note = {Reissued by PMLR on 04 October 2026.} }
Endnote
%0 Conference Paper %T Metrics for Probabilistic Geometry %A Alessandra Tosi UPC %A Søren Hauberg Technical University of Denmar %A Alfredo Vellido UPC %A Neil Lawrence %B Proceedings of the 30th Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2014 %E Nevin L. Zhang %E Jin Tian %F pmlr-vR12-upc14a %I PMLR %P 549--557 %U https://proceedings.mlr.press/r12/upc14a.html %V R12 %X We investigate the geometrical structure of probabilistic generative dimensionality reduction models using the tools of Riemannian geometry. We explicitly define a distribution over the natu- ral metric given by the models. We provide the necessary algorithms to compute expected metric tensors where the distribution over mappings is given by a Gaussian process. We treat the corre- sponding latent variable model as a Riemannian manifold and we use the expectation of the met- ric under the Gaussian process prior to define in- terpolating paths and measure distance between latent points. We show how distances that respect the expected metric lead to more appropriate gen- eration of new data. %Z Reissued by PMLR on 04 October 2026.
APA
UPC, A.T., Denmar, S.H.T.U.o., UPC, A.V. & Lawrence, N.. (2014). Metrics for Probabilistic Geometry. Proceedings of the 30th Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research R12:549-557 Available from https://proceedings.mlr.press/r12/upc14a.html. Reissued by PMLR on 04 October 2026.

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