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 natural 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 corresponding latent variable model as a Riemannian manifold and we use the expectation of the metric under the Gaussian process prior to define interpolating paths and measure distance between latent points. We show how distances that respect the expected metric lead to more appropriate generation 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 natural 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 corresponding latent variable model as a Riemannian manifold and we use the expectation of the metric under the Gaussian process prior to define interpolating paths and measure distance between latent points. We show how distances that respect the expected metric lead to more appropriate generation 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 natural 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 corresponding latent variable model as a Riemannian manifold and we use the expectation of the metric under the Gaussian process prior to define interpolating paths and measure distance between latent points. We show how distances that respect the expected metric lead to more appropriate generation 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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