Metrics for Probabilistic Geometries

Alessandra Tosi, Søren Hauberg, Alfredo Vellido, Neil D. Lawrence
,  30:800-808, 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-v-tosi-metrics14, title = {Metrics for Probabilistic Geometries}, author = {Alessandra Tosi and Søren Hauberg and Alfredo Vellido and Neil D. Lawrence}, pages = {800--808}, year = {}, editor = {}, volume = {30}, url = {http://inverseprobability.com/publications/tosi-metrics14.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.} }
Endnote
%0 Conference Paper %T Metrics for Probabilistic Geometries %A Alessandra Tosi %A Søren Hauberg %A Alfredo Vellido %A Neil D. Lawrence %B %C Proceedings of Machine Learning Research %D %E %F pmlr-v-tosi-metrics14 %I PMLR %J Proceedings of Machine Learning Research %P 800--808 %U http://inverseprobability.com %V %W PMLR %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.
RIS
TY - CPAPER TI - Metrics for Probabilistic Geometries AU - Alessandra Tosi AU - Søren Hauberg AU - Alfredo Vellido AU - Neil D. Lawrence BT - PY - DA - ED - ID - pmlr-v-tosi-metrics14 PB - PMLR SP - 800 DP - PMLR EP - 808 L1 - UR - http://inverseprobability.com/publications/tosi-metrics14.html AB - 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. ER -
APA
Tosi, A., Hauberg, S., Vellido, A. & Lawrence, N.D.. (). Metrics for Probabilistic Geometries. , in PMLR :800-808

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