Metrics for Probabilistic Geometries

Alessandra Tosi; Søren Hauberg; Alfredo Vellido; Neil D. Lawrence

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Metrics for Probabilistic Geometries

Alessandra Tosi, Søren Hauberg, Alfredo Vellido, Neil D. Lawrence

Uncertainty in Artificial Intelligence, AUAI Press 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.

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BibTeX


@InProceedings{Tosi-metrics14,
  title = 	 {Metrics for Probabilistic Geometries},
  author = 	 {Tosi, Alessandra and Hauberg, Søren and Vellido, Alfredo and Lawrence, Neil D.},
  booktitle = 	 {Uncertainty in Artificial Intelligence},
  pages = 	 {800--808},
  year = 	 {2014},
  editor = 	 {Zhang, Nevin and Tian, Jin},
  volume = 	 {30},
  publisher =    {AUAI Press},
  pdf = 	 {http://auai.org/uai2014/proceedings/individuals/171.pdf},
  url = 	 {/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 Uncertainty in Artificial Intelligence
%D 2014
%E Nevin Zhang
%E Jin Tian	
%F Tosi-metrics14
%I AUAI Press
%P 800--808
%U /publications/tosi-metrics14.html
%V 30
%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  - Uncertainty in Artificial Intelligence
DA  - 2014/07/23
ED  - Nevin Zhang
ED  - Jin Tian	
ID  - Tosi-metrics14
PB  - AUAI Press
VL  - 30
SP  - 800
EP  - 808
L1  - http://auai.org/uai2014/proceedings/individuals/171.pdf
UR  - /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.. (2014). Metrics for Probabilistic Geometries. Uncertainty in Artificial Intelligence 30:800-808 Available from /publications/tosi-metrics14.html.