# Metrics for Probabilistic Geometries

Alessandra Tosi, University of Oxford
Søren Hauberg
Alfredo Vellido
Neil D. Lawrence, University of Sheffield

in Uncertainty in Artificial Intelligence 30, pp 800-808

#### 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.

  @InProceedings{tosi-metrics14, title = {Metrics for Probabilistic Geometries}, author = {Alessandra Tosi and Søren Hauberg and Alfredo Vellido and Neil D. Lawrence}, booktitle = {Uncertainty in Artificial Intelligence}, pages = {800}, year = {2014}, editor = {Nevin Zhang and Jin Tian}, volume = {30}, month = {00}, publisher = {AUAI Press}, edit = {https://github.com/lawrennd//publications/edit/gh-pages/_posts/2014-07-23-tosi-metrics14.md}, 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.}, crossref = {Zhang:uai14}, key = {Tosi:metrics14}, linkpdf = {http://auai.org/uai2014/proceedings/individuals/171.pdf}, OPTgroup = {} }
 %T Metrics for Probabilistic Geometries %A Alessandra Tosi and Søren Hauberg and Alfredo Vellido and Neil D. Lawrence %B %C Uncertainty in Artificial Intelligence %D %E Nevin Zhang and Jin Tian %F tosi-metrics14 %I AUAI Press %P 800--808 %R %U http://inverseprobability.com/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. 
 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 PY - 2014/07/23 DA - 2014/07/23 ED - Nevin Zhang ED - Jin Tian ID - tosi-metrics14 PB - AUAI Press SP - 800 EP - 808 L1 - http://auai.org/uai2014/proceedings/individuals/171.pdf 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 - 
 Tosi, A., Hauberg, S., Vellido, A. & Lawrence, N.D.. (2014). Metrics for Probabilistic Geometries. Uncertainty in Artificial Intelligence 30:800-808