# Topologically-Constrained Latent Variable Models

Raquel Urtasun, University of Toronto
David J. Fleet
Andreas Geiger
Jovan Popović
Trevor J. Darrell
Neil D. Lawrence, University of Sheffield

in Proceedings of the International Conference in Machine Learning 25, pp 1080-1087

#### Abstract

In dimensionality reduction approaches, the data are typically embedded in a Euclidean latent space. However for some data sets this is inappropriate. For example, in human motion data we expect latent spaces that are cylindrical or a toroidal, that are poorly captured with a Euclidean space. In this paper, we present a range of approaches for embedding data in a non-Euclidean latent space. Our focus is the Gaussian Process latent variable model. In the context of human motion modeling this allows us to (a) learn models with interpretable latent directions enabling, for example, style/content separation, and (b) generalise beyond the data set enabling us to learn transitions between motion styles even though such transitions are not present in the data.

  @InProceedings{urtasun-topology08, title = {Topologically-Constrained Latent Variable Models}, author = {Raquel Urtasun and David J. Fleet and Andreas Geiger and Jovan Popović and Trevor J. Darrell and Neil D. Lawrence}, booktitle = {Proceedings of the International Conference in Machine Learning}, pages = {1080}, year = {2008}, editor = {Sam Roweis and Andrew McCallum}, volume = {25}, month = {00}, publisher = {Omnipress}, edit = {https://github.com/lawrennd//publications/edit/gh-pages/_posts/2008-01-01-urtasun-topology08.md}, url = {http://inverseprobability.com/publications/urtasun-topology08.html}, abstract = {In dimensionality reduction approaches, the data are typically embedded in a Euclidean latent space. However for some data sets this is inappropriate. For example, in human motion data we expect latent spaces that are cylindrical or a toroidal, that are poorly captured with a Euclidean space. In this paper, we present a range of approaches for embedding data in a non-Euclidean latent space. Our focus is the Gaussian Process latent variable model. In the context of human motion modeling this allows us to (a) learn models with interpretable latent directions enabling, for example, style/content separation, and (b) generalise beyond the data set enabling us to learn transitions between motion styles even though such transitions are not present in the data.}, crossref = {Roweis:icml08}, key = {Urtasun:topology08}, doi = {10.1145/1390156.1390292}, linkpdf = {ftp://ftp.dcs.shef.ac.uk/home/neil/topology.pdf}, group = {} }
 %T Topologically-Constrained Latent Variable Models %A Raquel Urtasun and David J. Fleet and Andreas Geiger and Jovan Popović and Trevor J. Darrell and Neil D. Lawrence %B %C Proceedings of the International Conference in Machine Learning %D %E Sam Roweis and Andrew McCallum %F urtasun-topology08 %I Omnipress %P 1080--1087 %R 10.1145/1390156.1390292 %U http://inverseprobability.com/publications/urtasun-topology08.html %V 25 %X In dimensionality reduction approaches, the data are typically embedded in a Euclidean latent space. However for some data sets this is inappropriate. For example, in human motion data we expect latent spaces that are cylindrical or a toroidal, that are poorly captured with a Euclidean space. In this paper, we present a range of approaches for embedding data in a non-Euclidean latent space. Our focus is the Gaussian Process latent variable model. In the context of human motion modeling this allows us to (a) learn models with interpretable latent directions enabling, for example, style/content separation, and (b) generalise beyond the data set enabling us to learn transitions between motion styles even though such transitions are not present in the data. 
 TY - CPAPER TI - Topologically-Constrained Latent Variable Models AU - Raquel Urtasun AU - David J. Fleet AU - Andreas Geiger AU - Jovan Popović AU - Trevor J. Darrell AU - Neil D. Lawrence BT - Proceedings of the International Conference in Machine Learning PY - 2008/01/01 DA - 2008/01/01 ED - Sam Roweis ED - Andrew McCallum ID - urtasun-topology08 PB - Omnipress SP - 1080 EP - 1087 DO - 10.1145/1390156.1390292 L1 - ftp://ftp.dcs.shef.ac.uk/home/neil/topology.pdf UR - http://inverseprobability.com/publications/urtasun-topology08.html AB - In dimensionality reduction approaches, the data are typically embedded in a Euclidean latent space. However for some data sets this is inappropriate. For example, in human motion data we expect latent spaces that are cylindrical or a toroidal, that are poorly captured with a Euclidean space. In this paper, we present a range of approaches for embedding data in a non-Euclidean latent space. Our focus is the Gaussian Process latent variable model. In the context of human motion modeling this allows us to (a) learn models with interpretable latent directions enabling, for example, style/content separation, and (b) generalise beyond the data set enabling us to learn transitions between motion styles even though such transitions are not present in the data. ER - 
 Urtasun, R., Fleet, D.J., Geiger, A., Popović, J., Darrell, T.J. & Lawrence, N.D.. (2008). Topologically-Constrained Latent Variable Models. Proceedings of the International Conference in Machine Learning 25:1080-1087