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DP-GP-LVM A Bayesian Non-Parametric Model for Learning Multivariate Dependency Structures Andrew R. Lawrence 1 Carl Henrik Ek 2 Neill D. F. Campbell 1 1 University of Bath, UK 2 University of Bristol, UK Proceedings of the 36 th International

  1. DP-GP-LVM A Bayesian Non-Parametric Model for Learning Multivariate Dependency Structures Andrew R. Lawrence 1 Carl Henrik Ek 2 Neill D. F. Campbell 1 1 University of Bath, UK 2 University of Bristol, UK Proceedings of the 36 th International Conference on Machine Learning Long Beach, California, USA 12 June 2019 Lawrence, Ek, Campbell DP-GP-LVM ICML 2019 1 / 11

  2. Motivation Lawrence, Ek, Campbell DP-GP-LVM ICML 2019 2 / 11

  3. Motivation Lawrence, Ek, Campbell DP-GP-LVM ICML 2019 2 / 11

  4. Motivation Lawrence, Ek, Campbell DP-GP-LVM ICML 2019 3 / 11

  5. Motivation Lawrence, Ek, Campbell DP-GP-LVM ICML 2019 3 / 11

  6. Background Factorized Gaussian Process Latent Variable Model (GP-LVM) [1, 2] X (1) X (1 , 2) X (2) F (1) F (2) Y (1) Y (2) Lawrence, Ek, Campbell DP-GP-LVM ICML 2019 4 / 11

  7. Background Factorized Gaussian Process Latent Variable Model (GP-LVM) [1, 2] X (1) Y (1) X (1) X (1 , 2) X (2) F (1) X (1 , 3) X (1 , 2) F (1) F (2) X (1 , 2 , 3) X (3) X (2) F (3) X (2 , 3) F (2) Y (1) Y (2) Y (3) Y (2) Lawrence, Ek, Campbell DP-GP-LVM ICML 2019 4 / 11

  8. Background Manifold Relevance Determination (MRD) [3] X ... F (1) F (2) F ( T ) ... Y (1) Y (2) Y ( T ) Lawrence, Ek, Campbell DP-GP-LVM ICML 2019 5 / 11

  9. Background Manifold Relevance Determination (MRD) [3] x n X f n,t θ t ... F (1) F (2) F ( T ) t ∈ [1 , T ] ... y n,d Y (1) Y (2) z d Y ( T ) d ∈ [1 , D ] n ∈ [1 , N ] Lawrence, Ek, Campbell DP-GP-LVM ICML 2019 5 / 11

  10. Background Fully Independent MRD (fi-MRD) [3] X ... f 1 f 2 f D ... y 1 y 2 y D Lawrence, Ek, Campbell DP-GP-LVM ICML 2019 6 / 11

  11. Background Fully Independent MRD (fi-MRD) [3] x n X f n,d θ d ... f 1 f 2 f D ... y n,d z d y 1 y 2 y D d ∈ [1 , D ] n ∈ [1 , N ] Lawrence, Ek, Campbell DP-GP-LVM ICML 2019 6 / 11

  12. DP-GP-LVM X ... F (1) F (2) F ( T ) ... y 1 y 2 y 3 y D Lawrence, Ek, Campbell DP-GP-LVM ICML 2019 7 / 11

  13. DP-GP-LVM x n X f n,t θ t ... t ∈ [1 , T ] DP F (1) F (2) F ( T ) ... y n,d y 1 y 2 y 3 y D z d d ∈ [1 , D ] n ∈ [1 , N ] Lawrence, Ek, Campbell DP-GP-LVM ICML 2019 7 / 11

  14. Experiments PoseTrack [4] – Two People Lawrence, Ek, Campbell DP-GP-LVM ICML 2019 8 / 11

  15. Experiments PoseTrack [4] – Two People Lawrence, Ek, Campbell DP-GP-LVM ICML 2019 8 / 11

  16. Experiments PoseTrack [4] – Two People Lawrence, Ek, Campbell DP-GP-LVM ICML 2019 8 / 11

  17. Experiments PoseTrack [4] – Two People Lawrence, Ek, Campbell DP-GP-LVM ICML 2019 8 / 11

  18. Experiments PoseTrack [4] – Missing Data ˜ N Missing (%) 10 10 10 20 20 20 ˜ D Missing (%) 10 20 30 10 20 30 PoseTrack 2 Person BGP-LVM [5] − 50 . 02 ± 12 . 32 − 93 . 39 ± 18 . 84 − 157 . 67 ± 32 . 20 − 61 . 64 ± 13 . 95 − 134 . 08 ± 22 . 73 − 191 . 90 ± 33 . 57 fi-MRD [3] − 40 . 62 ± 10 . 64 − 109 . 73 ± 17 . 36 − 138 . 67 ± 14 . 84 − 54 . 92 ± 8 . 51 − 149 . 36 ± 11 . 36 − 248 . 89 ± 36 . 79 DP-GP-LVM − 18 . 11 ± 0 . 48 − 35 . 83 ± 0 . 44 − 54 . 07 ± 0 . 49 − 52 . 28 ± 0 . 29 − 104 . 76 ± 0 . 47 − 158 . 4 ± 0 . 81 PoseTrack 4 Person BGP-LVM [5] − 121 . 06 ± 24 . 10 − 189 . 52 ± 34 . 73 − 358 . 36 ± 52 . 06 − 102 . 98 ± 24 . 06 − 209 . 79 ± 37 . 35 − 322 . 79 ± 59 . 27 fi-MRD [3] − 28 . 55 ± 0 . 73 − 56 . 45 ± 1 . 34 − 123 . 47 ± 27 . 63 − 73 . 89 ± 1 . 23 − 147 . 42 ± 2 . 03 − 270 . 04 ± 45 . 07 DP-GP-LVM − 28 . 14 ± 0 . 92 − 55 . 44 ± 1 . 69 − 107 . 41 ± 2 . 61 − 71 . 80 ± 1 . 87 − 143 . 06 ± 3 . 26 − 311 . 39 ± 1 . 31 Lawrence, Ek, Campbell DP-GP-LVM ICML 2019 9 / 11

  19. References Carl Henrik Ek, J Rihan, Phil H. S. Torr, G Rogez, and Neil D Lawrence. Ambiguity modeling in latent spaces. Int. Conference on Machine Learning for Multimodal Interaction , 2008. Mathieu Salzmann, Carl Henrik Ek, Raquel Urtasun, and Trevor Darrell. Factorized orthogonal latent spaces. In Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics , pages 701–708, 2010. Andreas C. Damianou, Carl Henrik Ek, Michalis K. Titsias, and Neil D. Lawrence. Manifold relevance determination. In International Conference on Machine Learning (ICML) , 2012. M. Andriluka, U. Iqbal, E. Ensafutdinov, L. Pishchulin, A. Milan, J. Gall, and Schiele B. PoseTrack: A benchmark for human pose estimation and tracking. In CVPR , 2018. Andreas C. Damianou, Michalis K. Titsias, and Neil D. Lawrence. Variational inference for latent variables and uncertain inputs in Gaussian processes. Journal of Machine Learning Research , 17(1):1425–1486, January 2016. Lawrence, Ek, Campbell DP-GP-LVM ICML 2019 10 / 11

  20. Thank You Poster #221 Pacific Ballroom - 12 June 2019 - 18:30–21:00 Acknowledgements We would like to acknowledge the European Union’s Horizon 2020 research and innovation programme under the Marie Sk� lodowska-Curie grant agreement No 665992, the UK’s EPSRC Centre for Doctoral Training in Digital Entertainment (CDE, EP/L016540/1), the RCUK-funded Centre for the Analysis of Motion, Entertainment Research and Applications (CAMERA, EP/M023281/1), and the Royal Society for supporting this research. Lawrence, Ek, Campbell DP-GP-LVM ICML 2019 11 / 11

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