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Relational Learning with Gaussian Processes By  Wei Chu, Vikas Sindhwani, Zoubin Ghahramani, S.Sathiya Keerthi (Columbia, Chicago, Cambridge, Yahoo!) Presented by Nesreen Ahmed, Nguyen Cao,  Sebastian Moreno, Philip Schatz
Outline ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],12/02/08 CS590M: Statistical Machine Learning - Fall 2008
Introduction ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],12/02/08 CS590M: Statistical Machine Learning - Fall 2008
Introduction ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],12/02/08 CS590M: Statistical Machine Learning - Fall 2008
[object Object],Gaussian Processes 12/02/08 CS590M: Statistical Machine Learning - Fall 2008  where
[object Object],[object Object],Relational Gaussian Processes 12/02/08 CS590M: Statistical Machine Learning - Fall 2008  x j x i ε ij
[object Object],Relational Gaussian Processes 12/02/08 CS590M: Statistical Machine Learning - Fall 2008  i,j runs over the set of observed undirected linkages EP algorithm approximates where as : is a 2x2 symmetric matrix
Relational Gaussian Processes 12/02/08 CS590M: Statistical Machine Learning - Fall 2008  where is a nxn matrix with four non-zero entries augmented from
[object Object],Relational Gaussian Processes 12/02/08 CS590M: Statistical Machine Learning - Fall 2008  where elements of covariance matrix are given by evaluating the following (covariance) kernel function:
Linkage Prediction ,[object Object],[object Object],12/02/08 CS590M: Statistical Machine Learning - Fall 2008
Semi supervised learning 12/02/08 CS590M: Statistical Machine Learning - Fall 2008  ? ? ? ? ? ? ? ? -1 1 1 ? -1
Semi supervised learning 12/02/08 CS590M: Statistical Machine Learning - Fall 2008  ? ? ? ? ? ? ? ? -1 1 1 ? -1 Nearest Neighborhood  K=1
Semi supervised learning 12/02/08 CS590M: Statistical Machine Learning - Fall 2008  ? ? ? ? ? ? ? ? -1 1 1 ? -1 Nearest Neighborhood  K=2
Semi supervised learning ,[object Object],[object Object],[object Object],12/02/08 CS590M: Statistical Machine Learning - Fall 2008
Semi supervised learning ,[object Object],[object Object],12/02/08 CS590M: Statistical Machine Learning - Fall 2008
Experiments ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],12/02/08 CS590M: Statistical Machine Learning - Fall 2008
12/02/08 CS590M: Statistical Machine Learning - Fall 2008  30 Samples collected from a gaussian mixture with two components on the x-axis.  Two labeled samples indicated by diamond and circle. K=3 Best  value =0.4  based on approximate model evidence Results
12/02/08 CS590M: Statistical Machine Learning - Fall 2008  Posterior Covariance matrix of RGP learnt from the data It captures the density information of unlabelled data Using the posterior covariance matrix learnt from the data as the new prior, supervised learning is carried out  Curves represent predictive distribution for each class Results
Results ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],12/02/08 CS590M: Statistical Machine Learning - Fall 2008
Results ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],12/02/08 CS590M: Statistical Machine Learning - Fall 2008  Student or Not Other or Not Univ. GPC LapSVM RGP GPC LapSVM RGP Corn. 0.825±0.016 0.987±0.008 0.989±0.009 0.708±0.021 0.865±0.038 0.884±0.025 Texa. 0.899±0.016 0.994±0.007 0.999±0.001 0.799±0.021 0.932±0.026 0.906±0.026 Wash. 0.839±0.018 0.957±0.014 0.961±0.009 0.782±0.023 0.828±0.025 0.877±0.024 Wisc 0.883±0.013 0.976±0.029 0.992±0.008 0.839±0.014 0.812±0.030 0.899±0.015
Conclusion ,[object Object],[object Object],[object Object],[object Object],[object Object],12/02/08 CS590M: Statistical Machine Learning - Fall 2008
Discussion ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],12/02/08 CS590M: Statistical Machine Learning - Fall 2008
12/02/08 CS590M: Statistical Machine Learning - Fall 2008  Thanks Questions ?

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PPT slides

  • 1. Relational Learning with Gaussian Processes By Wei Chu, Vikas Sindhwani, Zoubin Ghahramani, S.Sathiya Keerthi (Columbia, Chicago, Cambridge, Yahoo!) Presented by Nesreen Ahmed, Nguyen Cao, Sebastian Moreno, Philip Schatz
  • 2.
  • 3.
  • 4.
  • 5.
  • 6.
  • 7.
  • 8. Relational Gaussian Processes 12/02/08 CS590M: Statistical Machine Learning - Fall 2008 where is a nxn matrix with four non-zero entries augmented from
  • 9.
  • 10.
  • 11. Semi supervised learning 12/02/08 CS590M: Statistical Machine Learning - Fall 2008 ? ? ? ? ? ? ? ? -1 1 1 ? -1
  • 12. Semi supervised learning 12/02/08 CS590M: Statistical Machine Learning - Fall 2008 ? ? ? ? ? ? ? ? -1 1 1 ? -1 Nearest Neighborhood K=1
  • 13. Semi supervised learning 12/02/08 CS590M: Statistical Machine Learning - Fall 2008 ? ? ? ? ? ? ? ? -1 1 1 ? -1 Nearest Neighborhood K=2
  • 14.
  • 15.
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  • 17. 12/02/08 CS590M: Statistical Machine Learning - Fall 2008 30 Samples collected from a gaussian mixture with two components on the x-axis. Two labeled samples indicated by diamond and circle. K=3 Best value =0.4 based on approximate model evidence Results
  • 18. 12/02/08 CS590M: Statistical Machine Learning - Fall 2008 Posterior Covariance matrix of RGP learnt from the data It captures the density information of unlabelled data Using the posterior covariance matrix learnt from the data as the new prior, supervised learning is carried out Curves represent predictive distribution for each class Results
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  • 23. 12/02/08 CS590M: Statistical Machine Learning - Fall 2008 Thanks Questions ?