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Advanced Artificial Intelligence

Advanced Artificial Intelligence. Lecture 8: Advance machine learning. Outline. Clustering K-Means EM Spectral Clustering Dimensionality Reduction. The unsupervised learning problem. Many data points, no labels. K-Means. Many data points, no labels. Choose a fixed number of clusters

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Advanced Artificial Intelligence

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  1. Advanced Artificial Intelligence Lecture 8: Advance machine learning

  2. Outline • Clustering • K-Means • EM • Spectral Clustering • Dimensionality Reduction

  3. The unsupervised learning problem Many data points, no labels

  4. K-Means Many data points, no labels

  5. Choose a fixed number of clusters Choose cluster centers and point-cluster allocations to minimize error can’t do this by exhaustive search, because there are too many possible allocations. Algorithm fix cluster centers; allocate points to closest cluster fix allocation; compute best cluster centers x could be any set of features for which we can compute a distance (careful about scaling) K-Means

  6. K-Means

  7. K-Means * From Marc Pollefeys COMP 256 2003

  8. K-Means • Is an approximation to EM • Model (hypothesis space): Mixture of N Gaussians • Latent variables: Correspondence of data and Gaussians • We notice: • Given the mixture model, it’s easy to calculate the correspondence • Given the correspondence it’s easy to estimate the mixture models

  9. Expectation Maximzation: Idea • Data generated from mixture of Gaussians • Latent variables: Correspondence between Data Items and Gaussians

  10. Generalized K-Means (EM)

  11. Gaussians

  12. ML Fitting Gaussians

  13. E-Step Learning a Gaussian Mixture(with known covariance) M-Step

  14. Expectation Maximization • Converges! • Proof [Neal/Hinton, McLachlan/Krishnan]: • E/M step does not decrease data likelihood • Converges at local minimum or saddle point • But subject to local minima

  15. Practical EM • Number of Clusters unknown • Suffers (badly) from local minima • Algorithm: • Start new cluster center if many points “unexplained” • Kill cluster center that doesn’t contribute • (Use AIC/BIC criterion for all this, if you want to be formal)

  16. Spectral Clustering

  17. Spectral Clustering

  18. Spectral Clustering: Overview Data Similarities Block-Detection

  19. Eigenvectors and Blocks • Block matrices have block eigenvectors: • Near-block matrices have near-block eigenvectors: [Ng et al., NIPS 02] l2= 2 l3= 0 l1= 2 l4= 0 eigensolver l3= -0.02 l1= 2.02 l2= 2.02 l4= -0.02 eigensolver

  20. Spectral Space • Can put items into blocks by eigenvectors: • Resulting clusters independent of row ordering: e1 e2 e1 e2 e1 e2 e1 e2

  21. The Spectral Advantage • The key advantage of spectral clustering is the spectral space representation:

  22. Measuring Affinity Intensity Distance Texture

  23. Scale affects affinity

  24. Dimensionality Reduction

  25. Dimensionality Reduction with PCA

  26. Linear: Principal Components • Fit multivariate Gaussian • Compute eigenvectors of Covariance • Project onto eigenvectors with largest eigenvalues

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