graphs geometry and semi supervised learning
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Graphs, Geometry and Semi-supervised Learning Mikhail Belkin The Ohio State University, Dept of Computer Science and Engineering and Dept of Statistics Collaborators: Partha Niyogi, Vikas Sindhwani Ubiquity of manifolds In many domains


  1. Graphs, Geometry and Semi-supervised Learning Mikhail Belkin The Ohio State University, Dept of Computer Science and Engineering and Dept of Statistics Collaborators: Partha Niyogi, Vikas Sindhwani

  2. Ubiquity of manifolds � In many domains (e.g., speech, some vision problems) data explicitly lies on a manifold.

  3. Ubiquity of manifolds � In many domains (e.g., speech, some vision problems) data explicitly lies on a manifold. � For all sources of high-dimensional data, true dimensionality is much lower than the number of features.

  4. Ubiquity of manifolds � In many domains (e.g., speech, some vision problems) data explicitly lies on a manifold. � For all sources of high-dimensional data, true dimensionality is much lower than the number of features. � Much of the data is highly nonlinear.

  5. Manifold Learning Important point: only small distances are meaningful. In fact, all large distances are (almost) the same.

  6. Manifold Learning Important point: only small distances are meaningful. In fact, all large distances are (almost) the same. Manifolds (Riemannian manifolds with a measure + noise) provide a natural mathematical language for thinking about high-dimensional data.

  7. Manifold Learning Learning when data ∼ M ⊂ R N � Clustering: M → { 1 , . . . , k } connected components, min cut, normalized cut � Classification/Regression: M → {− 1 , +1 } or M → R P on M × {− 1 , +1 } or P on M × R � Dimensionality Reduction: f : M → R n n << N � M unknown: what can you learn about M from data? e.g. dimensionality, connected components holes, handles, homology curvature, geodesics

  8. Graph-based methods Data ——– Probability Distribution Graph ——– Manifold

  9. Graph-based methods Data ——– Probability Distribution Graph ——– Manifold

  10. Graph-based methods Data ——– Probability Distribution Graph ——– Manifold Graph extracts underlying geometric structure.

  11. Problems of machine learning � Classification / regression. � Data representation / dimensionality reduction. � Clustering. Common intuition – similar objects have similar labels.

  12. Intuition

  13. Intuition

  14. Intuition

  15. Intuition Geometry of data changes our notion of similarity.

  16. Manifold assumption

  17. Manifold assumption Geometry is important.

  18. Manifold assumption Manifold/geometric assumption: functions of interest are smooth with respect to the underlying geometry.

  19. Manifold assumption Manifold/geometric assumption: functions of interest are smooth with respect to the underlying geometry. Probabilistic setting: Map X → Y . Probability distribution P on X × Y . Regression/(two class)classification: X → R .

  20. Manifold assumption Manifold/geometric assumption: functions of interest are smooth with respect to the underlying geometry. Probabilistic setting: Map X → Y . Probability distribution P on X × Y . Regression/(two class)classification: X → R . Probabilistic version: conditional distributions P ( y | x ) are smooth with respect to the marginal P ( x ) .

  21. What is smooth? Function f : X → R . Penalty at x ∈ X : � 1 ( f ( x ) − f ( x + δ )) 2 p ( x ) d δ ≈ �∇ f � 2 p ( x ) δ k small δ Total penalty – Laplace operator: � �∇ f � 2 p ( x ) = � f, ∆ p f � X X

  22. What is smooth? Function f : X → R . Penalty at x ∈ X : � 1 ( f ( x ) − f ( x + δ )) 2 p ( x ) d δ ≈ �∇ f � 2 p ( x ) δ k small δ Total penalty – Laplace operator: � �∇ f � 2 p ( x ) = � f, ∆ p f � X X Two-class classification – conditional P (1 | x ) . Manifold assumption: � P (1 | x ) , ∆ p P (1 | x ) � X is small.

  23. Laplace operator Laplace operator is a fundamental geometric object. k ∂ 2 f � ∆ f = − ∂x 2 i i =1 The only differential operator invariant under translations and rotations. Heat, Wave, Schroedinger equations. Fourier analysis.

  24. Laplacian on the circle φ − d 2 f dφ 2 = λf where f (0) = f (2 π ) Same as in R with periodic boundary conditions. Eigenvalues: λ n = n 2 Eigenfunctions: sin( nφ ) , cos( nφ ) Fourier analysis.

  25. Laplace-Beltrami operator ��������������������������������������� ��������������������������������������� x p ��������������������������������������� ��������������������������������������� 1 ��������������������������������������� ��������������������������������������� ��������������������������������������� ��������������������������������������� x ��������������������������������������� ��������������������������������������� f : M k → R ��������������������������������������� ��������������������������������������� 2 ��������������������������������������� ��������������������������������������� ��������������������������������������� ��������������������������������������� ��������������������������������������� ��������������������������������������� ��������������������������������������� ��������������������������������������� ��������������������������������������� ��������������������������������������� exp p : T p M k → M k ��������������������������������������� ��������������������������������������� ∂ 2 f (exp p ( x )) � ∆ M f ( p ) = − ∂x 2 i i Generalization of Fourier analysis.

  26. Key learning question Machine learning: manifold is unknown. How to do Fourier analysis/reconstruct Laplace operator on an unknown manifold?

  27. Algorithmic framework

  28. Algorithmic framework

  29. Algorithmic framework � xi − xj � 2 W ij = e − [justification: heat equation] t � xi − xj � 2 � xi − xj � 2 � � e − f ( x j ) e − Lf ( x i ) = f ( x i ) − t t j j � xi − xj � 2 � f t L f = 2 e − ( f i − f j ) 2 t i ∼ j

  30. Data representation f : G → R Minimize � i ∼ j w ij ( f i − f j ) 2 Preserve adjacency. Solution: Lf = λf (slightly better Lf = λDf ) Lowest eigenfunctions of L ( ˜ L ). Laplacian Eigenmaps Related work: LLE: Roweis, Saul 00; Isomap: Tenenbaum, De Silva, Langford 00 Hessian Eigenmaps: Donoho, Grimes, 03; Diffusion Maps: Coifman, et al, 04

  31. Laplacian Eigenmaps � Visualizing spaces of digits and sounds. Partiview, Ndaona, Surendran 04 � Machine vision: inferring joint angles. Corazza, Andriacchi, Stanford Biomotion Lab, 05, Partiview, Surendran Isometrically invariant representation. [link] � Reinforcement Learning: value function approximation. Mahadevan, Maggioni, 05

  32. Semi-supervised learning Learning from labeled and unlabeled data. � Unlabeled data is everywhere. Need to use it. � Natural learning is semi-supervised.

  33. Semi-supervised learning Learning from labeled and unlabeled data. � Unlabeled data is everywhere. Need to use it. � Natural learning is semi-supervised. Labeled data: ( x 1 , y 1 ) , . . . , ( x l , y l ) ∈ R N × R Unlabeled data: x l +1 , . . . , x l + u ∈ R N Need to reconstruct f L,U : R N → R

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