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Subspace Clustering

Subspace Clustering. Ali Sekmen and Ghan S. Bhatt Computer Science and Mathematical Sciences College of Engineering Tennessee State University. 1 st Annual Workshop on Data Sciences. Part I. Some Linear Algebra Spectral Analysis Singular Value Decomposition Presenter

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Subspace Clustering

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  1. Subspace Clustering Ali Sekmen and Ghan S. Bhatt Computer Science and Mathematical Sciences College of Engineering Tennessee State University 1st Annual Workshop on Data Sciences

  2. Part I • Some Linear Algebra • Spectral Analysis • Singular Value Decomposition • Presenter • Dr. Ghan S. Bhatt

  3. Definitions

  4. Range and Null Spaces

  5. Range and Null Spaces

  6. Definitions

  7. Eigenvalues - Eigenvectors

  8. Eigenvalues - Eigenvectors

  9. Eigenvalues - Eigenvectors

  10. Eigenvalues - Eigenvectors

  11. Symmetric Matrices

  12. Symmetric Matrices

  13. Projection on a Vector

  14. Projection on a Subspace

  15. Singular Value Decomposition

  16. Singular Value Decomposition

  17. Singular Value Decomposition

  18. Singular Value Decomposition

  19. Important Lemma

  20. Recall – Linear Mapping

  21. Recall – Linear Mapping

  22. General Matrix Norms

  23. An Intuitive Matrix Norm This satisfies the general matrix norm properties Although it is useful, it is not suitable for large set of problems and we need another definition of matrix norms

  24. Induced Matrix Norms

  25. Matrix p-Norm

  26. More on Matrix Norms

  27. Part II • Subspace Segmentation Problem • Motion Segmentation • Principal Component Analysis • Dimensionality Reduction • Spectral Clustering • Presenter • Dr. Ali Sekmen

  28. Subspace Segmentation • In many engineering and mathematics applications, data lives in a union of low dimensional subspaces • Motion segmentation • Facial images of a person with the same expression under different illumination approximately lie on the same subspace

  29. Face Recognition

  30. Problem Statement

  31. Problem Statement

  32. Problem Statement

  33. What are we trying to solve?

  34. Example – Motion Segmentation

  35. Motion Segmentation Motion segmentation problem can simply be defined as identifying independently moving rigid objects in a video.

  36. Motion Segmentation We will show that all trajectories lie in a 4-dim subspace of

  37. Motion Segmentation Z Z p z x Y Y X y X

  38. Motion Segmentation Z p z x Y X y

  39. Motion Segmentation Z p z x Y X y

  40. Motion Segmentation

  41. Motion Segmentation Y X

  42. Motion Segmentation Motion Segmentation Y X

  43. Motion Segmentation

  44. Motion Segmentation

  45. Principal Component Analysis • The goal is to reduce dimension of dataset with minimal loss of information • We project a feature space onto a smaller subspace that represent data well • Search for a subspace which maximizes the variance of projected points • This is equivalent to linear least square fitting • Minimize the sum of squared distances between points and subspace • We find directions (components) that maximizes variance in dataset • PCA can be done by • Eigenvalue decomposition of a data covariance matrix • Or SVD of a data matrix

  46. Least Square Approximation

  47. Principal Component Analysis

  48. Principal Component Analysis

  49. PCA with SVD Coordinates w.r.t. new basis

  50. Principal Component Analysis inch cm

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