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Subspace Clustering. Ali Sekmen Computer Science College of Engineering Tennessee State University. 1 st Annual Workshop on Data Sciences. Outline. Subspace Segmentation Problem Motion Segmentation Principal Component Analysis Dimensionality Reduction Spectral Clustering Presenter
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Subspace Clustering Ali Sekmen Computer Science College of Engineering Tennessee State University 1st Annual Workshop on Data Sciences
Outline • Subspace Segmentation Problem • Motion Segmentation • Principal Component Analysis • Dimensionality Reduction • Spectral Clustering • Presenter • Dr. Ali Sekmen
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
Motion Segmentation Motion segmentation problem can simply be defined as identifying independently moving rigid objects in a video.
Motion Segmentation We will show that all trajectories lie in a 4-dim subspace of
Motion Segmentation Z Z p z x Y Y X y X
Motion Segmentation Z p z x Y X y
Motion Segmentation Z p z x Y X y
Motion Segmentation Motion Segmentation Y X
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
PCA with SVD Coordinates w.r.t. new basis
Principal Component Analysis inch cm
Spectral Clustering • A very powerful clustering algorithm • Easy to implement • Outperforms traditional clustering algorithms • Example: k-means • It is not easy to understand why it works • Given a set of data points and some similarity measure between all pairs of data points, we divide data into groups • Points in the same group are similar • Points in different groups are dissimilar
Spectral Clustering • Most of subspace clustering algorithms employ spectral clustering as the last step
Spectral Clustering Example From Lecture Notes of Ulrike von Luxburg
Spectral Clustering Example From Lecture Notes of Ulrike von Luxburg
Spectral Clustering Example From Lecture Notes of Ulrike von Luxburg
Spectral Clustering Example From Lecture Notes of Ulrike von Luxburg
Spectral Clustering Example From Lecture Notes of Ulrike von Luxburg