Subspace Clustering Ali Sekmen Computer Science College of
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 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
Face Recognition
Problem Statement
Problem Statement
Problem Statement
What are we trying to solve?
Example – Motion Segmentation
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 X y
Motion Segmentation Z p z x Y X y
Motion Segmentation Z p z x Y X y
Motion Segmentation
Motion Segmentation Y X
Motion Segmentation Y X
Motion Segmentation
Motion Segmentation
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
Least Square Approximation
Principal Component Analysis
Principal Component Analysis
PCA with SVD Coordinates w. r. t. new basis
Principal Component Analysis inch cm
Principal Component Analysis inch cm 10 28 12 19 15 40 20 47 23 56 26 69
Solution with SVD
PCA: Pre-Processing 80 inch cm 10 28 12 19 15 40 20 47 20 23 56 10 26 69 70 60 Inch 50 40 30 0 0 Zero Mean, Unit Variance 30 20 0 -10 0 -20 -30 cm 5 10 inch 10 -5 20 cm Zero Mean -10 10 -1, 5 -1 2 1, 5 1 0, 5 0 -0, 5 0 -1 -1, 5 cm 0, 5 1 1, 5 30
PCA: Optimization
PCA: Reduce Dimensionality
PCA: Reduce Dimensionality
General PCA
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
Similarity
Spectral Clustering
Spectral Clustering
Spectral Clustering
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
Spectral Clustering Example
Spectral Clustering Example
Spectral Clustering Example
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