CS475/CS675 Lecture 23: July 19, 2016 Principal Component Analysis, - - PowerPoint PPT Presentation

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CS475/CS675 Lecture 23: July 19, 2016 Principal Component Analysis, Eigenfaces CS475/CS675 (c) 2016 P. Poupart 1 Principal Component Analysis (PCA) Data exploration technique: Dimensionality reduction Principal components are axes

SLIDE 1

CS475/CS675 Lecture 23: July 19, 2016

Principal Component Analysis, Eigenfaces

SLIDE 2

Principal Component Analysis (PCA)

Data exploration technique:

– Dimensionality reduction – Principal components are axes that preserve most of the variance in the data

Picture

SLIDE 3

Empirical Variance

Data:
Empirical mean:
Empirical covariance:
CS475/CS675 (c) 2016 P. Poupart

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Principal Component

Axis that preserves the most variance
When

, then

is

SLIDE 5

Principal Component Analysis

Eigendecomposition of the empirical covariance

matrix

Λ

Eigenvector: dimension (or basis function)
Eigenvalue: amount of variance preserved in that dimension

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Dimensionality Reduction

Problem: what is the smallest linear subspace (i.e., fewest

dimensions) that captures 95% of the variance?

Solution: retain eigenvectors of the largest eigenvalues such

that is the smallest integer that satisfies ∑

∑
0.95
Picture:

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Example: Eigenfaces

Turk and Pentland (1991):

– Image compression – Face detection

Solution:

– embed images in low dimensional eigenspace – Face detection: nearest neighbour in eigenspace

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Principal Component Analysis

Data:

( pixels images)

Covariance matrix:

( )

Eigendecomposition:
CS475/CS675 (c) 2016 P. Poupart

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Eigenfaces

Dataset Mean image Eigenfaces

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Face Detection

Project each image in the space if eigenfaces

– I.e., approximate each image as a linear combination of the eigenfaces

Face detection: find matching image in a database

– Nearest neighbour in space of eigenfaces ∗

CS475/CS675 (c) 2016 P. Poupart