Principal Component Analysis
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1 B: Chapter 1 HTF: Chapter 1.5 Principal Component Analysis Barnabás Póczos University of Alberta Nov, 009
2 Contents Motivation PCA algorithms Applications Face recognition Facial expression recognition PCA theory Kernel-PCA Some of these slides are taken from Karl Booksh Research group Tom Mitchell Ron Parr
3 PCA Applications Data Visualization Data Compression Noise Reduction Data Classification Trend Analysis Factor Analysis 3
4 Data Visualization Example: Given 53 blood and urine samples (features) from 5 people. How can we visualize the measurements?
5 Data Visualization Matrix format (5x53) Instances H -W B C H -R B C H -H g b H -H c t H -M C V H -M C H H -M C H C A A A A A A A A A Features Difficult to see the correlations between the features... 5
6 Data Visualization Spectral format (5 curves, one for each person) Value measurement Measurement Difficult to compare the different patients
7 Data Visualization Spectral format (53 pictures, one for each feature) H-Bands Person Difficult to see the correlations between the features... 7
8 Data Visualization Tri-variate M-EPI C-LDH Bi-variate C-LDH 00 C-Triglycerides C-Triglycerides How can we visualize the other variables??? difficult to see in or higher dimensional spaces... 8
9 Data Visualization Is there a representation better than the coordinate axes? Is it really necessary to show all the 53 dimensions? what if there are strong correlations between the features? How could we find the smallest subspace of the 53-D space that keeps the most information about the original data? A solution: Principal Component Analysis 9
10 Principle Component Analysis PCA: Orthogonal projection of data onto lower-dimension linear space that... maximizes variance of projected data (purple line) minimizes mean squared distance between data point and projections (sum of blue lines)
11 Principle Components Analysis Idea: Given data points in a d-dimensional space, project into lower dimensional space while preserving as much information as possible Eg, find best planar approximation to 3D data Eg, find best 1-D approximation to -D data In particular, choose projection that minimizes squared error in reconstructing original data 11
12 The Principal Components Vectors originating from the center of mass Principal component #1 points in the direction of the largest variance. Each subsequent principal component is orthogonal to the previous ones, and points in the directions of the largest variance of the residual subspace 1
13 D Gaussian dataset 13
14 1st PCA axis 1
15 nd PCA axis 15
16 PCA algorithm I (sequential) Given the centered data {x1,, xm}, compute the principal vectors: 1 m w1 = arg max {( w T x i ) } w =1 m i= 1 1st PCA vector We maximize the variance of projection of x k 1 1 m w k = arg max {[ w T (x i w j w Tj x i )] } w =1 m i= 1 j= 1 kth PCA vector x PCA reconstruction We maximize the variance of the projection in the residual subspace w x w1(w1tx) w1 w(wtx) w x =w1(w1tx)+w(wtx) 1
17 PCA algorithm II (sample covariance matrix) Σ Given data {x1,, xm}, compute covariance matrix 1 m T Σ = (x i x)(x x) m i= 1 where 1 m x = xi m i= 1 PCA basis vectors = the eigenvectors of Σ Larger eigenvalue more important eigenvectors 17
18 PCA algorithm II PCA algorithm(x, k): top k eigenvalues/eigenvectors % X = N m data matrix, % each data point xi = column vector, i=1..m 1 m x= xi m i=1 X subtract mean x from each column vector xi in X Σ X XT covariance matrix of X { λi, ui }i=1..n = eigenvectors/eigenvalues of Σ... λ1 λ λn Return { λi, ui }i=1..k % top k principle components 18
19 PCA algorithm III (SVD of the data matrix) Singular Value Decomposition of the centered data matrix X. Xfeatures samples = USVT X U = S samples noise significant sig. VT significant noise noise 19
20 PCA algorithm III Columns of U the principal vectors, { u(1),, u(k) } orthogonal and has unit norm so UTU = I Can reconstruct the data using linear combinations of { u(1),, u(k) } Matrix S Diagonal Shows importance of each eigenvector Columns of VT The coefficients for reconstructing the samples 0
21 Face recognition
22 Challenge: Facial Recognition Want to identify specific person, based on facial image Robust to glasses, lighting, Can t just use the given 5 x 5 pixels
23 Applying PCA: Eigenfaces Method A: Build a PCA subspace for each person and check which subspace can reconstruct the test image the best Method B: Build one PCA database for the whole dataset and then classify based on the weights. Example data set: Images of faces Famous Eigenface approach [Turk & Pentland], [Sirovich & Kirby] x1,, xm 5 x 5 real values X= m faces Each face x is 5 5 values (luminance at location) x in ℜ5 5 (view as K dim vector) Form X = [ x1,, xm ] centered data mtx Compute Σ = XXT Problem: Σ is K K HUGE!!! 3
24 Computational Complexity Suppose m instances, each of size N Eigenfaces: m=500 faces, each of size N=K Given N N covariance matrix Σ, can compute all N eigenvectors/eigenvalues in O(N3) first k eigenvectors/eigenvalues in O(k N) But if N=K, EXPENSIVE!
25 A Clever Workaround L v=γv XTX v = γ v X (XTX v) = X(γ v) = γ Xv (XXT)X v = γ (Xv) Σ (Xv) = γ (Xv) x1,, xm 5 x 5 real values Note that m<<k Use L=XTX instead of Σ=XXT If v is eigenvector of L then Xv is eigenvector of Σ X= Proof: m faces 5
26 Principle Components (Method B)
27 Reconstructing (Method B) faster if train with only people w/out glasses same lighting conditions 7
28 Shortcomings Requires carefully controlled data: All faces centered in frame Same size Some sensitivity to angle Alternative: Learn one set of PCA vectors for each angle Use the one with lowest error Method is completely knowledge free (sometimes this is good!) Doesn t know that faces are wrapped around 3D objects (heads) Makes no effort to preserve class distinctions 8
29 Facial expression recognition
30 Happiness subspace (method A) 30
31 Disgust subspace (method A) 31
32 Facial Expression Recognition Movies (method A) 3
33 Facial Expression Recognition Movies (method A) 33
34 Facial Expression Recognition Movies (method A) 3
35 Image Compression
36 Original Image Divide the original 37x9 image into patches: Each patch is an instance that contains 1x1 pixels on a grid View each as a 1-D vector 3
37 L error and PCA dim 37
38 PCA compression: 1D ) 0D 38
39 PCA compression: 1D ) 1D 39
40 1 most important eigenvectors
41 PCA compression: 1D ) D 1
42 most important eigenvectors
43 PCA compression: 1D ) 3D 3
44 3 most important eigenvectors
45 PCA compression: 1D ) 1D 5
46 0 most important eigenvectors Looks like the discrete cosine bases of JPG!...
47 D Discrete Cosine Basis 7
48 Noise Filtering
49 Noise Filtering, Auto-Encoder x x Ux 9
50 Noisy image 50
51 Denoised image using 15 PCA components 51
52 PCA Shortcomings
53 PCA, a Problematic Data Set PCA doesn t know labels! 53
54 PCA vs Fisher Linear Discriminant PCA maximizes variance, independent of class magenta FLD attempts to separate classes green line 5
55 PCA, a Problematic Data Set PCA cannot capture NON-LINEAR structure! 55
56 PCA Conclusions PCA finds orthonormal basis for data Sorts dimensions in order of importance Discard low significance dimensions Uses: Get compact description Ignore noise Improve classification (hopefully) Not magic: Doesn t know class labels Can only capture linear variations One of many tricks to reduce dimensionality! 5
57 PCA Theory
58 Justification of Algorithm II GOAL: 58
59 Justification of Algorithm II x is centered! 59
60 Justification of Algorithm II GOAL: Use Lagrange-multipliers for the constraints. 0
61 Justification of Algorithm II 1
62 Kernel PCA
63 Kernel PCA Performing PCA in the feature space Lemma Proof: 3
64 Kernel PCA Lemma
65 Kernel PCA Proof 5
66 Kernel PCA How to use α to calculate the projection of a new sample t? Where was I cheating? The data should be centered in the feature space, too! But this is manageable...
67 Input points before kernel PCA 7
68 Output after kernel PCA The three groups are distinguishable using the first component only 8
69 We haven t covered... Artificial Neural Network Implementations Mixture of Probabilistic PCA Online PCA, Regret Bounds 9
70 Thanks for the Attention! 70
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