Advanced Introduction to Machine Learning CMU-10715 Principal Component Analysis Barnabás Póczos
Contents Motivation PCA algorithms Applications Some of these slides are taken from • Karl Booksh Research group • Tom Mitchell • Ron Parr 2
Motivation 3
PCA Applications • Data Visualization • Data Compression • Noise Reduction 4
Data Visualization Example: • Given 53 blood and urine samples (features) from 65 people. • How can we visualize the measurements? 5
Data Visualization • Matrix format (65x53) H-WBC H-RBC H-Hgb H-Hct H-MCV H-MCH H-MCHC H-MCHC A1 8.0000 4.8200 14.1000 41.0000 85.0000 29.0000 34.0000 A2 7.3000 5.0200 14.7000 43.0000 86.0000 29.0000 34.0000 Instances A3 4.3000 4.4800 14.1000 41.0000 91.0000 32.0000 35.0000 A4 7.5000 4.4700 14.9000 45.0000 101.0000 33.0000 33.0000 A5 7.3000 5.5200 15.4000 46.0000 84.0000 28.0000 33.0000 A6 6.9000 4.8600 16.0000 47.0000 97.0000 33.0000 34.0000 A7 7.8000 4.6800 14.7000 43.0000 92.0000 31.0000 34.0000 A8 8.6000 4.8200 15.8000 42.0000 88.0000 33.0000 37.0000 A9 5.1000 4.7100 14.0000 43.0000 92.0000 30.0000 32.0000 Features Difficult to see the correlations between the features... 6
Data Visualization • Spectral format (65 curves, one for each person) 1000 900 800 700 600 Value 500 400 300 200 100 0 0 10 20 30 40 50 60 measurement Measurement Difficult to compare the different patients... 7
Data Visualization • Spectral format (53 pictures, one for each feature) 1.8 1.6 1.4 1.2 H-Bands 1 0.8 0.6 0.4 0.2 0 0 10 20 30 40 50 60 70 Person Difficult to see the correlations between the features... 8
Data Visualization Bi-variate Tri-variate 550 500 4 450 400 3 C-LDH M-EPI 350 2 300 1 250 200 0 600 150 100200300400500 400 100 200 C-LDH 50 0 0 0 50 150 250 350 450 C-Triglycerides C-Triglycerides How can we visualize the other variables??? … difficult to see in 4 or higher dimensional spaces... 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 10
PCA Algorithms 11
Principal Component Analysis PCA: Orthogonal projection of the data onto a lower-dimension linear space that... maximizes variance of projected data (purple line) minimizes the mean squared distance between • data point and • projections (sum of blue lines) 12
Principal Component Analysis Idea: Given data points in a d-dimensional space, project them into a lower dimensional space while preserving as much information as possible. • Find best planar approximation of 3D data • Find best 12-D approximation of 10 4 -D data In particular, choose projection that minimizes squared error in reconstructing the original data. 13
Principal Component Analysis Properties: PCA Vectors originate 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 14
2D Gaussian dataset 15
1 st PCA axis 16
2 nd PCA axis 17
PCA algorithm I (sequential) Given the centered data { x 1 , …, x m }, compute the principal vectors: m 1 T 2 1 st PCA vector w arg max {( w x ) } 1 i m w 1 i 1 To find w 1 , maximize the variance of projection of x m 1 T T 2 2 nd PCA vector w arg max {[ w ( x w w x )] } 2 i 1 1 i m w 1 i 1 x’ PCA reconstruction w To find w 2 , we maximize x x- x’ w 2 the variance of the w 1 projection in the residual subspace x’=w 1 ( w 1 T x ) w 18
PCA algorithm I (sequential) Given w 1 ,…, w k-1 , we calculate w k principal vector as before: Maximize the variance of projection of x m k 1 1 T T 2 w arg max {[ w ( x w w x )] } k i j j i m w 1 i 1 j 1 k th PCA vector x’ PCA reconstruction w We maximize the variance x of the projection in the w 1 residual subspace w 1 ( w 1 T x ) w 2 ( w 2 T x ) x’=w 1 ( w 1 T x ) +w 2 ( w 2 T x ) w 2 19
PCA algorithm II (sample covariance matrix) • Given data { x 1 , …, x m }, compute covariance matrix m m 1 1 where T x x ( x x )( x x ) i i m m i 1 i 1 • PCA basis vectors = the eigenvectors of • Larger eigenvalue more important eigenvectors 20
PCA algorithm II (sample covariance matrix) PCA algorithm( X , k ): top k eigenvalues/eigenvectors % X = N m data matrix, % … each data point x i = column vector, i=1..m m 1 x x • i m i 1 • X subtract mean x from each column vector x i in X • X X T … covariance matrix of X • { i , u i } i=1..N = eigenvectors/eigenvalues of ... 1 2 … N • Return { i , u i } i=1.. k % top k PCA components 21
Finding Eigenvectors with Power Iteration First eigenvector: Power iteration 1: v=Sigma*v; v=v/sqrt(v'*v); ) v PCA1 Second eigenvector: Power iteration 2: v PCA1T *Sigma*v PCA1 Sigma 2 =Sigma- *v PCA1 *v PCA1 T ; v=Sigma 2 *v; v=v/sqrt(v'*v); ) v PCA2 22
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PCA algorithm III (SVD of the data matrix) Singular Value Decomposition of the centered data matrix X . X features samples = USV T X U S V T = sig. significant significant noise noise noise 24 samples
PCA algorithm III • Columns of U • the principal vectors, { u (1) , …, u ( k) } • orthogonal and has unit norm – so U T U = I • Can reconstruct the data using linear combinations of { u (1) , …, u ( k) } • Matrix S • Diagonal • Shows importance of each eigenvector • Columns of V T • The coefficients for reconstructing the samples 25
Applications 26
Face Recognition Want to identify specific person, based on facial image Robust to glasses, lighting,… Can’t just use the given 256 x 256 pixels 27
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. 28
Applying PCA: Eigenfaces Example data set: Images of faces • Eigenface approach [Turk & Pentland], [Sirovich & Kirby] Each face x is … • 256 256 values (luminance at location) x 1 , …, x m • x in 256 256 (view as 64K dim vector) real values 256 x 256 Form X = [ x 1 , …, x m ] centered data X = mtx Compute = XX T Problem: is 64K 64K … HUGE!!! m faces 29
Computational Complexity Suppose m instances, each of size N • Eigenfaces: m=500 faces, each of size N=64K Given N N covariance matrix , can compute • all N eigenvectors/eigenvalues in O(N 3 ) • first k eigenvectors/eigenvalues in O(k N 2 ) If N=64K, this is EXPENSIVE! 30
A Clever Workaround • Note that m<<64K x 1 , …, x m • Use L =X T X instead of =XX T • If v is eigenvector of L, real values 256 x 256 then Xv is eigenvector of X = Proof: L v = v X T X v = v X (X T X v) = X( v) = Xv m faces (XX T )X v = ( Xv ) ( Xv) = ( Xv ) 31
Principle Components (Method B) 32
Shortcomings Requires carefully controlled data: • All faces centered in frame • Same size • Some sensitivity to angle 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 33
Happiness subspace 34
Disgust subspace 35
Facial Expression Recognition Movies 36
Facial Expression Recognition Movies 37
Facial Expression Recognition Movies 38
Image Compression
Original Image Divide the original 372x492 image into patches: • Each patch is an instance that contains 12x12 pixels on a grid Consider each as a 144-D vector 40
L 2 error and PCA dim 41
PCA compression: 144D ) 60D 42
PCA compression: 144D ) 16D 43
16 most important eigenvectors 2 2 2 2 4 4 4 4 6 6 6 6 8 8 8 8 10 10 10 10 12 12 12 12 2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12 2 2 2 2 4 4 4 4 6 6 6 6 8 8 8 8 10 10 10 10 12 12 12 12 2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12 2 2 2 2 4 4 4 4 6 6 6 6 8 8 8 8 10 10 10 10 12 12 12 12 2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12 2 2 2 2 4 4 4 4 6 6 6 6 8 8 8 8 10 10 10 10 12 12 12 12 2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12 44
PCA compression: 144D ) 6D 45
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