Robot Image Credit: Viktoriya Sukhanova 123RF.com. Dimensionality Reduction

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1 Robot Image Credit: Viktoriya Sukhanova 13RF.com Dimensionality Reduction

2 Feature Selection vs. Dimensionality Reduction Feature Selection (last time) Select a subset of features. When classifying novel patterns, only a small number of features need to be computed (i.e., faster classification). The measurement units (length, weight, etc.) of the features are preserved. Dimensionality Reduction (this time) Transform features into a smaller set. When classifying novel patterns, all features need to be computed. The measurement units (length, weight, etc.) of the features are lost.

3 How Can We Visualize High Dimensional Data? E.g., 53 blood and urine tests for 5 patients Instances H-WBC H-RBC H-Hgb H-Hct H-MCV H-MCH H-MCHC A A A A A A A A A Features Difficult to see the correlations between the features... 3

4 Data Visualization Is there a representation better than the raw features? Is it really necessary to show all the 53 dimensions? what if there are strong correlations between the features? Could we find the smallest subspace of the 53-D space that keeps the most information about the original data? One solution: Principal Component Analysis

5 Principle Component Analysis 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) 5

6 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

7 D Gaussian Dataset 7

8 1 st PCA axis

9 nd PCA axis 9

10 PCA Algorithm Given data {x 1,, x n }, compute covariance matrix S X is the n x d data matrix Compute data mean (average over all rows of X) Subtract mean from each row of X (centering the data) Compute covariance matrix S = X T X ( S is d x d ) PCA basis vectors are given by the eigenvectors of S Q,Λ = numpy.linalg.eig(s) {q i, l i } i=1..n are the eigenvectors/eigenvalues of S... l 1 ³ l ³ ³ l n Larger eigenvalue Þ more important eigenvectors

11 Dimensionality Reduction Can ignore the components of lesser significance 5 0 Variance (%) PC1 PC PC3 PC PC5 PC PC7 PC PC9 PC You do lose some information, but if the eigenvalues are small, you don t lose much choose only the first k eigenvectors, based on their eigenvalues final data set has only k dimensions 11

12 X = PCA X has d columns Slide by Eric Eaton Q = Q are the eigenvectors of Σ; columns are ordered by importance! Q is d x d

13 X = PCA Each row of Q corresponds to a feature; keep only first k columns of Q Slide by Eric Eaton Q =

14 PCA Each column of Q gives weights for a linear combination of the original features Q = = 0.3 feature feature 0. feature Slide by Eric Eaton 1

15 PCA We can apply these formulas to get the new representation for each instance x ^ X = x 3 Q = The new D representation for x 3 is given by: ^ x 31 = 0.3(0) + 0.0(0) - 0.(1) +... ^ x 3 = 0.3(0) (0) (1) +... ^ The re-projected data matrix is given by X = XQ Slide by Eric Eaton ^ 15

16 PCA Example Data Center Data Covariance Matrix Eigenvectors Eigenvalues

17 PCA Visualization of MNIST Digits 17

18 Challenge: Facial Recognition Want to identify specific person, based on facial image Robust to glasses, lighting, Þ Can t just use the given 5 x 5 pixels 1

19 PCA applications - Eigenfaces Eigenfaces are the eigenvectors of the covariance matrix of the probability distribution of the vector space of human faces Eigenfaces are the standardized face ingredients derived from the statistical analysis of many pictures of human faces A human face may be considered to be a combination of these standard face ingredients

20 PCA applications -Eigenfaces To generate a set of eigenfaces: 1. Large set of digitized images of human faces is taken under the same lighting conditions.. The images are normalized to line up the eyes and mouths. 3. The eigenvectors of the covariance matrix of the statistical distribution of face image vectors are then extracted.. These eigenvectors are called eigenfaces.

21 PCA applications -Eigenfaces the principal eigenface looks like a bland androgynous average human face

22 Eigenfaces

23 Eigenfaces Face Recognition When properly weighted, eigenfaces can be summed together to create an approximate gray-scale rendering of a human face. Remarkably few eigenvector terms are needed to give a fair likeness of most people's faces Hence eigenfaces provide a means of applying data compression to faces for identification purposes. Similarly, Expert Object Recognition in Video

24 Experiment and Results Eigenfaces Data used here are from the ORL database of faces. Facial images of 1 people each with views are used. - Training set contains 1 7 images. - Test set contains 1 3 images. First three eigenfaces :

25 Classification Using Nearest Neighbor Save average coefficients for each person. Classify new face as the person with the closest average. Recognition accuracy increases with number of eigenfaces until ~15. Best recognition rates Training set 99% Test set 9%

26 Image Compression

27 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 7

28 L error and PCA dim

29 PCA compression: 1D à 0D 9

30 PCA compression: 1D à 1D 30

31 1 most important eigenvectors

32 PCA compression: 1D à D 3

33 most important eigenvectors

34 PCA compression: 1D à 3D 3

35 3 most important eigenvectors

36 PCA compression: 1D à 1D 3

37 0 most important eigenvectors Looks like the discrete cosine bases of JPG!... 37

38 D Discrete Cosine Basis 3