ECE 5984: Introduction to Machine Learning

Transcription

1 ECE 5984: Introduction to Machine Learning Topics: Unsupervised Learning: Kmeans, GMM, EM Readings: Barber Dhruv Batra Virginia Tech

2 Midsem Presentations Graded Mean 8/10 = 80% Min: 3 Max: 10 (C) Dhruv Batra 2

3 Tasks Supervised Learning x Classification y Discrete x Regression y Continuous Unsupervised Learning x Clustering c Discrete ID x Dimensionality z Continuous Reduction (C) Dhruv Batra 3

4 Unsupervised Learning Learning only with X Y not present in training data Some example unsupervised learning problems: Clustering / Factor Analysis Dimensionality Reduction / Embeddings Density Estimation with Mixture Models (C) Dhruv Batra 4

5 New Topic: Clustering Slide Credit: Carlos Guestrin 5

6 Synonyms Clustering Vector Quantization Latent Variable Models Hidden Variable Models Mixture Models Algorithms: K-means Expectation Maximization (EM) (C) Dhruv Batra 6

7 Some Data (C) Dhruv Batra Slide Credit: Carlos Guestrin 7

8 K-means 1. Ask user how many clusters they d like. (e.g. k=5) (C) Dhruv Batra Slide Credit: Carlos Guestrin 8

9 K-means 1. Ask user how many clusters they d like. (e.g. k=5) 2. Randomly guess k cluster Center locations (C) Dhruv Batra Slide Credit: Carlos Guestrin 9

10 K-means 1. Ask user how many clusters they d like. (e.g. k=5) 2. Randomly guess k cluster Center locations 3. Each datapoint finds out which Center it s closest to. (Thus each Center owns a set of datapoints) (C) Dhruv Batra Slide Credit: Carlos Guestrin 10

11 K-means 1. Ask user how many clusters they d like. (e.g. k=5) 2. Randomly guess k cluster Center locations 3. Each datapoint finds out which Center it s closest to. 4. Each Center finds the centroid of the points it owns (C) Dhruv Batra Slide Credit: Carlos Guestrin 11

12 K-means 1. Ask user how many clusters they d like. (e.g. k=5) 2. Randomly guess k cluster Center locations 3. Each datapoint finds out which Center it s closest to. 4. Each Center finds the centroid of the points it owns 5. and jumps there 6. Repeat until (C) Dhruv Batra terminated! Slide Credit: Carlos Guestrin 12

13 K-means Randomly initialize k centers µ (0) = µ 1 (0),, µ k (0) Assign: Assign each point i {1, n} to nearest center: C(i) argmin j x i µ j 2 Recenter: µ j becomes centroid of its points (C) Dhruv Batra Slide Credit: Carlos Guestrin 13

14 K-means Demo AppletKM.html (C) Dhruv Batra 14

15 What is K-means optimizing? Objective F(µ,C): function of centers µ and point allocations C: F (µ,c)= NX x i µ C(i) 2 i=1 1-of-k encoding F (µ, a) = NX kx a ij x i µ j 2 i=1 j=1 Optimal K-means: min µ min a F(µ,a) (C) Dhruv Batra 15

16 Coordinate descent algorithms Want: min a min b F(a,b) Coordinate descent: fix a, minimize b fix b, minimize a repeat Converges!!! if F is bounded to a (often good) local optimum as we saw in applet (play with it!) K-means is a coordinate descent algorithm! (C) Dhruv Batra Slide Credit: Carlos Guestrin 16

17 K-means as Co-ordinate Descent Optimize objective function: min µ 1,...,µ k min a 1,...,a N F (µ, a) = Fix µ, optimize a (or C) min µ 1,...,µ k min a 1,...,a N NX i=1 kx a ij x i µ j 2 j=1 (C) Dhruv Batra Slide Credit: Carlos Guestrin 17

18 K-means as Co-ordinate Descent Optimize objective function: min µ 1,...,µ k min a 1,...,a N F (µ, a) = Fix a (or C), optimize µ min µ 1,...,µ k min a 1,...,a N NX i=1 kx a ij x i µ j 2 j=1 (C) Dhruv Batra Slide Credit: Carlos Guestrin 18

19 One important use of K-means Bag-of-word models in computer vision (C) Dhruv Batra 19

20 Bag of Words model aardvark 0 about 2 all 2 Africa 1 apple 0 anxious 0... gas 1... oil 1 Zaire 0 (C) Dhruv Batra Slide Credit: Carlos Guestrin 20

21 Object Bag of words Fei- Fei Li

22 Fei- Fei Li

23 Interest Point Features Compute SIFT descriptor [Lowe 99] Normalize patch Detect patches [Mikojaczyk and Schmid 02] [Matas et al. 02] [Sivic et al. 03] Slide credit: Josef Sivic

24 Patch Features Slide credit: Josef Sivic

25 dictionary formation Slide credit: Josef Sivic

26 Clustering (usually k-means) Vector quantization Slide credit: Josef Sivic

27 Clustered Image Patches Fei-Fei et al. 2005

28 Visual synonyms and polysemy Visual Polysemy. Single visual word occurring on different (but locally similar) parts on different object categories. Visual Synonyms. Two different visual words representing a similar part of an object (wheel of a motorbike). Andrew Zisserman

29 Image representation frequency.. codewords Fei- Fei Li

30 (One) bad case for k-means Clusters may overlap Some clusters may be wider than others GMM to the rescue! (C) Dhruv Batra Slide Credit: Carlos Guestrin 30

31 GMM (C) Dhruv Batra Figure Credit: Kevin Murphy 31

32 Recall Multi-variate Gaussians (C) Dhruv Batra 32

33 GMM (C) Dhruv Batra Figure Credit: Kevin Murphy 33

34 Hidden Data Causes Problems #1 Fully Observed (Log) Likelihood factorizes Marginal (Log) Likelihood doesn t factorize All parameters coupled! (C) Dhruv Batra 34

35 GMM vs Gaussian Joint Bayes Classifier On Board Observed Y vs Unobserved Z Likelihood vs Marginal Likelihood (C) Dhruv Batra 35

36 Hidden Data Causes Problems # (C) Dhruv Batra Figure Credit: Kevin Murphy 36

37 Hidden Data Causes Problems #2 Identifiability µ µ 1 (C) Dhruv Batra Figure Credit: Kevin Murphy 37

38 Hidden Data Causes Problems #3 Likelihood has singularities if one Gaussian collapses p(x) (C) Dhruv Batra x 38

39 Special case: spherical Gaussians and hard assignments If P(X Z=k) is spherical, with same σ for all classes: # P(x i z = j) exp 1 $ % 2σ 2 x i µ j 2 & ' ( If each x i belongs to one class C(i) (hard assignment), marginal likelihood: N k N % P(x i, y = j) exp 1 & ' 2σ 2 i=1 j=1 i=1 x i µ C(i) 2 ( ) * M(M)LE same as K-means!!! (C) Dhruv Batra Slide Credit: Carlos Guestrin 39

40 The K-means GMM assumption There are k components Component i has an associated mean vector µ ι µ 2 µ 1 µ 3 (C) Dhruv Batra Slide Credit: Carlos Guestrin 40

41 The K-means GMM assumption There are k components Component i has an associated mean vector µ ι Each component generates data from a Gaussian with mean m i and covariance matrix σ 2 Ι Each data point is generated according to the following recipe: µ 1 µ 2 µ 3 (C) Dhruv Batra Slide Credit: Carlos Guestrin 41

42 The K-means GMM assumption There are k components Component i has an associated mean vector µ ι Each component generates data from a Gaussian with mean m i and covariance matrix σ 2 Ι Each data point is generated according to the following recipe: 1. Pick a component at random: Choose component i with probability P(y=i) µ 2 (C) Dhruv Batra Slide Credit: Carlos Guestrin 42

43 The K-means GMM assumption There are k components Component i has an associated mean vector µ ι Each component generates data from a Gaussian with mean m i and covariance matrix σ 2 Ι Each data point is generated according to the following recipe: 1. Pick a component at random: Choose component i with probability P(y=i) 2. Datapoint Ν(µ ι, σ 2 Ι ) µ 2 x (C) Dhruv Batra Slide Credit: Carlos Guestrin 43

44 The General GMM assumption There are k components Component i has an associated mean vector m i Each component generates data from a Gaussian with mean m i and covariance matrix Σ i Each data point is generated according to the following recipe: 1. Pick a component at random: Choose component i with probability P(y=i) 2. Datapoint ~ N(m i, Σ i ) (C) Dhruv Batra Slide Credit: Carlos Guestrin 44 µ 1 µ 2 µ 3

45 K-means vs GMM K-Means AppletKM.html GMM (C) Dhruv Batra 45