Clustering K-means. Machine Learning CSEP546 Carlos Guestrin University of Washington February 18, Carlos Guestrin
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1 Clustering K-means Machine Learning CSEP546 Carlos Guestrin University of Washington February 18, 2014 Carlos Guestrin
2 Clustering images Set of Images [Goldberger et al.] Carlos Guestrin
3 Clustering web search results Carlos Guestrin
4 Example (Taken from Kevin Murphy s ML textbook) Data: gene expression levels Goal: cluster genes with similar expression trajectories Carlos Guestrin
5 Some Data Carlos Guestrin
6 K-means 1. Ask user how many clusters they d like. (e.g. k=5) Carlos Guestrin
7 K-means 1. Ask user how many clusters they d like. (e.g. k=5) 2. Randomly guess k cluster Center locations Carlos Guestrin
8 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) Carlos Guestrin
9 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 Carlos Guestrin
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. 4. Each Center finds the centroid of the points it owns 5. and jumps there 6. Repeat until terminated! Carlos Guestrin
11 K-means Randomly initialize k centers µ (0) = µ 1 (0),, µ k (0) Classify: Assign each point j {1, N} to nearest center: Recenter: µ i becomes centroid of its point: Equivalent to µ i average of its points! Carlos Guestrin
12 What is K-means optimizing? Potential function F(µ,C) of centers µ and point allocations C: N Optimal K-means: min µ min C F(µ,C) Carlos Guestrin
13 Does K-means converge??? Part 1 Optimize potential function: Fix µ, optimize C Carlos Guestrin
14 Does K-means converge??? Part 2 Optimize potential function: Fix C, optimize µ Carlos Guestrin
15 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!) (For LASSO it converged to the global optimum, because of convexity) K-means is a coordinate descent algorithm! Carlos Guestrin
16 Mixtures of Gaussians Machine Learning CSEP546 Carlos Guestrin University of Washington February 18, 2014 Carlos Guestrin
17 (One) bad case for k-means Clusters may overlap Some clusters may be wider than others Carlos Guestrin
18 Nonspherical data Carlos Guestrin
19 Quick Review of Gaussians Univariate and multivariate Gaussians Carlos Guestrin
20 Two-Dimensional Gaussians 5 spherical 10 diagonal 6 full full spherical 0.2 diagonal Carlos 10 Guestrin
21 Gaussians in d Dimensions P(x) = 1 (2π ) d/2 Σ exp # 1 1/2 $ % 2 x µ ( ) T Σ 1 ( x µ ) & ' ( Carlos Guestrin
22 Learning Gaussians P(x) = 1 (2π ) d/2 Σ exp # 1 1/2 $ % 2 x µ ( ) T Σ 1 ( x µ ) & ' ( Given data: MLE for mean: MLE for covariance: Carlos Guestrin
23 When the world is not Gaussian Distribution of male heights in US Distribution of male heights in Sweden What if we mix these together? Carlos Guestrin
24 Gaussian Mixture Model Most commonly used mixture model Observations: Parameters: Cluster indicator: (a) Per-cluster likelihood: Ex. z i = country of origin, x i = height of i th person k th mixture component = distribution of heights in country k Carlos Guestrin
25 Generative Model We can think of sampling observations from the model For each observation i, Sample a cluster assignment (a) Sample the observation from the selected Gaussian Carlos Guestrin
26 Density Estimation Estimate a density based on x 1,,x N Carlos Guestrin
27 Density Estimation Contour Plot of Joint Density Carlos Guestrin
28 Density as Mixture of Gaussians Approximate density with a mixture of Gaussians Mixture of 3 Gaussians Contour Plot of Joint Density Carlos Guestrin
29 Density as Mixture of Gaussians Approximate density with a mixture of Gaussians Mixture of 3 Gaussians p(x i,µ, ) = Carlos Guestrin
30 Density as Mixture of Gaussians Approximate with density with a mixture of Gaussians Mixture of 3 Gaussians Our actual observations (b) C. Bishop, Pattern Recognition & Machine Learning Carlos Guestrin
31 Clustering our Observations Imagine we have an assignment of each x i to a Gaussian Our actual observations 1 (a) 1 (b) Complete data labeled by true cluster assignments C. Bishop, Pattern Recognition & Machine Learning Carlos Guestrin
32 Clustering our Observations Imagine we have an assignment of each x i to a Gaussian 1 (a) Introduce latent cluster indicator variable z i 0.5 Then we have p(x i z i,,µ, ) = Complete data labeled by true cluster assignments C. Bishop, Pattern Recognition & Machine Learning Carlos Guestrin
33 Clustering our Observations We must infer the cluster assignments from the observations (c) Posterior probabilities of assignments to each cluster *given* model parameters: r ik = p(z i = k x i,,µ, ) = Soft assignments to clusters C. Bishop, Pattern Recognition & Machine Learning Carlos Guestrin
34 Unsupervised Learning: not as hard as it looks Sometimes easy Sometimes impossible and sometimes in between Carlos Guestrin
35 Summary of GMM Concept Estimate a density based on x 1,,x N p(x i, µ, ) = 1 (a) KX z i =1 z in (x i µ z i, z i) Complete data labeled by true cluster assignments Surface Plot of Joint Density, Marginalizing Cluster Assignments Carlos Guestrin
36 Summary of GMM Components Observations x i 2 R d, x i i =1, 2,...,N Hidden cluster labels z i 2{1, 2,...,K}, i =1, 2,...,N Hidden mixture means µ k 2 R d, k =1, 2,...,K Hidden mixture covariances Hidden mixture probabilities k 2 R d d, k =1, 2,...,K k, KX k =1 k=1 Gaussian mixture marginal and conditional likelihood : KX p(x i, µ, ) = z i p(x i z i,µ, ) z i =1 p(x i z i,µ, ) =N (x i µ z i, z i) Carlos Guestrin
37 Application to Document Modeling Machine Learning CSEP546 Carlos Guestrin University of Washington February 18, 2014 Carlos Guestrin
38 Cluster Documents Cluster documents based on topic Carlos Guestrin
39 Document Representation Bag of words model Carlos Guestrin
40 Issues with Document Representation Words counts are bad for standard similarity metrics Term Frequency Inverse Document Frequency (tf-idf) Increase importance of rare words Carlos Guestrin
41 TF-IDF Term frequency: tf(t, d) = Could also use {0, 1}, 1 + logt f(t, d),... Inverse document frequency: idf(t, D) = tf-idf: tfidf(t, d, D) = High for document d with high frequency of term t (high term frequency ) and few documents containing term t in the corpus (high inverse doc frequency ) Carlos Guestrin
42 A Generative Model Documents: Associated topics: Parameters simple mixture of Gaussians: Carlos Guestrin
43 What you get from mixture model for documents Words give topic: Topic proportions: Topic distribution of each document: Carlos Guestrin
44 Results from Wikipedia data using similar model (LDA) Carlos Guestrin
45 Expectation Maximization Machine Learning CSEP546 Carlos Guestrin University of Washington February 18, 2014 Carlos Guestrin
46 Next back to Density Estimation What if we want to do density estimation with multimodal or clumpy data? Carlos Guestrin
47 Learning Model Parameters Want to learn model parameters Mixture of 3 Gaussians Our actual observations (b) C. Bishop, Pattern Recognition & Machine Learning Carlos Guestrin
48 ML Estimate of Mixture Model Params Log likelihood L x ( ), log p({x i } ) = X i log X z i p(x i,z i ) Want ML estimate ˆ ML = Neither convex nor concave and local optima Carlos Guestrin
49 Complete Data Imagine we have an assignment of each x i to a cluster Our actual observations 1 (a) 1 (b) Complete data labeled by true cluster assignments C. Bishop, Pattern Recognition & Machine Learning Carlos Guestrin
50 If complete data were observed z i Assume class labels were observed in addition to L x,z ( ) = X log p(x i,z i ) i x i Compute ML estimates Separates over clusters k! Example: mixture of Gaussians (MoG) = { k,µ k, k } K k=1 Carlos Guestrin
51 Cluster Responsibilities We must infer the cluster assignments from the observations (c) Posterior probabilities of assignments to each cluster *given* model parameters: r ik = p(z i = k x i,, )= Soft assignments to clusters C. Bishop, Pattern Recognition & Machine Learning Carlos Guestrin
52 Iterative Algorithm Motivates a coordinate ascent-like algorithm: 1. Infer missing values given estimate of parameters 2. Optimize parameters to produce new given filled in data 3. Repeat z i ˆ ˆ z i Example: MoG 1. Infer responsibilities r ik = p(z i = k x i, ˆ (t 1) )= 2. Optimize parameters max w.r.t. k : max w.r.t. k : Carlos Guestrin
53 E.M. Convergence EM is coordinate ascent on an interesting potential function Coord. ascent for bounded pot. func. è convergence to a local optimum guaranteed This algorithm is REALLY USED. And in high dimensional state spaces, too. E.G. Vector Quantization for Speech Data Carlos Guestrin
54 Gaussian Mixture Example: Start Carlos Guestrin
55 After first iteration Carlos Guestrin
56 After 2nd iteration Carlos Guestrin
57 After 3rd iteration Carlos Guestrin
58 After 4th iteration Carlos Guestrin
59 After 5th iteration Carlos Guestrin
60 After 6th iteration Carlos Guestrin
61 After 20th iteration Carlos Guestrin
62 Some Bio Assay data Carlos Guestrin
63 GMM clustering of the assay data Carlos Guestrin
64 Resulting Density Estimator Carlos Guestrin
65 Initialization In mixture model case where there are many ways to initialize the EM algorithm Examples: y i = {z i,x i } Choose K observations at random to define each cluster. Assign other observations to the nearest centriod to form initial parameter estimates Pick the centers sequentially to provide good coverage of data Grow mixture model by splitting (and sometimes removing) clusters until K clusters are formed Can be quite important to convergence rates in practice Carlos Guestrin
66 Label switching Color = label does not matter Can switch labels and likelihood is unchanged Carlos Guestrin
67 What you should know K-means for clustering: algorithm converges because it s coordinate ascent EM for mixture of Gaussians: How to learn maximum likelihood parameters (locally max. like.) in the case of unlabeled data Remember, E.M. can get stuck in local minima, and empirically it DOES EM is coordinate ascent Carlos Guestrin
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