Clustering Lecture 14
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1 Clustering Lecture 14 David Sontag New York University Slides adapted from Luke Zettlemoyer, Vibhav Gogate, Carlos Guestrin, Andrew Moore, Dan Klein
2 Clustering: Unsupervised learning Clustering Requires data, but no labels Detect patterns e.g. in Group s or search results Customer shopping patterns Regions of images Useful when don t know what you re looking for But: can get gibberish
3 Clustering Basic idea: group together similar instances Example: 2D point patterns
4 Clustering Basic idea: group together similar instances Example: 2D point patterns
5 Clustering Basic idea: group together similar instances Example: 2D point patterns What could similar mean? One option: small Euclidean distance (squared) dist(~x, ~y) = ~x ~y 2 2 Clustering results are crucially dependent on the measure of similarity (or distance) between points to be clustered
6 Clustering algorithms 8+'%(%(,) +"*,'(%-.$ 9:"+%; <.&+)$ A2&0%'+"!"#$%&'()* /(&'+'0-(0+" +"*,'(%-.$ 1,%%,. #2 3 +**",.&'+%(4& 5,2 6,7) 3 6(4($(4&
7 Clustering examples Image segmenta3on Goal: Break up the image into meaningful or perceptually similar regions [Slide from James Hayes]
8 Clustering examples Clustering gene expression data Eisen et al, PNAS 1998
9 Cluster news ar3cles Clustering examples
10 Clustering examples Cluster people by space and 3me [Image from Pilho Kim]
11 Clustering languages Clustering examples [Image from scienceinschool.org]
12 Clustering examples Clustering languages [Image from dhushara.com]
13 Clustering examples Clustering species ( phylogeny ) [Lindblad-Toh et al., Nature 2005]
14 Clustering search queries Clustering examples
15 An iterative clustering algorithm Initialize: Pick K random points as cluster centers Alternate: 1. Assign data points to closest cluster center 2. Change the cluster center to the average of its assigned points Stop when no points assignments change K-Means
16 An iterative clustering algorithm Initialize: Pick K random points as cluster centers Alternate: 1. Assign data points to closest cluster center 2. Change the cluster center to the average of its assigned points Stop when no points assignments change K-Means
17 K- means clustering: Example Pick K random points as cluster centers (means) Shown here for K=2 17
18 K- means clustering: Example Iterative Step 1 Assign data points to closest cluster center 18
19 K- means clustering: Example Iterative Step 2 Change the cluster center to the average of the assigned points 19
20 K- means clustering: Example Repeat undl convergence 20
21 K- means clustering: Example 21
22 K- means clustering: Example 22
23 K- means clustering: Example 23
24 ProperDes of K- means algorithm Guaranteed to converge in a finite number of iteradons Running Dme per iteradon: 1. Assign data points to closest cluster center O(KN) time 2. Change the cluster center to the average of its assigned points O(N)
25 !"#$%& '(%)#*+#%,#!"#$%&'($ -. /01 2 (340"05#!" 6. /01!# (340"05# 7$8# 3$*40$9 :#*0)$40)# (; $%:  4( 5#*(2 <# =$)#!"#$ % &' ()#*+,!"#$ - &' ()#*+,!"#$%& 4$8#& $% $94#*%$40%+ (340"05$40(% $33*($,=2 #$,= &4#3 0& +>$*$%4##: 4( :#,*#$&# 4=# (?@#,40)# A4=>& +>$*$%4##: 4(,(%)#*+# [Slide from Alan Fern]
26 Example: K-Means for Segmentation K=2 K =2 K =3 K = 10 Goal of Segmentation is Original image to partition an image into regions each of which has reasonably homogenous visual appearance.
27 Example: K-Means for Segmentation K=2 K =2 K=3 K =3 K=10 K = 10 Original image
28 Example: K-Means for Segmentation K=2 K =2 K=3 K =3 K=10 K = 10 Original image
29 Example: Vector quantization FIGURE Sir Ronald A. Fisher ( ) was one of the founders of modern day statistics, to whom we owe maximum-likelihood, sufficiency, and many other fundamental concepts. The image on the left is a grayscale image at 8 bits per pixel. The center image is the result of 2 2 block VQ, using 200 code vectors, with a compression rate of 1.9 bits/pixel. The right image uses only four code vectors, with a compression rate of 0.50 bits/pixel [Figure from Hastie et al. book]
30 Initialization K-means algorithm is a heuristic Requires initial means It does matter what you pick! What can go wrong? Various schemes for preventing this kind of thing: variance-based split / merge, initialization heuristics
31 K-Means Getting Stuck A local optimum: Would be better to have one cluster here and two clusters here
32 K-means not able to properly cluster Y X
33 Changing the features (distance function) can help R θ
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