Berkeley COMPSCI 188  Lecture 25: Clustering (39 pages)
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Lecture 25: Clustering
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 Compsci 188  Introduction to Artificial Intelligence
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CS 188 Artificial Intelligence Fall 2006 Lecture 25 Clustering 11 30 2006 Dan Klein UC Berkeley Announcements Project 4 is up Pacman contest Watch the web page for final review info after last class Properties of Perceptrons Separability some parameters get the training set perfectly correct Separable Convergence if the training is separable perceptron will eventually converge binary case Mistake Bound the maximum number of mistakes binary case related to the margin or degree of separability Non Separable Non Linear Separators Data that is linearly separable with some noise works out great x 0 But what are we going to do if the dataset is just too hard x 0 How about mapping data to a higher dimensional space x2 0 x This and next few slides adapted from Ray Mooney UT Non Linear Separators General idea the original feature space can always be mapped to some higher dimensional feature space where the training set is separable x x Support Vector Machines Several related problems with perceptron Can thrash around No telling which separator you get Once you make an update can t retract it SVMs address these problems Converge to globally optimal parameters Good choice of separator maximum margin Find sparse vectors dual sparsity Linear Separators Binary classification can be viewed as the task of separating classes in feature space w x 0 w x 0 w x 0 Linear Separators Which of the linear separators is optimal Classification Margin Distance from example xi to the separator is r Examples closest to the hyperplane are support vectors Margin of the separator is the distance between support vectors r Support Vector Machines Maximizing the margin good according to intuition and PAC theory Only support vectors matter other training examples are ignorable Support vector machines SVMs find the separator with max margin Mathematically gives a quadratic program which calculates alphas Basically SVMs are perceptrons with smarter update counts Today Clustering K means Similarity Measures Agglomerative clustering Case based reasoning K nearest neighbors Collaborative filtering Recap Classification Classification systems Supervised learning Make a rational prediction given evidence We ve seen several methods for this Useful when you have labeled data or can get it Clustering Clustering systems Unsupervised learning Detect patterns in unlabeled data E g group emails or search results E g find categories of customers E g detect anomalous program executions Useful when don t know what you re looking for Requires data but no labels Often get gibberish Clustering Basic idea group together similar instances Example 2D point patterns What could similar mean One option small squared Euclidean distance K Means An iterative clustering algorithm Pick K random points as cluster centers means Alternate Assign data instances to closest mean Assign each mean to the average of its assigned points Stop when no points assignments change K Means Example K Means as Optimization Consider the total distance to the means means points assignments Each iteration reduces phi Two stages each iteration Update assignments fix means c change assignments a Update means fix assignments a change means c Phase I Update Assignments For each point re assign to closest mean Can only decrease total distance phi Phase II Update Means Move each mean to the average of its assigned points Also can only decrease total distance Why Fun fact the point y with minimum squared Euclidean distance to a set of points x is their mean Initialization K means is nondeterministic 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 K Means Getting Stuck A local optimum Why doesn t this work out like the earlier example with the purple taking over half the blue K Means Questions Will K means converge To a global optimum Will it always find the true patterns in the data If the patterns are very very clear Will it find something interesting Do people ever use it How many clusters to pick Clustering for Segmentation Quick taste of a simple vision algorithm Idea break images into manageable regions for visual processing object recognition activity detection etc http www cs washington edu research imagedatabase demo kmcluster Representing Pixels Basic representation of pixels 3 dimensional color vector r g b Ranges r g b in 0 1 What will happen if we cluster the pixels in an image using this representation Improved representation for segmentation 5 dimensional vector r g b x y Ranges x in 0 M y in 0 N Bigger M N makes position more important How does this change the similarities Note real vision systems use more sophisticated encodings which can capture intensity texture shape and so on K Means Segmentation Results depend on initialization Why Note best systems use graph segmentation algorithms Other Uses of K Means Speech recognition can use to quantize wave slices into a small number of types SOTA work with multivariate continuous features Document clustering detect similar documents on the basis of shared words SOTA use probabilistic models which operate on topics rather than words Agglomerative Clustering Agglomerative clustering First merge very similar instances Incrementally build larger clusters out of smaller clusters Algorithm Maintain a set of clusters Initially each instance in its own cluster Repeat Pick the two closest clusters Merge them into a new cluster Stop when there s only one cluster left Produces not one clustering but a family of clusterings represented by a dendrogram Agglomerative Clustering How should we define closest for clusters with multiple elements Many options Closest pair single link clustering Farthest pair complete link clustering Average of all pairs Distance between centroids broken Ward s method my pick like kmeans Different choices create different clustering behaviors Agglomerative Clustering Complete Link farthest vs Single Link closest Back to Similarity K means naturally operates in Euclidean space why Agglomerative clustering didn t require any mention of averaging Can use any function which takes two instances and returns a similarity Kernelized clustering if your similarity function has the right properties can adapt k means too Kinds of similarity functions Euclidian dot product Weighted Euclidian Edit distance between strings Anything else Collaborative Filtering Ever wonder how online
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