K-Means Clustering 3/3/17
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1 K-Means Clustering 3/3/17
2 Unsupervised Learning We have a collection of unlabeled data points. We want to find underlying structure in the data. Examples: Identify groups of similar data points. Clustering Find a better basis to represent the data. Principal component analysis Compress the data to a shorter representation. Auto-encoders
3 Unsupervised Learning We have a collection of unlabeled data points. We want to find underlying structure in the data. Applications: Generating the input representation for another AI or ML algorithm. Clusters could lead to states in a state space search or MDP model. A new basis could be the input to a classification or regression algorithm. Making data easier to understand, by identifying what s important and/or discarding what isn t.
4 The Goal of Clustering Given a bunch of data, we want to come up with a representation that will simplify future reasoning. Key idea: group similar points into clusters. Examples: Identifying objects in sensor data Detecting communities in social networks Constructing phylogenetic trees of species Making recommendations from similar users
5 EM Algorithm E step: expectation terrible name Classify the data using the current model. M step: maximization slightly less terrible name Generate the best model using the current classification of the data. Initialize the model, then alternate E and M steps until convergence. Note: The EM algorithm has many variations, including some that have nothing to do with clustering.
6 K-Means Algorithm Model: k clusters each represented by a centroid. E step: Assign each point to the closest centroid. M step: Move each centroid to the mean of the points assigned to it. Convergence: we ran an E step where no points had their assignment changed.
7 K-Means Example
8 Initializing K-Means Reasonable options: 1. Start with a random E step. Randomly assign each point to a cluster in {1, 2,, k}. 2. Start with a random M step. a) Pick random centroids within the maximum range of the data. b) Pick random data points to use as initial centroids.
9 K-Means in Action
10 Another EM Example: GMMs GMM: Gaussian mixture model A Gaussian distribution is a multivariate generalization of a normal distribution (the classic bell curve). A Gaussian mixture is a distribution comprised of several independent Gaussians. If we model our data as a Gaussian mixture, we re saying that each data point was a random draw from one of several Gaussian distributions (but we may not know which).
11 EM for Gaussian Mixture Models Model: data drawn from a mixture of k Gaussians E step: Compute the (log) likelihood of the data Each point s probability of being drawn from each Gaussian. M step: Update the mean and covariance of each Gaussian. Weighted by how responsible that Gaussian was for each data point.
12 How do we pick K? There s no hard rule. Sometimes the application for which the clusters will be used dictates k. If k can be flexible, then we need to consider the tradeoffs: Higher k will always decrease the error (increase the likelihood). Lower k will always produce a simpler model.
13 Hierarchical Clustering Organizes data points into a hierarchy. Every level of the binary tree splits the points into two subsets. Points in a subset should be more similar than points in different subsets. The resulting clustering can be represented by a dendrogram.
14 Direction of Clustering Agglomerative (bottom-up) Each point starts in its own cluster. Repeatedly merge the two most-similar clusters until only one remains. Divisive (top-down) All points start in a single cluster. Repeatedly split the data into the two most self-similar subsets. Either version can stop early if a specific number of clusters is desired.
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