Clustering Paolo Ferragina Dipartimento di Informatica Università di Pisa This is a mix of slides taken from several presentations, plus my touch !

Objectives of Cluster Analysis  Finding groups of objects such that the objects in a group will be similar (or related) to one another and different from (or unrelated to) the objects in other groups Inter-cluster distances are maximized Intra-cluster distances are minimized Competing objectives The commonest form of unsupervised learning

Yahoo! Hierarchy isn’t clustering but is the kind of output you want from clustering dairy crops agronomy forestry AI HCI craft missions botany evolution cell magnetism relativity courses agriculturebiologyphysicsCSspace... … (30)

Google News: automatic clustering gives an effective news presentation metaphor

For improving search recall  Cluster hypothesis - Documents in the same cluster behave similarly with respect to relevance to information needs  Therefore, to improve search recall:  Cluster docs in corpus a priori  When a query matches a doc D, also return other docs in the cluster containing D  Hope if we do this: The query “car” will also return docs containing automobile Sec. 16.1

For better visualization/navigation of search results Sec. 16.1

Issues for clustering  Representation for clustering  Document representation  Vector space? Normalization?  Need a notion of similarity/distance  How many clusters?  Fixed a priori?  Completely data driven? Sec. 16.2

Notion of similarity/distance  Ideal: semantic similarity  Practical: term-statistical similarity  Docs as vectors  We will use cosine similarity.  For many algorithms, easier to think in terms of a distance (rather than similarity) between docs.

Clustering Algorithms  Flat algorithms  Create a set of clusters  Usually start with a random (partial) partitioning  Refine it iteratively  K means clustering  Hierarchical algorithms  Create a hierarchy of clusters (dendogram)  Bottom-up, agglomerative  Top-down, divisive

Hard vs. soft clustering  Hard clustering: Each document belongs to exactly one cluster  More common and easier to do  Soft clustering: Each document can belong to more than one cluster.  Makes more sense for applications like creating browsable hierarchies  News is a proper example  Search results is another example

Flat & Partitioning Algorithms  Given: a set of n documents and the number K  Find: a partition in K clusters that optimizes the chosen partitioning criterion  Globally optimal  Intractable for many objective functions  Ergo, exhaustively enumerate all partitions  Locally optimal  Effective heuristic methods: K-means and K-medoids algorithms

K-Means  Assumes documents are real-valued vectors.  Clusters based on centroids (aka the center of gravity or mean) of points in a cluster, c:  Reassignment of instances to clusters is based on distance to the current cluster centroids. Sec. 16.4

K-Means Algorithm Select K random docs {s 1, s 2,… s K } as seeds. Until clustering converges (or other stopping criterion): For each doc d i : Assign d i to the cluster c r such that dist(d i, s r ) is minimal. For each cluster c j s j =  (c j ) Sec. 16.4

K Means Example (K=2) Pick seeds Reassign clusters Compute centroids x x Reassign clusters x x x x Compute centroids Reassign clusters Converged! Sec. 16.4

Termination conditions  Several possibilities, e.g.,  A fixed number of iterations.  Doc partition unchanged.  Centroid positions don’t change. Sec. 16.4

Convergence  Why should the K-means algorithm ever reach a fixed point?  K-means is a special case of a general procedure known as the Expectation Maximization (EM) algorithm  EM is known to converge  Number of iterations could be large  But in practice usually isn’t Sec. 16.4

Convergence of K-Means  Define goodness measure of cluster c as sum of squared distances from cluster centroid:  G(c,s) = Σ i (d i – s c ) 2 (sum over all d i in cluster c)  G(C,s) = Σ c G(c,s)  Reassignment monotonically decreases G  It is a coordinate descent algorithm (opt one component at a time)  At any step we have some value for G(C,s) 1) Fix s, optimize C  assign d to the closest centroid  G(C’,s) < G(C,s) 2) Fix C’, optimize s  take the new centroids  G(C’,s’) < G(C’,s) < G(C,s) The new cost is smaller than the original one  local minimum Sec. 16.4

Time Complexity  The centroids are K  Each doc/centroid consists of M dimensions  Computing distance btw vectors is O(M) time.  Reassigning clusters: Each doc compared with all centroids, O(KNM) time.  Computing centroids: Each doc gets added once to some centroid, O(NM) time. Assume these two steps are each done once for I iterations: O(IKNM). Sec. 16.4

Seed Choice  Results can vary based on random seed selection.  Some seeds can result in poor convergence rate, or convergence to sub-optimal clusterings.  Select good seeds using a heuristic  doc least similar to any existing centroid  According to a probability distribution that depends inversely-proportional on the distance from the other current centroids In the above, if you start with B and E as centroids you converge to {A,B,C} and {D,E,F} If you start with D and F you converge to {A,B,D,E} {C,F} Example showing sensitivity to seeds Sec. 16.4

How Many Clusters?  Number of clusters K is given  Partition n docs into predetermined number of clusters  Finding the “right” number of clusters is part of the problem  Can usually take an algorithm for one flavor and convert to the other.

Bisecting K-means Variant of K-means that can produce a partitional or a hierarchical clustering

Bisecting K-means Example

K-means Pros  Simple  Fast for low dimensional data  It can find pure sub-clusters if large number of clusters is specified (but, over-partitioning) Cons  K-Means cannot handle non-globular data of different sizes and densities  K-Means will not identify outliers  K-Means is restricted to data which has the notion of a center (centroid)

Hierarchical Clustering  Build a tree-based hierarchical taxonomy (dendrogram) from a set of documents  One approach: recursive application of a partitional clustering algorithm animal vertebrate fish reptile amphib. mammal worm insect crustacean invertebrate Ch. 17

Strengths of Hierarchical Clustering  No assumption of any particular number of clusters  Any desired number of clusters can be obtained by ‘cutting’ the dendogram at the proper level  They may correspond to meaningful taxonomies  Example in biological sciences (e.g., animal kingdom, phylogeny reconstruction, …)

Hierarchical Agglomerative Clustering (HAC)  Starts with each doc in a separate cluster  Then repeatedly joins the closest pair of clusters, until there is only one cluster.  The history of mergings forms a binary tree or hierarchy. Sec. 17.1

Closest pair of clusters  Single-link  Similarity of the closest points, the most cosine-similar  Complete-link  Similarity of the farthest points, the least cosine-similar  Centroid  Clusters whose centroids are the closest (or most cosine- similar)  Average-link  Clusters whose average distance/cosine between pairs of elements is the smallest Sec. 17.2

How to Define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1p2p3p4p Similarity?  Single link (MIN)  Complete link (MAX)  Centroids  Average Proximity Matrix

How to Define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1p2p3p4p Proximity Matrix  MIN  MAX  Centroids  Average

How to Define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1p2p3p4p Proximity Matrix  MIN  MAX  Centroids  Average

How to define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1p2p3p4p Proximity Matrix  MIN  MAX  Centroids  Average 

How to Define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1p2p3p4p Proximity Matrix  MIN  MAX  Centroids  Average

Starting Situation  Start with clusters of individual points and a proximity matrix p1 p3 p5 p4 p2 p1p2p3p4p Proximity Matrix

Intermediate Situation  After some merging steps, we have some clusters C1 C4 C2 C5 C3 C2C1 C3 C5 C4 C2 C3C4C5 Proximity Matrix

Intermediate Situation  We want to merge the two closest clusters (C2 and C5) and update the proximity matrix. C1 C4 C2 C5 C3 C2C1 C3 C5 C4 C2 C3C4C5 Proximity Matrix

After Merging  The question is “How do we update the proximity matrix?” C1 C4 C2 U C5 C3 ? ? ? ? ? C2 U C5 C1 C3 C4 C2 U C5 C3C4 Proximity Matrix

Cluster Similarity: MIN or Single Link  Similarity of two clusters is based on the two most similar (closest) points in the different clusters  Determined by one pair of points, i.e., by one link in the proximity graph

Hierarchical Clustering: MIN Nested ClustersDendrogram

Strength of MIN Original Points Two Clusters Can handle non-elliptical shapes

Limitations of MIN Original Points Two Clusters Sensitive to noise and outliers

Cluster Similarity: MAX or Complete Linkage  Similarity of two clusters is based on the two least similar (most distant) points in the different clusters  Determined by all pairs of points in the two clusters 12345

Hierarchical Clustering: MAX Nested ClustersDendrogram

Strength of MAX Original Points Two Clusters Less susceptible to noise and outliers

Limitations of MAX Original Points Two Clusters Tends to break large clusters Biased towards globular clusters

Cluster Similarity: Average  Proximity of two clusters is the average of pairwise proximity between points in the two clusters

Hierarchical Clustering: Average Nested ClustersDendrogram

Hierarchical Clustering: Comparison Average MIN MAX

How to evaluate clustering quality ? Assesses a clustering with respect to ground truth … requires labeled data Produce the gold standard is costly !! Sec. 16.3

A different approach: Co-Clustering  Co-occurrence Matrices Characteristics  Data sparseness  High dimension  Noise  Is it possible to combine the document and term clustering together?  Yes, Co-Clustering  Simultaneously cluster the rows and columns of the co- occurrence matrix.

Information-Theoretic Co-Clustering  View co-occurrence matrix as a joint probability distribution between row & column random variables  We seek a hard-clustering of both dimensions such that loss in “Mutual Information” is minimized given a fixed no. of row & col. clusters

Example  Mutual Information btw random variables X and Y: It can be verified that this is the minimum mutual information loss Column grouping identifies the key terms of a cluster !!