Weighted Clustering Margareta Ackerman Work with Shai BenDavid

Weighted Clustering Margareta Ackerman Work with Shai Ben-David, Simina Branzei, and David Loker

The Theory-Practice Gap Clustering is one of the most widely used tools for exploratory data analysis. Social Sciences Biology Astronomy Computer Science …. All apply clustering to gain a first understanding of the structure of large data sets. 2

The Theory-Practice Gap “While the interest in and application of cluster analysis has been rising rapidly, the abstract nature of the tool is still poorly understood” (Wright, 1973) “There has been relatively little work aimed at reasoning about clustering independently of any particular algorithm, objective function, or generative data model” (Kleinberg, 2002) Both statements still apply today. 3

Inherent Obstacles: Clustering is ill-defined Clustering aims to assign data into groups of similar items Beyond that, there is very little consensus on the definition of clustering 4

Inherent Obstacles • Clustering is inherently ambiguous – There may be multiple reasonable clusterings – There is usually no ground truth • There are many clustering algorithms with different (often implicit) objective functions • Different algorithms have radically different input-output behaviour 5

Differences in Input/Output Behavior of Clustering Algorithms 6

Differences in Input/Output Behavior of Clustering Algorithms 7

Clustering Algorithm Selection There a wide variety of clustering algorithms, which can produce very different clusterings. How should a user decide which algorithm to use for a given application? 8

Clustering Algorithm Selection Users rely on cost related considerations: running times, space usage, software purchasing costs, etc… There is inadequate emphasis on input-output behaviour 9

Our Framework for Algorithm Selection We propose a framework that lets a user utilize prior knowledge to select an algorithm • Identify properties that distinguish between different input-output behaviour of clustering paradigms • The properties should be: 1) Intuitive and “user-friendly” 2) Useful for distinguishing clustering algorithms 10

Our Framework for Algorithm Selection In essence, our goal is to understand fundamental differences between clustering methods, and convey them formally, clearly, and as simply as possible. 11

Previous Work • Axiomatic perspective • Impossibility Result: Kleinberg (NIPS, 2003) • Consistent axioms for quality measures: Ackerman & Ben-David (NIPS, 2009) • Axioms in the weighted setting: Wright (Pattern Recognition, 1973) 12

Previous Work • Characterizations of Single-Linkage • Partitional Setting: Bosah Zehad and Ben-David (UAI, 2009) • Hierarchical Setting: Jarvis and Sibson (Mathematical Taxonomy, 1981) and Carlsson and Memoli (JMLR, 2010). • Characterizations of Linkage-Based Clustering • Partitional Setting: Ackerman, Ben-David, and Loker (COLT, 2010). • Hierarchical Setting: Ackerman & Ben-David (IJCAI, 2011). 13

Previous Work • Classifications of clustering methods • Fischer and Van Ness (Biometrica, 1971) • Ackerman, Ben-David, and Loker (NIPS, 2010) 14

What’s Left To Be Done? Despite much work on clustering properties, some basic questions remain unanswered. Consider some of the most popular clustering methods: k-means, single-linkage, average-linkage, etc… • What are the advantages of k-means over other methods? • Previous classifications are missing key properties. 15

Our Contributions (at a high level) We indentify 3 fundamental categories that clearly delineate some essential differences between common clustering methods The strength of these categories lies in their simplicity. We hope this gives insight into core differences between popular clustering methods. To define these categories, we first present the weighted clustering setting. 16

Outline • • Formal framework Categories and classification A result from each category Conclusions and future work

Weighted Clustering Every element is associated with a real valued weight, representing its mass or importance. Generalizes the notion of element duplication. Algorithm design, particularly design of approximation algorithms, is often done in this framework. 18

Other Reasons to Add Weight: An Example • Apply clustering to facility allocation, such as the placement of police stations in a new district. • The distribution of stations should enable quick access to most areas in the district. • Accessibility of different institutions to a station may have varying importance. • The weighted setting enables a convenient method for prioritizing certain landmarks. 19

Algorithms in the Weighted Clustering Setting Traditional clustering algorithms can be readily translated into the weighted setting by considering their behavior on data containing element duplicates. 20

Formal Setting • For a finite domain set X, a weight function w: X →R+ defines the weight of every element. • For a finite domain set X, a distance function d: X x X →R + u {0} is the distance defined between the domain points.

Formal Setting Partitional Clustering Algorithm (X, d) denotes unweighted data (w[X], d) denotes weighted data A Partitional Algorithm maps Input: (w[X], d, k) to Output: a k-partition (k-clustering) of X

Formal Setting Hierarchical Clustering Algorithm A Hierarchical Algorithm maps Input: (w[X], d) to Output: dendrogram of X A dendrogram of (X, d) is a strictly binary tree whose leaves correspond to elements of X C appears in A(w[X], d) if its clusters are in the dendrogram

Our Contributions • We utilize the weighted framework to indentify 3 fundamental categories, describing how algorithms respond to weight. • Classify traditional algorithms according to these categories • Fully characterize when different algorithms react to weight 24

Towards Basic Categories Range(X, d) PARTITIONAL: Range(A(X, d, k)) = {C | ∃ w s. t. C=A(w[X], d)} The set of clusterings that A outputs on (X, d) over all possible weight functions. HIERARCHICAL: Range(A(X, d)) = {D | ∃ w s. t. D=A(w[X], d)} The set of dendrograms that A outputs on (X, d) over all possible weight functions.

Outline • • Formal framework Categories and classification A result from each category Conclusions and future work

Categories: Weight Robust A is weight-robust if for all (X, d), |Range(X, d)| = 1. A never responds to weight. 27

Categories: Weight Sensitive A is weight-sensitive if for all (X, d) |Range(X, d)| > 1. A always responds to weight. 28

Categories: Weight Considering An algorithm A is weight-considering if 1) There exists (X, d) where |Range(X, d)|=1. 2) There exists (X, d) where |Range(X, d)|>1 A responds to weight on some data sets, but not others. 29

Summary of Categories Range(A(X, d)) = {C | ∃ w such that A(w[X], d) = C} Range(A(X, d)) = {D | ∃ w such that A(w[X], d) = D} Weight-robust: for all (X, d), |Range(X, d)| = 1. Weight-sensitive: for all (X, d), |Range(X, d)| > 1. Weight-considering: 1) ∃ (X, d) where |Range(X, d)|=1. 2) ∃ (X, d) where |Range(X, d)|>1 30

Outline Connecting To Applications The desired category depends on the application. In the facility allocation example above, a weightsensitive algorithm may be preferred. In phylogeny, where sampling procedures can be highly biased, some degree of weight robustness may be desired.

Classification Partitional Hierarchical Weight Robust Min Diameter K-center Single Linkage Complete Linkage Weight Sensitive K-means, k-medoids, k-median, min-sum Ward’s Method Bisecting K-means Weight Considering Ratio Cut Average Linkage For the weight considering algorithms, we fully characterize when they are sensitive to weight.

Outline • • • Formal framework Categories and classification A result from each category Classification of heuristics Conclusions and future work

Classification Partitional Hierarchical Weight Robust Min Diameter Single Linkage Complete Linkage Weight Sensitive K-means Ward’s Method Bisecting K-means Weight Considering Ratio Cut Average Linkage K-center k-medoids k-median, min-sum

Zooming Into: Weight Sensitive Algorithms We show that k-means is weight-sensitive. A is weight-separable if for any data set (X, d) and subset S of X with at most k points, ∃ w so that A(w[X], d, k) separates all points of S. Fact: Every algorithm that is weight-separable is also weight-sensitive. 35

K-means is Weight-Sensitive Theorem: k-means is weight-sensitive. Proof: • Show that k-means is weight-separable • Consider any (X, d) and S⊂X on at least k points • Increase weight of points in S until each belongs to a distinct cluster. 36

Zooming Into: Weight Considering Algorithms • We show that Average-Linkage is Weight Considering. • Characterize the precise conditions under which it is sensitive to weight. Recall: An algorithm A is weight-considering if 1) There exists (X, d) where |Range(X, d)|=1. 2) There exists (X, d) where |Range(X, d)|>1 37

Average Linkage • Average-Linkage is a hierarchical algorithm. • It starts by creating a leaf for every element. • It then repeatedly merges the “closest” clusters using the following linkage function: Average weighted distance between clusters 38

Average Linkage is Weight Considering (X, d) where Range(X, d) =1: A B C D A The same dendrogram is output for every weight function. B C D

Average Linkage is Weight Considering (X, d) where Range(X, d) >1: 1+ϵ 1 A B 1 C 2+2ϵ D E A 1+ ϵ 1 A B 1 C B C D E 2+2ϵ D E A B C D E

When is Average Linkage Sensitive to Weight? We showed that Average-Linkage is weightconsidering. Can we show when it is sensitive to weight? We provide a complete characterization of when Average-Linkage is sensitive to weight, and when it is not. 41

Nice Clustering A clustering is nice if every point is closer to all points within its cluster than to all other points. Nice 42

Nice Clustering A clustering is nice if every point is closer to all points within its cluster than to all other points. Nice 43

Nice Clustering A clustering is nice if every point is closer to all points within its cluster than to all other points. Not nice 44

Characterizing When Average Linkage is Sensitive to Weight A dendrogram is nice if all of its clusterings are nice. Theorem: Range(AL(X, d)) = 1 if and only if (X, d) has a nice dendrogram. 45

Characterizing When Average Linkage is Sensitive to Weight: Proof Theorem: Range(AL(X, d)) = 1 if and only if (X, d) has a nice dendrogram. Proof: Show that: 1) If there is a nice dendrogram for (X, d), (X, d) then Average-Linkage outputs it. 2) If a clustering that is not nice appears in dendrogram AL(w[X], d) for some w, then Range(AL(X, d)) > 1. 1 46

Characterizing When Average Linkage is Sensitive to Weight: Proof (cnt. ) Lemma: If there is a nice dendrogram for (X, d), (X, d) then Average-Linkage outputs it. Proof Sketch: 1) Assume that (w[X], d) has a nice dendrogram. 2) Main idea: Show that every nice clustering of the data appears in AL(w[X], d). 3) For that, we show that each cluster in a nice clustering is formed by the algorithm. 47

Characterizing When Average Linkage is Sensitive to Weight: Proof (cnt. ) Given a nice clustering C, it can be shown that for any clusters Ci and Cj of C, any disjoint subsets Y and Z of Ci, and any subset W of Cj, Y and Z are closer than Y and W. This implies that C appears in the dendrogram. 48

Characterizing When Average Linkage Responds to Weight: Proof (cnt. ) Lemma: If a clustering C that is not nice appears in AL(w[X], d), AL(w[X], d) then range(X, d)>1 Proof: • Since C is not nice, there exist points x, y, and z, so that • x and y are belong to the same cluster in C • x and z belong to difference clusters • yet d(x, z) < d(x, y) • If x, y and z are sufficiently heavier than all other points, then x and z will be merged before x and y, so C will not be formed. 49

Characterizing When Average Linkage is Sensitive to Weight Theorem: Range(AL(X, d)) = 1 if and only if (X, d) has a nice dendrogram. Average Linkage is robust to weight whenever there is a dendrogram of (X, d) consisting of only nice clusterings, and it is sensitive to weight otherwise. 50

Zooming Into: Weight Robust Algorithms These algorithms are invariant to element duplication. Ex. Min-Diameter returns a clustering that minimizes the length of the longest within-cluster edge. As this quantity is not effected by the number of points (or weight) at any location, Min-Diameter is weight robust. 51

Outline • • Introduce framework Present categories and classification Show several results from different categories Conclusions and future work

Conclusions • We introduced three basic categories describing how algorithms respond to weights • We characterize the precise conditions under which algorithms respond to weights • The same results apply in the non-weighted setting for data duplicates • This classification can be used to help select clustering algorithms for specific applications

Future Directions • Capture differences between objective functions similar to k-means (ex. k-medians, k-medoids, min-sum) • Show bounds on the size of the Range of weight considering and weight sensitive methods • Analyze clustering algorithms for categorical data • Analyze clustering algorithms with a noise bucket • Indentify properties that are significant for specific clustering applications (some previous work in this directions by Ackerman, Brown, and Loker (ICCABS, 2012)).

Supplementary Material

Characterizing When Ratio Cut is Weight Responsive Ratio-cut is a similarity based clustering function A clustering is perfect if all within-cluster distances are shorter than all between-cluster distances. A clustering is separation uniform if all betweencluster distances are equal. 56

Characterizing When Ratio Cut is Weight Responsive Theorem: Given a clustering C of (X, s) where every cluster has more than one element, ratio-cut is weight-responsive on C if and only if either • C is not perfect, or • C is not separation-uniform. 57

What about heuristics? • Our analysis for k-means and similar methods is for their corresponding objective functions. • Unfortunately, optimal partitions are NP-hard to find. In practice, heuristics such as the Lloyd method are used. • We analyze several variations of the Lloyd method for k-means, as well as the Partitional Around Medoids (PAM) algorithm for k-medoids. 58

Heuristics Classification Partitional Weight Sensitive • Lloyd with random initialization • K-means++ • PAM Weight Considering • The Lloyd Method with Further Centroid Initialization Note that the more popular heuristics respond to weights in the same way as the k-means and kmedoids objective functions.

Heuristics Classification Partitional Weight Sensitive • Lloyd with random initialization • K-means++ • PAM Weight Considering • The Lloyd Method with Further Centroid Initialization Just like Average-Linkage, the Lloyd Method with Furthest Centroid initialization responds to weight only on clusterings that are not nice.

Classification Partitional Hierarchical Weight Robust • Min Diameter • K-center • Single Linkage • Complete Linkage Weight Sensitive • K-means • k-medoids • k-median • Min-Sum • Randomized Lloyd • k-means++ • Ward’s Method • Bisecting K-means Weight Considering • Ratio Cut • Lloyd with Furthest Centroid • Average Linkage

Wright’s Axioms In 1973, Wright proposed axioms requiring in this setting • Points with zero mass can be treated as nonexistent • Multiple points with mass at the same location are equivalent to one point whose weight is the sum of these masses 62

Characterizing When Average Linkage Responds to Weight: Proof (cnt. ) We use the following lemma to show that if there is a nice dendrogram, then it is output by Avergae-Linkage. Logic: Every nice clustering is output by AL. So if there is a nice dendrog, consider the set of clusters it has. All of them have to appear in any dendrogam output by AL. Because a binary tree always have the same number of nodes, and every node corresponds to a unique cluster, every dendrog produces the same clusters, which means it has the same clusterings as the nice dendrogam. Show by induction that every level in a level-condensed dendrogram has the same clusters. Lemma: Consider any data set (w[X], d). d) If a clustering C of X is nice, then C appears in the dendrogram Average-Linkage(w[X], d) 63

Previous Work • Classifications of clustering methods • Fischer and Van Ness (Biometrica, 1971) 64