Formal Foundations of Clustering Margareta Ackerman Work with Shai Ben-David, Simina Branzei, and David Loker

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. The Theory-Practice Gap 2

“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 The Theory-Practice Gap

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 ill-defined Inherent Obstacles: Clustering is ill-defined

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 Inherent Obstacles

6 Differences in Input/Output Behavior of Clustering Algorithms

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

12 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)

13 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).

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

15 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.

16 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 is 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.

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

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 Weighted Clustering

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. 19 Other Reasons to Add Weight: An Example Other Reasons to Add Weight: An Example Accessibility of different institutions to a station may have varying importance. The weighted setting enables a convenient method for prioritizing certain landmarks.

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

X For a finite domain set X, a weight function w: X →R + defines the weight of every element. X 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

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

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

24 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

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

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

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

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

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

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

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

PartitionalHierarchical 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 CutAverage Linkage Classification 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 Outline

PartitionalHierarchical Weight RobustMin Diameter K-center Single Linkage Complete Linkage Weight SensitiveK-means k-medoids k-median, min-sum Ward’s Method Bisecting K-means Weight Considering Ratio Cut Average Linkage Classification

35 Zooming Into: Weight Sensitive Algorithms Zooming Into: Weight Sensitive Algorithms We show that k-means is weight-sensitive. A(X, d) SXkw A(w[X],d,k) S 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.

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

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

38 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

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

Average Linkage is Weight Considering (X,d) Range(X,d) >1: (X,d) where Range(X,d) >1: ABCD A BCD E 2+2ϵ11 1+ϵ1+ϵ ABCDE ϵ E A B CDE

41 When is Average Linkage Sensitive to Weight? When is Average Linkage Sensitive to Weight? We showed that Average-Linkage is weight- considering. 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.

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

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

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

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

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

47 Characterizing When Average Linkage is Sensitive to Weight: Proof (cnt.) (X,d) Lemma: If there is a nice dendrogram for (X,d), then Average-Linkage outputs it. Proof Sketch: (w[X],d) 1)Assume that (w[X],d) has a nice dendrogram. AL(w[X],d). 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.

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

49 Proof: Cxyz Since C is not nice, there exist points x, y, and z, so that xyC x and y are belong to the same cluster in C xz x and z belong to difference clusters d(x,z) < d(x,y) yet d(x,z) < d(x,y) x y z xzxyC 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. C AL(w[X],d)range(X,d)>1 Lemma: If a clustering C that is not nice appears in AL(w[X],d), then range(X,d)>1. Characterizing When Average Linkage Responds to Weight: Proof (cnt.)

50 Characterizing When Average Linkage is Sensitive to Weight Characterizing When Average Linkage is Sensitive to Weight (X,d) 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. Range(AL(X,d)) = 1 (X,d) Theorem: Range(AL(X,d)) = 1 if and only if (X,d) has a nice dendrogram.

51 Zooming Into: Weight Robust Algorithms 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.

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

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

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)). Future Directions