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### Formal Foundations of Clustering

Margareta Ackerman

mackerma@caltech.edu

Work with Shai Ben-David, SiminaBranzei, 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.

“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.

- 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

- 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

Clustering Algorithm Selection

There are 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?

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

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”

- Useful for distinguishing clustering algorithms

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.

- 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)

- Characterizations of Single-Linkage
- Partitional Setting: BosahZehad 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).

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

- 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.

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

- Outline

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

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.

- 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.

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.

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 xX →R + u {0}

is the distance defined between the domain points.

- 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

- 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 appearsin A(w[X],d) if its clusters are in the dendrogram

- 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

Towards Basic CategoriesRange(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

- Outline

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

- Weight Robust

A is weight-robustif for all (X, d), |Range(X,d)| = 1.

- A never responds to weight.

- Weight Sensitive

- A is weight-sensitive if for all (X, d),|Range(X,d)| > 1.
- A always respondsto weight.

- Weight Considering

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

- 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:
- ∃ (X, d) where |Range(X,d)|=1.
- ∃ (X, d) where |Range(X,d)|>1.

Outline

- Connecting To Applications

- The desired category depends on the application.

In the facility allocation example above, a weight-sensitive algorithm may be preferred.

- In phylogeny, where sampling procedures can be highly biased, some degree of weight robustness may be desired.

For the weight considering algorithms, we fully characterize when they are sensitive to weight.

Outline

- Outline

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

- Weight Sensitive Algorithms

- We show that k-means is weight-sensitive.
- Ais weight-separableif for any data set (X, d) and subset S of X with at most k points, ∃ wso that A(w[X],d,k) separates all points of S.
- Fact: Every algorithm that is weight-separable is also weight-sensitive.

Proof:

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

- Theorem:k-means is weight-sensitive.

- 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
- There exists (X, d) where |Range(X,d)|=1.
- There exists (X, d) where |Range(X,d)|>1.

- 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) where Range(X,d) =1:
- The same dendrogram is output
- for every weight function.

A

B

C

D

B

C

D

A

Average Linkage is Weight Considering

- (X,d) where Range(X,d) >1:

1+ϵ

1

1

2+2ϵ

A

B

C

D

E

B

C

D

A

E

1+ϵ

1

1

2+2ϵ

A

B

C

D

E

B

A

C

D

E

- 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.

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

Nice

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

Nice

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

Not nice

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.

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:
- If there is a nice dendrogram for (X,d),then
- Average-Linkage outputs it.
- If a clustering that is not nice appears in dendrogramAL(w[X],d) for some w, then Range(AL(X,d)) > 1.

Characterizing When Average Linkage is Sensitive to Weight: Proof (cnt.)

- Lemma:If there is a nice dendrogram for (X,d),then Average-Linkage outputs it.

- Proof Sketch:
- Assume that (w[X],d) has a nice dendrogram.
- Main idea: Show that every nice clustering of the data appears in AL(w[X],d).
- For that, we show that each cluster in a nice clustering is formed by the algorithm.

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.

Characterizing When Average Linkage Responds to Weight: Proof (cnt.)

- Lemma:If a clustering Cthat is not nice appears in 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 andz are sufficiently heavier than all other points, then x and z will be merged before x and y, so C will not be formed.

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.

- 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

- Outline

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

- 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)).

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