CS728Web Clustering II

Lecture 14

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.

(Or one can equivalently phrase it in terms of similarities)

cx

xc

||

1(c)μ

K-Means Algorithm

Let d be the distance measure between instances.Select k random instances {s1, s2,… sk} as seeds.Until clustering converges or other stopping criterion: For each instance xi: Assign xi to the cluster cj such that d(xi, sj) is minimal. (Update the seeds to the centroid of each cluster) For each cluster cj

sj = (cj)

K Means Example(K=2)

Pick seeds

Reassign clusters

Compute centroids

xx

Reassign clusters

xx xx Compute centroids

Reassign clusters

Converged!

Termination conditions

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

Does this mean that the docs in a cluster are

unchanged?

Convergence

Why should the K-means algorithm ever reach a fixed point? A state in which clusters don’t change.

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.

Convergence of K-Means

Define goodness measure of cluster k as sum of squared distances from cluster centroid: Gk = Σi (vi – ck)2 (sum all vi in cluster k)

G = Σk Gk

Reassignment monotonically decreases G since each vector is assigned to the closest centroid.

Recomputation monotonically decreases each Gk since: (mk is number of members in cluster) Σ (vin – a)2 reaches minimum for:

Σ –2(vin – a) = 0

Convergence of K-Means

Σ –2(vin – a) = 0

Σ vin = Σ a

mk a = Σ vin

a = (1/ mk) Σ vin = ckn

K-means typically converges quite quickly But, convergence only to local minimum

Linear Time Complexity

Assume computing distance between two instances is O(m) where m is the dimensionality of the vectors.

Reassigning clusters: O(kn) distance computations, or O(knm).

Computing centroids: Each instance vector gets added once to some centroid: O(nm).

Assume these two steps are each done once for i iterations: O(iknm).

Linear in all relevant factors, assuming a fixed number of iterations, more efficient than hierarchical agglomerative methods

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 (e.g., doc least similar to any existing mean)

Try out multiple starting points Initialize with the results of

another method.

In the above, if you startwith B and E as centroidsyou converge to {A,B,C}and {D,E,F}If you start with D and Fyou converge to {A,B,D,E} {C,F}

Example showingsensitivity to seeds

Exercise: find good approach for finding good starting points

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 Given docs, partition into an “appropriate” number

of subsets. E.g., for query results - ideal value of k not known

up front - though UI may impose limits. Can usually take an algorithm for one flavor and

convert to the other.

k not specified in advance

Say, the results of a query. Solve an optimization problem: penalize having

lots of clusters application dependent, e.g., compressed summary

of search results list. Tradeoff between having more clusters (better

focus within each cluster) and having too many clusters

k not specified in advance

Given a clustering, define the Benefit for a doc to be the cosine similarity to its centroid

Define the Total Benefit to be the sum of the individual doc Benefits.

Why is there always a clustering of Total Benefit n?

Penalize lots of clusters

For each cluster, we have a Cost C. Thus for a clustering with k clusters, the Total

Cost is kC. Define the Value of a clustering to be =

Total Benefit - Total Cost. Find the clustering of highest value, over all

choices of k. Total benefit increases with increasing K. But can

stop when it doesn’t increase by “much”. The Cost term enforces this.

K-means issues, variations, etc.

Recomputing the centroid after every assignment (rather than after all points are re-assigned) can improve speed of convergence of K-means

Assumes clusters are spherical in vector space Sensitive to coordinate changes, weighting etc.

Disjoint and exhaustive Doesn’t have a notion of “outliers”

Soft Clustering

Clustering typically assumes that each instance is given a “hard” assignment to exactly one cluster.

Does not allow uncertainty in class membership or for an instance to belong to more than one cluster.

Soft clustering gives probabilities that an instance belongs to each of a set of clusters.

Each instance is assigned a probability distribution across a set of discovered categories (probabilities of all categories must sum to 1).

Expectation Maximization-Background

Assume data comes from distribution model.

How to classify points and estimate parameters of the models in a mixture at the same time?

(Chicken and egg problem)

In mixture of models, two targets are twisted:

1. The parameters of the models

2. The assignment of each data point to the process that generate it

Intuition behind EM

Each of the step is easy assuming the other is solved

Know the assignment of each data points, we can estimate the parameters

Know the parameters of the distributions, we can assign each point to a model ( eg. by MLE)

This is what K-Means does

Key Factor in EM

Adaptive hard clustering: k-mean. Assign at each point to only one class at each step.

Adaptive soft clustering: EM. Data is assigned to each class with a probability equal to the relative likelihood of that point belonging to the class.

Structure of EM Algorithm

Really a large class of algorithms Initialization: Pick start values for parameters Iteratively process until parameters converge

Expectation (E) step: Calculate weights for every data point by running the responsibilities (weights)

Maximization (M) step: Maximize a loglikelihood function with the weights given by E step to update the parameters of the models

Model based clustering and EM

Gives a soft variant of the K-means algorithm Assume k clusters: {c1, c2,… ck}

Assume a probabilistic model of categories that allows computing P(ci | D) for each category, ci, for a given example document, D.

For text, typically assume a naïve Bayes category model. Model Parameters

P(ci) – percentage of docs in class

P(wj | ci) – chance of seeing word in doc in class ci

Naïve Bayes assumes conditional independence to simplify combined evidence calculations

EM Algorithm for text clustering

Iterative method for learning probabilistic categorization model from unsupervised data.

Initially assume random assignment of docs to categories.

Learn an initial probabilistic model by estimating model parameters from this randomly labeled data.

Iterate following two steps until convergence: Expectation (E-step): Compute P(ci | D) for each example given

the current model, and probabilistically re-label the examples based on these posterior probability estimates.

Maximization (M-step): Re-estimate the model parameters, , from the probabilistically re-labeled data.

EM Experiment on Web Docs [Soumen Chakrabarti]

Semi-supervised: corpus of labeled and unlabeled data

Take labeled corpus D, and randomly select a subset as DK as a test set.

Use the set of unlabeled documents in the EM procedure.

Correct classification of a document<=> concealed class label = class with largest probability

Accuracy with unlabeled documents > accuracy without unlabeled documents

Keeping labeled set of same size EM beats (supervised) naïve Bayes with same size

of labeled document set Largest boost for small size of labeled set Comparable or poorer performance of EM for large labeled sets

DDU

Increasing DU while holding DK fixed also shows the advantage of using large unlabeled sets in the EM-like algorithm.

Purity

Summary

Covered two types of clustering Flat, partitional clustering Hierarchical, agglomerative clustering

Not covered – Spectral clustering based on eigenvectors How many clusters? Key issues

Representation of data points Similarity/distance measure

HAC – simple but requires n^2 distances K-means: the basic partitional algorithm – linear time Model-based clustering and EM estimation

Scientific evaluation of clustering Perhaps the most substantive issue in CS

research, and clustering in particular: how do you measure goodness?

Most measures focus on computational efficiency Time and space

For application of clustering to web search: Measure retrieval effectiveness

Approaches to evaluation

Anecdotal User inspection Ground “truth” comparison

Cluster retrieval Purely quantitative measures

Probability of generating clusters found Average distance between cluster members

Microeconomic / utility

Anecdotal evaluation

Probably the commonest (and surely the easiest) “I wrote this clustering algorithm and look what it

found!” No benchmarks, no comparison possible Any clustering algorithm will pick up the easy

stuff like partition by languages Generally, unclear scientific value.

User inspection

Induce a set of clusters or a navigation tree Have subject matter experts evaluate the results

and score them some degree of subjectivity

Often combined with search results clustering Not clear how reproducible across tests. Expensive / time-consuming

Ground “truth” comparison

Take a union of docs from a taxonomy & cluster Yahoo!, ODP, newspaper sections …

Compare clustering results to baseline e.g., 80% of the clusters found map “cleanly” to

taxonomy nodes How would we measure this?

But is it the “right” answer? There can be several equally right answers

For the docs given, the static prior taxonomy may be incomplete/wrong in places the clustering algorithm may have gotten right

things not in the static taxonomy

“Subjective”

Ground truth comparison

Divergent goals Static taxonomy designed to be the “right”

navigation structure somewhat independent of corpus at hand

Clusters found have to do with vagaries of corpus

Also, docs put in a taxonomy node may not be the most representative ones for that topic cf Yahoo!

Microeconomic viewpoint

Any algorithm - including clustering - is only as good as the economic utility it provides

For clustering: net economic gain produced by an approach (vs. another approach)

Strive for a concrete optimization problem Carefully chosen performance metric

Examples recommendation systems

need similarity/distance measure clock time for interactive search

Expensive to test