CS728 Web Clustering II Lecture 14. K-Means Assumes documents are real-valued vectors. Clusters...
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Transcript of CS728 Web Clustering II Lecture 14. K-Means Assumes documents are real-valued vectors. Clusters...
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