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CS246: Mining Massive Datasets Jure Leskovec, Stanford University
http://cs246.stanford.edu
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Given a set of points, with a notion of distance between points, group the points into some number of clusters, so that Members of a cluster are close/similar to each other Members of different clusters are dissimilar
Usually: Points are in a high-dimensional space Similarity is defined using a distance measure Euclidean, Cosine, Jaccard, edit distance, …
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x x x x x x x x x x
x x x x x
x xx x
x x x x x
x x x x
x
x x x x x x x x x
x
x
x
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Clustering in two dimensions looks easy. Clustering small amounts of data looks easy. And in most cases, looks are not deceiving.
Many applications involve not 2, but 10 or
10,000 dimensions. High-dimensional spaces look different:
almost all pairs of points are at about the same distance
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Assume random points within a bounding box e.g., values between 0 and 1 in each dimension
In 2 dimensions: A variety of distances between 0 and 1.41.
In 10,000 dimensions: The difference in any one dimension is distributed as a triangle
The law of large numbers applies Actual distance between two random points
is the sqrt of the sum of squares of essentially the same set of differences
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A catalog of 2 billion “sky objects” represents objects by their radiation in 7 dimensions (frequency bands)
Problem: Cluster into similar objects, e.g., galaxies, nearby stars, quasars, etc.
Sloan Sky Survey is a newer, better version of this
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Intuitively: Music divides into categories, and customers prefer a few categories But what are categories really?
Represent a CD by the customers who bought it
Similar CDs have similar sets of customers, and vice-versa
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Space of all CDs: Think of a space with one dimension for each
customer Values in a dimension may be 0 or 1 only A movie is a point in this space is (x1, x2,…, xk),
where xi = 1 iff the i th customer bought the CD Compare with boolean matrix: rows = customers; cols. = CDs
For Amazon, the dimension is tens of millions An alternative: use minhashing/LSH to get Jaccard
similarity between “close” CD’s 1 minus Jaccard similarity can serve as a distance
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Represent a document by a vector (x1, x2,…, xk), where xi = 1 iff the i th word (in some order) appears in the document It actually doesn’t matter if k is infinite; i.e., we
don’t limit the set of words Documents with similar sets of words
may be about the same topic
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As with CD’s we have a choice when we think of documents as sets of words or shingles: Sets as vectors: measure similarity by the
cosine distance. Sets as sets: measure similarity by the Jaccard
distance. Sets as points: measure similarity by
Euclidean distance.
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Hierarchical: Agglomerative (bottom up): Initially, each point is a cluster Repeatedly combine the two
“nearest” clusters into one.
Divisive (top down): Start with one cluster and recursively split it
Point assignment: Maintain a set of clusters Points belong to “nearest” cluster
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Key operation: Repeatedly combine two nearest clusters
Three important questions: How do you represent a cluster of more than one
point? How do you determine the “nearness” of clusters? When to stop combining clusters?
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Key problem: As you build clusters, how do you represent the location of each cluster, to tell which pair of clusters is closest?
Euclidean case: each cluster has a centroid = average of its points Measure cluster distances by distances of
centroids
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(5,3) o (1,2) o o (2,1) o (4,1) o (0,0) o (5,0)
x (1.5,1.5)
x (4.5,0.5) x (1,1)
x (4.7,1.3)
Data: o … datapoint x … centroid Dendrogram
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The only “locations” we can talk about are the points themselves i.e., there is no “average” of two points
Approach 1: clustroid = point “closest” to other points Treat clustroid as if it were centroid, when
computing intercluster distances
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Possible meanings of “closest”: Smallest maximum distance to the other points. Smallest average distance to other points. Smallest sum of squares of distances to other
points. For distance metric d clustroid c of cluster C is:
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∑∈Cxc
cxd 2),(min
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Approach 2: Intercluster distance = minimum of the distances between any two points, one from each cluster.
Approach 3: Pick a notion of “cohesion” of clusters, e.g., maximum distance from the clustroid Merge clusters whose union is most cohesive
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Approach 1: Use the diameter of the merged cluster = maximum distance between points in the cluster
Approach 2: Use the average distance between points in the cluster
Approach 3: Use a density-based approach Take the diameter or avg. distance, e.g., and divide
by the number of points in the cluster Perhaps raise the number of points to a power
first, e.g., square-root.
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Naïve implementation of hierarchical clustering: At each step, compute pairwise distances
between all pairs of clusters, then merge O(N3)
Careful implementation using priority queue can reduce time to O(N2 log N) Still too expensive for really big datasets
that do not fit in memory
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Assumes Euclidean space/distance
Start by picking k, the number of clusters
Initialize clusters by picking one point per cluster. Example: pick one point at random, then k-1
other points, each as far away as possible from the previous points
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For each point, place it in the cluster whose current centroid it is nearest
After all points are assigned, fix the centroids of the k clusters.
Optional: reassign all points to their closest centroid Sometimes moves points between clusters
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1
2
3
4
5
6
7 8 x
x
Clusters after first round
Reassigned points
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How to select k? Try different k, looking at the change in the
average distance to centroid, as k increases. Average falls rapidly until right k, then
changes little
24
k
Average distance to centroid
Best value of k
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x x x x x x x x x x
x x x x x
x xx x
x x x x x
x x x x
x
x x x x x x x x x
x
x
x
Too few; many long distances to centroid.
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x x x x x x x x x x
x x x x x
x xx x
x x x x x
x x x x
x
x x x x x x x x x
x
x
x
Just right; distances rather short.
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x x x x x x x x x x
x x x x x
x xx x
x x x x x
x x x x
x
x x x x x x x x x
x
x
x
Too many; little improvement in average distance.
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BFR [Bradley-Fayyad-Reina] is a variant of k-means designed to handle very large (disk-resident) data sets
It assumes that clusters are normally distributed around a centroid in a Euclidean space Standard deviations in different dimensions
may vary
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Points are read one main-memory-full at a time.
Most points from previous memory loads are summarized by simple statistics
To begin, from the initial load we select the initial k centroids by some sensible approach
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Possible initialization strategies of the k cluster centers: Take a small random sample and cluster optimally Take a sample; pick a random point, and then
k–1 more points, each as far from the previously selected points as possible (As you will learn in HW2, picking random set of k
points does not work too well)
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Discard set (DS): Points close enough to a centroid to be
summarized. Compression set (CS): Groups of points that are close together but not
close to any centroid. These points are summarized, but not assigned to
a cluster Retained set (RS): Isolated points
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A cluster. Its points are in the DS.
The centroid
Compressed sets. Their points are in the CS.
Points in the RS
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For each cluster, the discard set is summarized by: The number of points, N The vector SUM, whose ith component is the
sum of the coordinates of the points in the ith dimension
The vector SUMSQ: ith component = sum of squares of coordinates in ith dimension
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2d + 1 values represent any size cluster d = number of dimensions
Averages in each dimension (the centroid) can be calculated as SUMi /N SUMi = i th component of SUM
Variance of a cluster’s discard set in dimension i is: (SUMSQi /N ) – (SUMi /N )2
And standard deviation is the square root of that
Q: Why use this representation of clusters?
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Processing the “Memory-Load” of points: Find those points that are “sufficiently close”
to a cluster centroid; Add those points to that cluster and the DS
Use any main-memory clustering algorithm to cluster the remaining points and the old RS Clusters go to the CS; outlying points to the RS
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Processing the “Memory-Load” of points (2): Adjust statistics of the clusters to account for
the new points. Add Ns, SUMs, SUMSQs
Consider merging compressed sets in the CS
If this is the last round, merge all compressed sets in the CS and all RS points into their nearest cluster
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How do we decide if a point is “close enough” to a cluster that we will add the point to that cluster?
How do we decide whether two compressed sets deserve to be combined into one?
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We need a way to decide whether to put a new point into a cluster
BFR suggest two ways: The Mahalanobis distance is less than a threshold Low likelihood of the currently nearest centroid
changing
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Normalized Euclidean distance from centroid
For point (x1,…,xd) and centroid (c1,…,cd) 1. Normalize in each dimension: yi = (xi - ci)/σi
2. Take sum of the squares of the yi 3. Take the square root
𝑑 𝑥, 𝑐 = �𝑥𝑖 − 𝑐𝑖𝜎𝑖
𝑑
𝑖
2
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σi … standard deviation of points in the cluster in the ith dimension
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If clusters are normally distributed in d dimensions, then after transformation, one standard deviation = 𝑑 i.e., 68% of the points of the cluster will have a
Mahalanobis distance < 𝑑
Accept a point for a cluster if its M.D. is < some threshold, e.g. 4 standard deviations
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Should 2 CS subclusters be combined? Compute the variance of the combined
subcluster N, SUM, and SUMSQ allow us to make that
calculation quickly Combine if the variance is below some
threshold
Many alternatives: treat dimensions differently, consider density.
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Problem with BFR/k-means: Assumes clusters are normally
distributed in each dimension And axes are fixed – ellipses at
an angle are not OK
CURE: Assumes a Euclidean distance Allows clusters to assume
any shape
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Pick a random sample of points that fit in main memory
1) Initial clusters: Cluster these points hierarchically – group nearest
points/clusters 2) Pick disperse points: For each cluster, pick a sample of points, as
dispersed as possible From the sample, pick representatives by moving
them (say) 20% toward the centroid of the cluster
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Pick (say) 4 remote points for each cluster.
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Move points (say) 20% toward the centroid.
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Now, visit each point p in the data set
Place it in the “closest cluster” Normal definition of “closest”: that cluster with
the closest (to p) among all the sample points of all the clusters.
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