Data Structures in High-Dimensional Space
undefined
Mining of Massive Datasets
Jure Leskovec, Anand Rajaraman, Jeff Ullman
Stanford University
http://www.mmds.org
N
o
t
e
t
o
o
t
h
e
r
t
e
a
c
h
e
r
s
a
n
d
u
s
e
r
s
o
f
t
h
e
s
e
s
l
i
d
e
s
:
W
e
w
o
u
l
d
b
e
d
e
l
i
g
h
t
e
d
i
f
y
o
u
f
o
u
n
d
t
h
i
s
o
u
r
m
a
t
e
r
i
a
l
u
s
e
f
u
l
i
n
g
i
v
i
n
g
y
o
u
r
o
w
n
l
e
c
t
u
r
e
s
.
F
e
e
l
f
r
e
e
t
o
u
s
e
t
h
e
s
e
s
l
i
d
e
s
v
e
r
b
a
t
i
m
,
o
r
t
o
m
o
d
i
f
y
t
h
e
m
t
o
f
i
t
y
o
u
r
o
w
n
n
e
e
d
s
.
I
f
y
o
u
m
a
k
e
u
s
e
o
f
a
s
i
g
n
i
f
i
c
a
n
t
p
o
r
t
i
o
n
o
f
t
h
e
s
e
s
l
i
d
e
s
i
n
y
o
u
r
o
w
n
l
e
c
t
u
r
e
,
p
l
e
a
s
e
i
n
c
l
u
d
e
t
h
i
s
m
e
s
s
a
g
e
,
o
r
a
l
i
n
k
t
o
o
u
r
w
e
b
s
i
t
e
:
gro.sdmm.www//:ptth
Given a cloud of data points we want to
understand its structure
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
2
3
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, …
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
4
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
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
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
Outlier
Cluster
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
5
6
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
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
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 Digital Sky Survey
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
7
Intuitively:
Music divides into categories, and
customers prefer a few categories
But what are categories really?
Represent a CD by a set of customers who
bought it:
Similar CDs have similar sets of customers,
and vice-versa
8
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
Space of all CDs:
Think of a space with one dim. for each
customer
Values in a dimension may be 0 or 1 only
A CD is a point in this space (
x
1
,
x
2
,…,
x
k
),
where
x
i
= 1 iff the
i
th
customer bought the CD
For Amazon, the dimension is tens of millions
Task:
Find clusters of similar CDs
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
9
Finding topics:
Represent a document by a vector
(
x
1
,
x
2
,…,
x
k
), where
x
i
= 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
10
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
As with CDs 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
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
11
12
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
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
Key operation:
Repeatedly combine
two nearest clusters
Three important questions:
1)
How do you represent a cluster of more
than one point?
2)
How do you determine the “nearness” of
clusters?
3)
When to stop combining clusters?
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
13
Key operation:
Repeatedly combine two
nearest clusters
(1) How to represent a cluster of many points?
Key problem:
As you merge 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 (data)points
(2) How to determine “nearness” of clusters?
Measure cluster distances by distances of centroids
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
14
15
(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)
D
a
t
a
:
o
… data point
x
… centroid
D
e
n
d
r
o
g
r
a
m
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
What about the Non-Euclidean case?
The only “locations” we can talk about are the
points themselves
i.e., there is no “average” of two points
Approach 1:
(1) How to represent a cluster of many points?
clustroid
= (data)point “
closest
” to other points
(2) How do you determine the “nearness” of
clusters?
Treat clustroid as if it were centroid, when
computing inter-cluster distances
16
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
(1) How to represent a cluster of many points?
clustroid
= point “
closest
” to other points
Possible meanings of “closest”:
Smallest maximum distance to 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:
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
17
C
e
n
t
r
o
i
d
i
s
t
h
e
a
v
g
.
o
f
a
l
l
(
d
a
t
a
)
p
o
i
n
t
s
i
n
t
h
e
c
l
u
s
t
e
r
.
T
h
i
s
m
e
a
n
s
c
e
n
t
r
o
i
d
i
s
a
n
“
a
r
t
i
f
i
c
i
a
l
”
p
o
i
n
t
.
C
l
u
s
t
r
o
i
d
i
s
a
n
e
x
i
s
t
i
n
g
(
d
a
t
a
)
p
o
i
n
t
t
h
a
t
i
s
“
c
l
o
s
e
s
t
”
t
o
a
l
l
o
t
h
e
r
p
o
i
n
t
s
i
n
t
h
e
c
l
u
s
t
e
r
.
X
Cluster on
3 datapoints
C
e
n
t
r
o
i
d
C
l
u
s
t
r
o
i
d
Datapoint
(2) How do you determine the “nearness” of
clusters?
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
18
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
Approach 3.1:
Use the
diameter
of the
merged cluster = maximum distance between
points in the cluster
Approach 3.2:
Use the
average distance
between points in the cluster
Approach 3.3:
Use a
density-based approach
Take the diameter or avg. distance, e.g., and divide
by the number of points in the cluster
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
19
Naïve implementation of hierarchical
clustering:
At each step, compute pairwise distances
between all pairs of clusters, then merge
O(
N
3
)
Careful implementation using priority queue
can reduce time to O(
N
2
log
N
)
Still too expensive for really big datasets
that do not fit in memory
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
20
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
22
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
1)
For each point, place it in the cluster whose
current centroid it is nearest
2)
After all points are assigned, update the
locations of centroids of the
k
clusters
3)
Reassign all points to their closest centroid
Sometimes moves points between clusters
Repeat 2 and 3 until convergence
Convergence:
Points don’t move between clusters
and centroids stabilize
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
23
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
24
x
x
x
x
x
x
x
x
x … data point
… centroid
x
x
x
C
l
u
s
t
e
r
s
a
f
t
e
r
r
o
u
n
d
1
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
25
x
x
x
x
x
x
x
x
x … data point
… centroid
x
x
x
C
l
u
s
t
e
r
s
a
f
t
e
r
r
o
u
n
d
2
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
26
x
x
x
x
x
x
x
x
x … data point
… centroid
x
x
x
C
l
u
s
t
e
r
s
a
t
t
h
e
e
n
d
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
27
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
28
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
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
29
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
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
30
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
Extension of
k
-means to large data
BFR
[Bradley-Fayyad-Reina]
is a
variant of
k
-means designed to
handle
very large
(disk-resident) data sets
Assumes
that clusters are normally distributed
around a centroid in a Euclidean space
Standard deviations in different
dimensions may vary
Clusters are axis-aligned ellipses
Efficient way to summarize clusters
(
required
memory O(clusters) and not O(data))
32
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
Points are read from disk 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:
Take
k
random points
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
33
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
3 sets of points which we keep track of:
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 existing centroid
These points are summarized, but not
assigned to a cluster
Retained set (RS):
Isolated points waiting to be assigned to a
compression set
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
34
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
35
D
i
s
c
a
r
d
s
e
t
(
D
S
)
:
C
l
o
s
e
e
n
o
u
g
h
t
o
a
c
e
n
t
r
o
i
d
t
o
b
e
s
u
m
m
a
r
i
z
e
d
C
o
m
p
r
e
s
s
i
o
n
s
e
t
(
C
S
)
:
S
u
m
m
a
r
i
z
e
d
,
b
u
t
n
o
t
a
s
s
i
g
n
e
d
t
o
a
c
l
u
s
t
e
r
R
e
t
a
i
n
e
d
s
e
t
(
R
S
)
:
I
s
o
l
a
t
e
d
p
o
i
n
t
s
For each cluster, the discard set (DS) is
summarized
by:
The number of points,
N
The vector
SUM
, whose
i
th
component is the
sum of the coordinates of the points in the
i
th
dimension
The vector
SUMSQ
:
i
th
component = sum of
squares of coordinates in
i
th
dimension
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
36
2
d
+ 1
values represent any size cluster
d
= number of dimensions
Average in
each dimension
(
the centroid
)
can be calculated as
SUM
i
/
N
SUM
i
=
i
th
component of SUM
Variance of a cluster’s discard set in
dimension
i
is:
(SUMSQ
i
/
N
) – (SUM
i
/
N
)
2
And standard deviation is the square root of that
Next step: Actual clustering
37
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
N
o
t
e
:
D
r
o
p
p
i
n
g
t
h
e
“
a
x
i
s
-
a
l
i
g
n
e
d
”
c
l
u
s
t
e
r
s
a
s
s
u
m
p
t
i
o
n
w
o
u
l
d
r
e
q
u
i
r
e
s
t
o
r
i
n
g
f
u
l
l
c
o
v
a
r
i
a
n
c
e
m
a
t
r
i
x
t
o
s
u
m
m
a
r
i
z
e
t
h
e
c
l
u
s
t
e
r
.
S
o
,
i
n
s
t
e
a
d
o
f
S
U
M
S
Q
b
e
i
n
g
a
d
-
d
i
m
v
e
c
t
o
r
,
i
t
w
o
u
l
d
b
e
a
d
x
d
m
a
t
r
i
x
,
w
h
i
c
h
i
s
t
o
o
b
i
g
!
Processing the “Memory-Load” of points (1):
1)
Find those points that are “
sufficiently
close
” to a cluster centroid and add those
points to that cluster and the
DS
These points are so close to the centroid that
they can be summarized and then discarded
2)
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
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
38
D
i
s
c
a
r
d
s
e
t
(
D
S
)
:
C
l
o
s
e
e
n
o
u
g
h
t
o
a
c
e
n
t
r
o
i
d
t
o
b
e
s
u
m
m
a
r
i
z
e
d
.
C
o
m
p
r
e
s
s
i
o
n
s
e
t
(
C
S
)
:
S
u
m
m
a
r
i
z
e
d
,
b
u
t
n
o
t
a
s
s
i
g
n
e
d
t
o
a
c
l
u
s
t
e
r
R
e
t
a
i
n
e
d
s
e
t
(
R
S
)
:
I
s
o
l
a
t
e
d
p
o
i
n
t
s
Processing the “Memory-Load” of points (2):
3) DS set:
Adjust statistics of the clusters to
account for the new points
Add
N
s,
SUM
s,
SUMSQ
s
4)
Consider merging compressed sets in the
CS
5)
If this is the last round, merge all compressed
sets in the
CS
and all
RS
points into their nearest
cluster
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
39
D
i
s
c
a
r
d
s
e
t
(
D
S
)
:
C
l
o
s
e
e
n
o
u
g
h
t
o
a
c
e
n
t
r
o
i
d
t
o
b
e
s
u
m
m
a
r
i
z
e
d
.
C
o
m
p
r
e
s
s
i
o
n
s
e
t
(
C
S
)
:
S
u
m
m
a
r
i
z
e
d
,
b
u
t
n
o
t
a
s
s
i
g
n
e
d
t
o
a
c
l
u
s
t
e
r
R
e
t
a
i
n
e
d
s
e
t
(
R
S
)
:
I
s
o
l
a
t
e
d
p
o
i
n
t
s
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
40
D
i
s
c
a
r
d
s
e
t
(
D
S
)
:
C
l
o
s
e
e
n
o
u
g
h
t
o
a
c
e
n
t
r
o
i
d
t
o
b
e
s
u
m
m
a
r
i
z
e
d
C
o
m
p
r
e
s
s
i
o
n
s
e
t
(
C
S
)
:
S
u
m
m
a
r
i
z
e
d
,
b
u
t
n
o
t
a
s
s
i
g
n
e
d
t
o
a
c
l
u
s
t
e
r
R
e
t
a
i
n
e
d
s
e
t
(
R
S
)
:
I
s
o
l
a
t
e
d
p
o
i
n
t
s
Q1) How do we decide if a point is “close
enough” to a cluster that we will add the
point to that cluster?
Q2) How do we decide whether two
compressed sets (CS) deserve to be
combined into one?
41
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
Q1) We need a way to decide whether to put
a new point into a cluster (and discard)
BFR suggests two ways:
The
Mahalanobis distance
is less than a threshold
High likelihood of the point belonging to
currently nearest centroid
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
42
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
43
σ
i
… standard deviation of points in
the cluster in the
i
th
dimension
44
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
Euclidean vs. Mahalanobis distance
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
45
C
o
n
t
o
u
r
s
o
f
e
q
u
i
d
i
s
t
a
n
t
p
o
i
n
t
s
f
r
o
m
t
h
e
o
r
i
g
i
n
U
n
i
f
o
r
m
l
y
d
i
s
t
r
i
b
u
t
e
d
p
o
i
n
t
s
,
E
u
c
l
i
d
e
a
n
d
i
s
t
a
n
c
e
N
o
r
m
a
l
l
y
d
i
s
t
r
i
b
u
t
e
d
p
o
i
n
t
s
,
E
u
c
l
i
d
e
a
n
d
i
s
t
a
n
c
e
N
o
r
m
a
l
l
y
d
i
s
t
r
i
b
u
t
e
d
p
o
i
n
t
s
,
M
a
h
a
l
a
n
o
b
i
s
d
i
s
t
a
n
c
e
Q2) 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 combined variance is
below some threshold
Many alternatives:
Treat dimensions
differently, consider density
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
46
Extension of
k
-means to clusters
of arbitrary shapes
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 (Clustering Using REpresentatives):
Assumes a
n
Euclidean distance
Allows clusters to assume any shape
Uses a collection of representative
points to represent clusters
48
V
s
.
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
49
e
e
e
e
e
e
e
e
e
e
e
h
h
h
h
h
h
h
h
h
h
h
h
h
salary
age
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
2 Pass algorithm. Pass 1:
0) Pick a random sample of points that fit in
main memory
1) Initial clusters:
Cluster these points hierarchically – group
nearest points/clusters
2) Pick representative 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
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
50
51
e
e
e
e
e
e
e
e
e
e
e
h
h
h
h
h
h
h
h
h
h
h
h
h
salary
age
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
52
e
e
e
e
e
e
e
e
e
e
e
h
h
h
h
h
h
h
h
h
h
h
h
h
salary
age
Pick (say) 4
remote points
for each
cluster.
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
53
e
e
e
e
e
e
e
e
e
e
e
h
h
h
h
h
h
h
h
h
h
h
h
h
salary
age
Move points
(say) 20%
toward the
centroid.
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
Pass 2:
Now, rescan the whole dataset and
visit each point
p
in the data set
Place it in the “
closest cluster
”
Normal definition of “
closest
”:
Find the closest representative to
p
and
assign it to representative’s cluster
54
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
p
Clustering:
Given a
set of points
, with a notion
of
distance
between points,
group the points
into some number of
clusters
Algorithms:
Agglomerative
hierarchical clustering
:
Centroid and clustroid
k
-means:
Initialization, picking
k
BFR
CURE
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
55
Explore the concept of clustering data points in high-dimensional spaces with distance measures like Euclidean, Cosine, Jaccard, and edit distance. Discover the challenges of clustering in dimensions beyond 2 and the importance of similarity in grouping objects. Dive into applications such as cataloging sky objects based on radiation frequencies in 7 dimensions. These insights are from the book "Mining of Massive Datasets" by J. Leskovec, A. Rajaraman, and J. Ullman.
Uploaded on Sep 21, 2024 | 0 Views
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. Download presentation by click this link. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
E N D
Presentation Transcript
Note to other teachers and users of these slides: We would be delighted if you found this our material useful in giving your own lectures. Feel free to use these slides verbatim, or to modify them to fit your own needs. If you make use of a significant portion of these slides in your own lecture, please include this message, or a link to our web site: http://www.mmds.org Mining of Massive Datasets Jure Leskovec, Anand Rajaraman, Jeff Ullman Stanford University http://www.mmds.org
Given a cloud of data points we want to understand its structure J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 2
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, J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 3
x x x xx 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 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Cluster Outlier J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 4
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 5
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 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 6
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 Digital Sky Survey J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 7
Intuitively:Music divides into categories, and customers prefer a few categories But what are categories really? Represent a CD by a set of customers who bought it: Similar CDs have similar sets of customers, and vice-versa J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 8
Space of all CDs: Think of a space with one dim. for each customer Values in a dimension may be 0 or 1 only A CD is a point in this space (x1, x2, , xk), where xi = 1 iff the i th customer bought the CD For Amazon, the dimension is tens of millions Task: Find clusters of similar CDs J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 9
Finding topics: 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 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 10
As with CDs 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 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 11
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 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 12
Key operation: Repeatedly combine two nearest clusters Three important questions: 1) How do you represent a cluster of more than one point? 2)How do you determine the nearness of clusters? 3) When to stop combining clusters? J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 13
Key operation: Repeatedly combine two nearest clusters (1) How to represent a cluster of many points? Key problem: As you merge 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 (data)points (2) How to determine nearness of clusters? Measure cluster distances by distances of centroids J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 14
(5,3) o (1,2) o x (1.5,1.5) x (1,1) x (4.7,1.3) o (2,1) o (4,1) x (4.5,0.5) o (0,0) o (5,0) Data: o data point x centroid Dendrogram J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 15
What about the Non-Euclidean case? The only locations we can talk about are the points themselves i.e., there is no average of two points Approach 1: (1) How to represent a cluster of many points? clustroid = (data)point closest to other points (2) How do you determine the nearness of clusters? Treat clustroid as if it were centroid, when computing inter-cluster distances J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 16
(1) How to represent a cluster of many points? clustroid= point closest to other points Possible meanings of closest : Smallest maximum distance to 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: Centroid Datapoint C x 2) min ( , d x c c Centroid is the avg. of all (data)points in the cluster. This means centroid is an artificial point. Clustroid is an existing (data)point that is closest to all other points in the cluster. X Clustroid Cluster on 3 datapoints J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 17
(2) How do you determine the nearness of clusters? 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 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 18
Approach 3.1: Use the diameter of the merged cluster = maximum distance between points in the cluster Approach 3.2: Use the average distance between points in the cluster Approach 3.3: Use a density-based approach Take the diameter or avg. distance, e.g., and divide by the number of points in the cluster J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 19
Nave 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 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 20
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 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 22
1) For each point, place it in the cluster whose current centroid it is nearest 2) After all points are assigned, update the locations of centroids of the k clusters 3) Reassign all points to their closest centroid Sometimes moves points between clusters Repeat 2 and 3 until convergence Convergence:Points don t move between clusters and centroids stabilize J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 23
x x x x x x x x x x x x data point centroid Clusters after round 1 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 24
x x x x x x x x x x x x data point centroid Clusters after round 2 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 25
x x x x x x x x x x x x data point centroid Clusters at the end J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 26
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 Best value of k Average distance to centroid k J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 27
Too few; many long distances to centroid. 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 x x x x x x x x x x x x x x J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 28
x Just right; distances rather short. 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 x x x x x x x x x x x x x x J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 29
Too many; little improvement in average distance. 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 x x x x x x x x x x x x x x J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 30
BFR [Bradley-Fayyad-Reina] is a variant of k-means designed to handle very large (disk-resident) data sets Assumes that clusters are normally distributed around a centroid in a Euclidean space Standard deviations in different dimensions may vary Clusters are axis-aligned ellipses Efficient way to summarize clusters (required memory O(clusters) and not O(data)) J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 32
Points are read from disk 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: Take k random points 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 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 33
3 sets of points which we keep track of: 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 existing centroid These points are summarized, but not assigned to a cluster Retained set (RS): Isolated points waiting to be assigned to a compression set J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 34
Points in the RS Compressed sets. Their points are in the CS. A cluster. Its points are in the DS. The centroid Discard set (DS): Close enough to a centroid to be summarized Compression set (CS): Summarized, but not assigned to a cluster Retained set (RS): Isolated points J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 35
For each cluster, the discard set (DS) 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 A cluster. All its points are in the DS. The centroid J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 36
2d + 1 values represent any size cluster d = number of dimensions Average in each dimension (the centroid) can be calculated as SUMi/ N SUMi = ith 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 Next step: Actual clustering Note:Dropping the axis-aligned clusters assumption would require storing full covariance matrix to summarize the cluster. So, instead of SUMSQ being a d-dim vector, it would be a dx d matrix, which is too big! J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 37
Processing the Memory-Load of points (1): 1) Find those points that are sufficiently close to a cluster centroid and add those points to that cluster and the DS These points are so close to the centroid that they can be summarized and then discarded 2) 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 Discard set (DS): Close enough to a centroid to be summarized. Compression set (CS): Summarized, but not assigned to a cluster Retained set (RS): Isolated points J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 38
Processing the Memory-Load of points (2): 3) DS set: Adjust statistics of the clusters to account for the new points Add Ns, SUMs, SUMSQs 4) Consider merging compressed sets in the CS 5) If this is the last round, merge all compressed sets in the CS and all RS points into their nearest cluster Discard set (DS): Close enough to a centroid to be summarized. Compression set (CS): Summarized, but not assigned to a cluster Retained set (RS): Isolated points J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 39
Points in the RS Compressed sets. Their points are in the CS. A cluster. Its points are in the DS. The centroid Discard set (DS): Close enough to a centroid to be summarized Compression set (CS): Summarized, but not assigned to a cluster Retained set (RS): Isolated points J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 40
Q1) How do we decide if a point is close enough to a cluster that we will add the point to that cluster? Q2) How do we decide whether two compressed sets (CS) deserve to be combined into one? J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 41
Q1) We need a way to decide whether to put a new point into a cluster (and discard) BFR suggests two ways: The Mahalanobis distance is less than a threshold High likelihood of the point belonging to currently nearest centroid J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 42
Normalized Euclidean distance from centroid For point (x1, , xd) and centroid (c1, , cd) 1. Normalize in each dimension: yi = (xi - ci) / i 2. Take sum of the squares of theyi 3. Take the square root 2 ? ?? ?? ?? ? ?,? = ?=1 i standard deviation of points in the cluster in the ith dimension J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 43
If clusters are normally distributed in d dimensions, then after transformation, one standard deviation = ? i.e., 68% of the points of the cluster will have a Mahalanobis distance < ? Accept a point for a cluster if its M.D. is < some threshold, e.g. 2 standard deviations J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 44
Euclidean vs. Mahalanobis distance Contours of equidistant points from the origin Uniformly distributed points, Euclidean distance Normally distributed points, Euclidean distance Normally distributed points, Mahalanobis distance J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 45
Q2) Should 2 CS subclusters be combined? Compute the variance of the combined subcluster N, SUM, and SUMSQallow us to make that calculation quickly Combine if the combined variance is below some threshold Many alternatives: Treat dimensions differently, consider density J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 46
Extension of k-means to clusters of arbitrary shapes
Vs. 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 (Clustering Using REpresentatives): Assumes an Euclidean distance Allows clusters to assume any shape Uses a collection of representative points to represent clusters J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 48
h h h e e e h e e e h e e e e h salary e h h h h h h h age J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 49
2 Pass algorithm. Pass 1: 0) Pick a random sample of points that fit in main memory 1) Initial clusters: Cluster these points hierarchically group nearest points/clusters 2) Pick representative 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 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 50
