Unsupervised learning and clustering.pdf

9.54
Class 13
Unsupervised learning
Clustering
Shimon Ullman + Tomaso Poggio
Danny Harari + Daneil Zysman + Darren Seibert

Outline
• Introduction to clustering
• K-means
• Bag of words (dictionary learning)
• Hierarchical clustering
• Competitive learning (SOM)

What is clustering?
• The organization of unlabeled data into
similarity groups called clusters.
• A cluster is a collection of data items which
are “similar” between them, and “dissimilar”
to data items in other clusters.

Historic application of clustering

Computer vision application:
Image segmentation

What do we need for clustering?

Distance (dissimilarity) measures
 They are special cases of Minkowski distance:
(p is a positive integer)
p
p
m
k jk
ik
j
i
p x
x
d
1
1
)
,
( 




 
  
x
x

Cluster evaluation (a hard problem)
• Intra-cluster cohesion (compactness):
– Cohesion measures how near the data points in a
cluster are to the cluster centroid.
– Sum of squared error (SSE) is a commonly used
measure.
• Inter-cluster separation (isolation):
– Separation means that different cluster centroids
should be far away from one another.
• In most applications, expert judgments are
still the key

Clustering techniques
Divisive

Clustering techniques
Divisive
K-means

K-Means clustering
• K-means (MacQueen, 1967) is a partitional clustering
algorithm
• Let the set of data points D be {x1, x2, …, xn},
where xi = (xi1, xi2, …, xir) is a vector in X  Rr, and r is the
number of dimensions.
• The k-means algorithm partitions the given data into
k clusters:
– Each cluster has a cluster center, called centroid.
– k is specified by the user

K-means algorithm
• Given k, the k-means algorithm works as follows:
1. Choose k (random) data points (seeds) to be the initial
centroids, cluster centers
2. Assign each data point to the closest centroid
3. Re-compute the centroids using the current cluster
memberships
4. If a convergence criterion is not met, repeat steps 2 and 3

K-means convergence (stopping)
criterion
• no (or minimum) re-assignments of data points to
different clusters, or
• no (or minimum) change of centroids, or
• minimum decrease in the sum of squared error (SSE),
– Cj is the jth cluster,
– mj is the centroid of cluster Cj (the mean vector of all the
data points in Cj),
– d(x, mj) is the (Eucledian) distance between data point x
and centroid mj.




k
j
C j
j
d
SSE
1
2
)
,
(
x
m
x

K-means clustering example:
step 1

K-means clustering example –
step 2

K-means clustering example –
step 3

Why use K-means?
• Strengths:
– Simple: easy to understand and to implement
– Efficient: Time complexity: O(tkn),
where n is the number of data points,
k is the number of clusters, and
t is the number of iterations.
– Since both k and t are small. k-means is considered a linear
algorithm.
• K-means is the most popular clustering algorithm.
• Note that: it terminates at a local optimum if SSE is used.
The global optimum is hard to find due to complexity.

Weaknesses of K-means
• The algorithm is only applicable if the mean is
defined.
– For categorical data, k-mode - the centroid is
represented by most frequent values.
• The user needs to specify k.
• The algorithm is sensitive to outliers
– Outliers are data points that are very far away
from other data points.
– Outliers could be errors in the data recording or
some special data points with very different values.

Dealing with outliers
• Remove some data points that are much further away
from the centroids than other data points
– To be safe, we may want to monitor these possible outliers over
a few iterations and then decide to remove them.
• Perform random sampling: by choosing a small subset of
the data points, the chance of selecting an outlier is
much smaller
– Assign the rest of the data points to the clusters by distance or
similarity comparison, or classification

Sensitivity to initial seeds
Random selection of seeds (centroids)
Iteration 1 Iteration 2
Random selection of seeds (centroids)
Iteration 1 Iteration 2

Special data structures
• The k-means algorithm is not suitable for discovering
clusters that are not hyper-ellipsoids (or hyper-spheres).

K-means summary
• Despite weaknesses, k-means is still the most
popular algorithm due to its simplicity and
efficiency
• No clear evidence that any other clustering
algorithm performs better in general
• Comparing different clustering algorithms is a
difficult task. No one knows the correct
clusters!

Application to visual object recognition:
Dictionary learning (Bag of Words)

Learning the visual vocabulary

Types of hierarchical clustering
• Divisive (top down) clustering
Starts with all data points in one cluster, the root, then
– Splits the root into a set of child clusters. Each child cluster is
recursively divided further
– stops when only singleton clusters of individual data points
remain, i.e., each cluster with only a single point
• Agglomerative (bottom up) clustering
The dendrogram is built from the bottom level by
– merging the most similar (or nearest) pair of clusters
– stopping when all the data points are merged into a single
cluster (i.e., the root cluster).

Divisive hierarchical clustering

Agglomerative hierarchical clustering

Single linkage or Nearest neighbor

Complete linkage or Farthest neighbor

Object category structure in
monkey inferior temporal (IT) cortex

Kiani et al., 2007
Object category structure in
monkey inferior temporal (IT) cortex

Hierarchical clustering of neuronal
response patterns in monkey IT cortex
Kiani et al., 2007

Competitive learning algorithm:
Kohonen Self Organization Maps (K-SOM)

K-SOM example
• Four input data points
(crosses) in 2D space.
• Four output nodes in a
discrete 1D output space
(mapped to 2D as circles).
• Random initial weights
start the output nodes at
random positions.

• Randomly pick one input data
point for training (cross in
circle).
• The closest output node is the
winning neuron (solid
diamond).
• This winning neuron is moved
towards the input data point,
while its two neighbors move
also by a smaller increment
(arrows).
K-SOM example

• Randomly pick another input
data point for training (cross
in circle).
• The closest output node is the
new winning neuron (solid
diamond).
• This winning neuron is moved
towards the input data point,
while its single neighboring
neuron move also by a
smaller increment (arrows).
K-SOM example

K-SOM example
• Continue to randomly pick
data points for training, and
move the winning neuron
and its neighbors (by a
smaller increment) towards
the training data points.
• Eventually, the whole
output grid unravels itself
to represent the input
space.

Competitive learning claimed effect

Hebbian vs. Competitive learning

Summary
• Clustering has a long history and still is in active research
– There are a huge number of clustering algorithms, among them:
Density based algorithm, Sub-space clustering, Scale-up methods,
Neural networks based methods, Fuzzy clustering, Co-clustering …
– More are still coming every year
• Clustering is hard to evaluate, but very useful in practice
• Clustering is highly application dependent (and to some
extent subjective)
• Competitive learning in neuronal networks performs
clustering analysis of the input data

Unsupervised learning and clustering.pdf

More Related Content

Similar to Unsupervised learning and clustering.pdf (20)

Recently uploaded (20)

Unsupervised learning and clustering.pdf