Unsupervised Learning
Clustering
K-Means
Recall:
Key Components of Intelligent Agents
Representation Language: Graph, Bayes Nets, Linear functions
Inference Mechanism: A*, variable elimination, Gibbs sampling
Learning Mechanism: Maximum Likelihood, Laplace Smoothing,
gradient descent, perceptron, k-Nearest Neighbor, many more: kmeans, EM, PCA, …
------------------------------------Evaluation Metric: Likelihood, quadratic loss (a.k.a. squared error),
regularized loss, margins, many more: 0-1 loss, conditional likelihood,
precision/recall, …
Supervised vs. Unsupervised Learning
Supervised Learning: “Labeled” Data
Unsupervised Learning: “Unlabeled” Data
X11
X12
…
X1N
Y1
X11
X12
…
X1N
?
X21
X22
…
X2N
Y2
X21
X22
…
X2N
?
…
…
…
…
…
…
…
…
…
…
XM1
XM2
…
XMN
YM
XM1
XM2
…
XMN
?
In supervised learning, the learning algorithm is given training examples that
contain inputs (the X values) and “labels” or “outputs” (the Y values).
In unsupervised learning, the learning algorithm is given training examples that
contain inputs (the X values), but no “labels” or “outputs” (no Y values).
It’s called “unsupervised” because there are no “labels” to help “supervise” the
learning algorithm during the learning process, to get it to the right model.
Example Unsupervised Problem 1
X2
Are these data points distributed
completely randomly, or do you see
some structure in them?
How many clusters do you see?
None
1
2
3
4
5
X1
Example Unsupervised Problem 1
Are these data points distributed
completely randomly, or do you see
some structure in them?
X2
Structured – there are clusters!
How many clusters do you see?
None
1
2
3
4
5
X1
Example Unsupervised Problem 2
There are 2 input variables, X1 and
X2, in this space. So this is called a
“2-dimensional space”.
X2
How many dimensions are actually
needed to describe this data?
0
1
2
3
X1
Example Unsupervised Problem 2
There are 2 input variables, X1 and
X2, in this space. So this is called a
“2-dimensional space”.
X2
How many dimensions are actually
needed to describe this data?
1 dimension captures most of the
variation in this data.
2 dimensions will capture
everything.
X1
Types of Unsupervised Learning
Density Estimation
- Clustering (Example 1)
- Dimensionality Reduction (Example 2)
Factor Analysis
- Blind signal separation
Example Open Problem in AI:
Unsupervised Image Segmentation
(and Registration)
Examples taken from (Felzenszwab and Huttenlocher, Int. Journal of Computer Vision, 59:2,
2004). http://cs.brown.edu/~pff/segment/.
The K-Means Clustering Algorithm
Inputs:
1) Some unlabeled (no outputs) training data
2) A number K, which must be greater than 1
Output:
A label between 1 and K for each data point,
indicating which cluster the data point belongs
to.
Visualization of K-Means
Data
Visualization of K-Means
1. Generate K random initial cluster centers, or “means”.
Visualization of K-Means
2. Assign each point to the closest “mean” point.
Visualization of K-Means
2. Assign each point to the closest “mean” point.
Visually, the mean points divide the space into a Voronoi diagram.
Visualization of K-Means
3. Recompute the “mean” (center) of each colored set of data.
Notice: “means” do not have to be at the same position as a data point,
although some times they might be.
Visualization of K-Means
3. Recompute the “mean” (center) of each colored set of data.
Notice: “means” do not have to be at the same position as a data point,
although some times they might be.
Visualization of K-Means
4. Repeat steps 2 & 3 until the “means” stop moving (convergence).
a. Repeat step 2 (assign each point to the nearest mean)
Visualization of K-Means
4. Repeat steps 2 & 3 until the “means” stop moving (convergence).
a. Repeat step 2 (assign each point to the nearest mean)
Visualization of K-Means
4. Repeat steps 2 & 3 until the “means” stop moving (convergence).
a. Repeat step 2 (assign each point to the nearest mean)
b. Repeat step 3 (recompute means)
Visualization of K-Means
Quiz: Where will the means be after the next iteration?
4. Repeat steps 2 & 3 until the “means” stop moving (convergence).
a. Repeat step 2 (assign each point to the nearest mean)
b. Repeat step 3 (recompute means)
Visualization of K-Means
Answer: Where will the means be after the next iteration?
4. Repeat steps 2 & 3 until the “means” stop moving (convergence).
a. Repeat step 2 (assign each point to the nearest mean)
b. Repeat step 3 (recompute means)
Visualization of K-Means
Quiz: Where will the means be after the next iteration?
4. Repeat steps 2 & 3 until the “means” stop moving (convergence).
a. Repeat step 2 (assign each point to the nearest mean)
b. Repeat step 3 (recompute means)
Visualization of K-Means
Answer: Where will the means be after the next iteration?
4. Repeat steps 2 & 3 until the “means” stop moving (convergence).
a. Repeat step 2 (assign each point to the nearest mean)
b. Repeat step 3 (recompute means)
Formal Description of the Algorithm
Input:
1) X11, …, X1N; … ; XM1, …, XMN
2) K
Output: Y1; …; YM, where each Yi is in {1, …, K}
Formal Description of the Algorithm
1. Init: For each k in {1, …, K}, create a random point Ck
2. Repeat until all Ck remain the same:
Assignment (aka Expectation):
For each Xi,
let C[Xi]  the k value for the closest Ck to Xi
Update (aka Maximization):
For each Ck,
let Dk{Xi |C[Xi] = k} (set of Xi assigned to cluster k)
if |Dk| = 0, let Ck  random new point
else let Ck 
1
|𝐷𝑘 |
𝑋𝑖 ∈𝐷𝑘 𝑋𝑖
3. Return C[Xi] for each Xi
(average of points in Dk)
Evaulation metric for K-means
LOSS Function (or Objective function) for Kmeans:
Within-cluster-sum-of-squares loss (WCSS):
WCSS(X1, …, XM, C1, …, CK)
𝐾
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒(𝑋𝑖 , 𝐶𝑘 )2
=
𝑘=1 𝑋𝑖 |𝐶 𝑋𝑖 =𝑘
Complexity of K-Means
Finding a globally-optimal solution to WCSS is known to be an NP-hard
problem.
K-means is known to converge to a local minimum of WCSS.
K-means is a “heuristic” or “greedy” algorithm, with no guarantee that
it will find the global optimum.
On real datasets, K-means usually converges very quickly. Often,
people run it multiple times with different random initializations, and
choose the best result.
In some cases, K-means will still take exponential time (assuming
P!=NP), even to find a local minimum. However, such cases are rare in
practice.
Quiz
Is K-means
Classification or Regression?
Generative or Discriminative?
Parametric or Nonparametric?
Answer
Is K-means
Classification or Regression?
- classification: output is a discrete value (cluster label) for
each point
Generative or Discriminative?
- discriminative: it has fixed input variables and output
variables.
Parametric or Nonparametric?
- parametric: the number of cluster centers (K) does not
change with the number of training data points
Quiz
Is K-means
Supervised or Unsupervised?
Online or batch?
Closed-form or iterative?
Answer
Is K-means
Supervised or Unsupervised?
- Unsupervised
Online or batch?
- batch: if you add a new data point, you need to revisit
all the training data to recompute the locally-optimal
model
Closed-form or iterative?
-iterative: training requires many passes through the data
Quiz
Which of the following problems might be
solved using K-Means? Check all that apply.
For those that work, explain what the inputs and
outputs (X and Y variables) would be.
• Segmenting an image
• Finding galaxies (dense groups of stars) in a
telescope’s image of the night sky
• Identify different species of bacteria from DNA
samples of bacteria in seawater
Answer
Which of the following problems might be solved using KMeans? Check all that apply.
For those that work, explain what the inputs and outputs
(X and Y variables) would be.
• Segmenting an image: Yes. Inputs are the pixel
intensities, outputs are segment labels.
• Finding galaxies (dense groups of stars) in a telescope’s
image of the night sky. Yes. Inputs are star locations,
outputs are galaxy labels
• Identify different species of bacteria from DNA samples
of bacteria in seawater. Yes. Inputs are gene
sequences, outputs are species labels.