Unsupervised Learning

Basics of Unsupervised Learning

Clustering

Given points \(\displaystyle \{x^{(1)},...,x^{(m)}\}\) we wish to group them into clusters 1,...,k.
Usually we are given the number of clusters, k.
The problem is to assign labels \(\displaystyle \{z^{(1)},...,z^{(m)}\}\) where \(\displaystyle z^{(i)} \in \{1,...,k\}\)

K-means

Here we wish to cluster them to minimize the distance to the mean of their cluster.
In our formulation we have k clusters.
The mean of each cluster \(\displaystyle \mu_i\) is called the centroid.</math>

Optimization

Let \(\displaystyle \mathbf{\mu}\) denote the centroids and let \(\displaystyle \mathbf{z}\) denote the cluster labels for our data.
Our loss function is \(\displaystyle L(\mu, \mathbf{z}) = \sum_{i} \Vert x^{(i)} - \mu_{z^{(i)}} \Vert ^2\).

Using coordinate-block descent

\(\displaystyle L(\mu, \mathbf{z}) = \sum_{i} \Vert x^{(i)} - \mu_{z^{(i)}} \Vert ^2\).
If we fix \(\displaystyle \mu\) then we have only need to minimize \(\displaystyle z^{(i)}\).
Since each term of the sum is independent, we simply choose the closest centroid for each point
If we fix \(\displaystyle \mathbf{z}\) then we need to minimize \(\displaystyle L(\mu, \mathbf{z})\) wrt \(\displaystyle \mu\).
Taking the gradient and setting it to 0 we get:
\(\displaystyle kmo \begin{align} \nabla L(\mu, \mathbf{z}) &= \nabla \sum_{i} \Vert x^{(i)} - \mu_{z^{(i)}} \Vert ^2\\ &= \nabla \sum_{j=1}{k} \sum_{i\mid z(i)=j} \Vert x^{(i)} - \mu_{z^{(i)}} \Vert ^2\\ &= \nabla \sum_{j=1}{k} \sum_{i\mid z(i)=j} \Vert x^{(i)} - \mu_{j} \Vert ^2\\ &= \sum_{j=1}{k} \sum_{i\mid z(i)=j} \nabla \Vert x^{(i)} - \mu_{j} \Vert ^2\\ &= \sum_{j=1}{k} \sum_{i\mid z(i)=j} \nabla \Vert x^{(i)} - \mu_{j} \Vert ^2\\ &= \sum_{j=1}{k} \sum_{i\mid z(i)=j} 2(x^{(i)} - \mu_{j})\\ \implies \mu_{j} &= (\sum_{i\mid z(i)=j} x^{(i)})/(\sum_{i\mid z(i)=j} 1) \quad \forall j \end{align} \)

Notes

This procedure will yield us a sequence of parameters and losses.
\(\displaystyle \mu^{0}, z^{0}, \mu^1, z^1, ...\)
\(\displaystyle L(0) \geq L(1) \geq L(2) \geq ...\)

• Since the loss is monotone decreasing and bounded below, it will converge by the monotone convergence theorem.
• However, this does not imply that the parameters \(\displaystyle \mathbf{\mu}\) and \(\displaystyle \mathbf{z}\) will converge.

Soft K-means

We will develop a model for how our data is generated:
Given \(\displaystyle k\) clusters, the probability of a point being from cluster k is \(\displaystyle \phi_k = P(z^{(i)} = k)\)