\(k\)-means cluster centers #

Background #

In the Faces and dimension reduction lab, you used the singular value decomposition to compute a low-rank approximation for a data set of images, replacing vectors \(\{\vec{x}_i\} \in \mathbb{R}^{2914}\) with vectors \(\{\vec{c}_i\} \in \mathbb{R}^m \) for some value of \(m\). Following convention, we will call \(\mathbb{R}^{2914}\) image space and \(\mathbb{R}^m \) feature space. The \(\vec{x}_i\) were encoded as the columns of a matrix \(X\) and the \(\vec{c}_i \) as the columns of a matrix \(C\). The following is central:

We followed one approach to answer this question in the lab. This problem asks you to evaluate another.

Cluster centers #

Start by choosing a set of images, encode them as columns of a matrix \(X\), and compute the low-rank approximation \(C\) as detailed above.

• Use the \(k\)-means algorithm to construct a reasonable number of clusters in feature space:

  from sklearn.cluster import KMeans
  kmeans  = KMeans(n_clusters=17).fit(C)
  centers = kmeans.cluster_centers_ 
  

  This algorithm follows usual Python syntax: vectors being clustered are the rows of the array. Take the transpose if appropriate.
• You will now have an array of cluster centers, each a vector in feature space. As they are vectors in feature space, they are difficut to interpret.
• Use the singular value decomposition you computed above to recast each center as a vector \(\vec{y}\) in image space. In order to know what to do, you will have to carefully understand the SVD of \(X\).
• Finally, plot \(\vec{y}.\) Mimic the commands for the lab.