k-fold Cross Validation for determining k in k-means?

Question

In a document clustering process, as a data pre-processing step, I first applied singular vector decomposition to obtain U, S and Vt and then by choosing a suitable number of eigen values I truncated Vt, which now gives me a good document-document correlation from what I read here. Now I am performing clustering on the columns of the matrix Vt to cluster similar documents together and for this I chose k-means and the initial results looked acceptable to me (with k = 10 clusters) but I wanted to dig a bit deeper on choosing the k value itself. To determine the number of clusters k in k-means, I was suggested to look at cross-validation.

Before implementing it I wanted to figure out if there is a built-in way to achieve it using numpy or scipy. Currently, the way I am performing kmeans is to simply use the function from scipy.

import numpy, scipy

# Preprocess the data and compute svd
U, S, Vt = svd(A) # A is the TFIDF representation of the original term-document matrix

# Obtain the document-document correlations from Vt
# This 50 is the threshold obtained after examining a scree plot of S
docvectors = numpy.transpose(self.Vt[0:50, 0:]) 

# Prepare the data to run k-means
whitened = whiten(docvectors)
res, idx = kmeans2(whitened, 10, iter=20)

Assuming my methodology is correct so far (please correct me if I am missing some step), at this stage, what is the standard way of using the output to perform cross-validation? Any reference/implementations/suggestions on how this would be applied to k-means would be greatly appreciated.

Answer 1

To run k-fold cross validation, you'd need some measure of quality to optimize for. This could be either a classification measure such as accuracy or F₁, or a specialized one such as the V-measure.

Even the clustering quality measures that I know of need a labeled dataset ("ground truth") to work; the difference with classification is that you only need part of your data to be labeled for the evaluation, while the k-means algorithm can make use all the data to determine the centroids and thus the clusters.

V-measure and several other scores are implemented in scikit-learn, as well as generic cross validation code and a "grid search" module that optimizes according to a specified measure of evaluation using k-fold CV. Disclaimer: I'm involved in scikit-learn development, though I didn't write any of the code mentioned.