When you cluster documents, you group them into sets so that documents in the same group are more similar to each other than to those in other groups. One of the most widely used clustering algorithms is k-means, which is especially popular for high-dimensional data like document-term matrices. The k-means algorithm seeks to partition your document vectors into k clusters by minimizing the sum of squared distances between each document and the center (centroid) of its assigned cluster. Geometrically, k-means assumes that clusters are roughly spherical and equally sized in the feature space, which means it works best when groups of documents form dense, well-separated clouds. Each iteration of the algorithm alternates between assigning each document to the nearest centroid and then updating the centroids to be the mean of the assigned documents. This process repeats until the assignments no longer change or a maximum number of iterations is reached.

The choice of distance measure is crucial in clustering. K-means traditionally uses Euclidean distance, but in document spaces—where vectors are high-dimensional and often sparse — cosine similarity or other measures may be more appropriate. The initial placement of centroids can also have a significant impact on the final clustering: poor initialization may lead to suboptimal partitions or slow convergence. Methods like k-means++ help by spreading out initial centroids, which often leads to better results.

Clustering in high-dimensional, sparse spaces like those created by document-term matrices poses unique challenges. As the number of dimensions increases, distances between points become less informative, and most document vectors end up being nearly equidistant from each other. This phenomenon, known as the curse of dimensionality, can make it difficult for clustering algorithms to find meaningful structure. Additionally, sparsity means that most entries in your vectors are zero, which can cause centroids to be unrepresentative of any actual document. To address these issues, you commonly use dimensionality reduction techniques, carefully choose distance metrics, or adjust preprocessing steps to improve the quality of clustering results.