# Clustering¶

 KMeans([n_clusters, init, ...]) Scalable KMeans for clustering SpectralClustering([n_clusters, ...]) Apply parallel Spectral Clustering

The dask_ml.cluster module implements several algorithms for clustering unlabeled data.

## Spectral Clustering¶

Spectral Clustering finds a low-dimensional embedding on the affinity matrix between samples. The embedded dataset is then clustered, typically with KMeans.

Typically, spectral clustering algorithms do not scale well. Computing the $$n\_samples \times n\_samples$$ affinity matrix becomes prohibitively expensive when the number of samples is large. Several algorithms have been proposed to work around this limitation.

In dask-ml, we use the Nyström method to approximate the large affinity matrix. This involves sampling n_components rows from the entire training set. The exact affinity is computed for this subset ( $$n\_components \times n\_components$$ ), and between this small subset and the rest of the data ( $$n\_components \times (n\_samples - n\_components)$$ ). We avoid the direct computation of the rest of the affinity matrix.

Let $$S$$ be our $$n \times n$$ affinity matrix. We can rewrite that as

$\begin{split}S_d = \left[ \begin{array}\ A & B \\ B^T & C \\ \end{array} \right]\end{split}$

Where $$A$$ is the $$n \times n$$ affinity matrix of the $$n\_components$$ that we sampled, and $$B$$ is the $$n \times (n - n\_components)$$ affinity matrix between the sample and the rest of the dataset. Instead of computing $$C$$ directly, we approximate it with $$B^T A^{-1} B$$.

See the spectral clustering benchmark for an example showing how dask_ml.cluster.SpectralClustering scales in the number of samples.