What are the main challenges and limitations of spectral clustering?

2 Choosing the number of clusters

One of the most difficult decisions when using spectral clustering is how to choose the number of clusters, k, that you want to find. Unlike some other clustering algorithms, such as k-means, spectral clustering does not have a clear criterion to optimize or a way to automatically determine the optimal k. Instead, you have to rely on heuristics, such as the gap statistic, the silhouette score, or the eigenvalue gap, to estimate the best k. However, these methods are not always reliable or consistent, and may depend on the choice of the similarity measure and the data distribution.

3 Solving the eigenproblem

Another challenge of spectral clustering is solving the eigenproblem, which involves finding the k smallest eigenvalues and their corresponding eigenvectors of the similarity matrix. This is a computationally expensive and numerically unstable task, especially for large and sparse matrices. To speed up the process and reduce the memory requirements, you can use techniques such as random projection, subspace iteration, or Lanczos method to approximate the eigenvectors. However, these techniques may introduce errors or biases that affect the quality of the clustering results.

4 Postprocessing the eigenvectors

After solving the eigenproblem, you need to postprocess the eigenvectors to obtain the final cluster assignments. This usually involves applying a simple clustering algorithm, such as k-means, to the rows of the matrix formed by the k eigenvectors. However, this step can also introduce some issues, such as sensitivity to initialization, local optima, or different scaling of the eigenvectors. To overcome these issues, you can use techniques such as normalization, orthogonalization, or spectral rotation to improve the clustering performance.

5 Dealing with complex data structures

Finally, spectral clustering may face some limitations when dealing with complex data structures, such as non-convex, overlapping, or hierarchical clusters. Spectral clustering assumes that the data points are connected by a graph, and that the clusters correspond to the connected components or the cuts of the graph. However, this assumption may not hold for some real-world data sets, where the clusters may have different shapes, densities, or levels of granularity. To address these challenges, you can use techniques such as multi-scale spectral clustering, co-clustering, or spectral embedding to capture the diverse and rich structures of the data.