Spectral clustering: cluster by the graph Laplacian's eigenvectors, not by distance — non-convex shapes unfold into blobs

1Key Takeaways

• Plain k-means assigns every point to the nearest centroid, so the boundary between two clusters is always a straight line — and it fits convex blobs beautifully.
• But real structure is often non-convex: two crescent moons that interlock, a ring inside a ring.
• No set of centroids carves those apart, so k-means (and GMM's ellipses) split them straight down the middle and get it badly wrong.
• Spectral clustering sidesteps geometry entirely — it turns the data into a graph and reads the answer off….