. What is the key difference between k-means and a Gaussian mixture model (GMM)?
GMM generalizes k-means by giving each cluster its own covariance shape and computing a probability of membership for every point, rather than a single hard label.
. Why does k-means fail on two concentric rings, landing at chance accuracy?
Assigning each point to its nearest center can only produce straight-line boundaries between clusters, which cannot separate one ring from another.
. In DBSCAN, what makes a point a core point?
A core point is defined purely by local density: enough neighbors within a fixed radius, with no notion of a cluster center at all.
. What can DBSCAN do that neither k-means nor GMM can, when clutter/noise points are added to a dataset?
k-means and GMM assign every point to some cluster no matter what; DBSCAN has an explicit noise category for points that aren't part of any sufficiently dense region.