Introduction
K-means and Gaussian mixture models (GMMs) are two classic clustering methods. Both try to summarize data with groups, but they make different assumptions.
K-means assigns each point to one cluster center. GMMs model data as a mixture of Gaussian distributions and produce soft probabilities for cluster membership.
K-Means
K-means tries to minimize within-cluster squared distance:
$$ \sum_{i=1}^{n} |x_i - \mu_{c_i}|^2 $$
where $\mu_{c_i}$ is the centroid of the assigned cluster.
Use K-means when:
- Clusters are roughly spherical.
- Features can be scaled meaningfully.
- Hard assignments are acceptable.
- You need a fast baseline.
Watch out:
- Sensitive to feature scale.
- Sensitive to initialization.
- Requires choosing $k$.
- Struggles with non-spherical clusters.
- Struggles with clusters of very different density.
Gaussian Mixture Models
A GMM assumes data comes from a mixture of Gaussian components:
$$ p(x) = \sum_{k=1}^{K} \pi_k N(x \mid \mu_k, \Sigma_k) $$
Each point receives a probability of belonging to each component.
Use GMMs when:
- Soft cluster membership is useful.
- Clusters may have elliptical shapes.
- You want a probabilistic model.
Watch out:
- More parameters than K-means.
- Sensitive to initialization.
- Can overfit with too many components.
- Covariance choices matter.
Choosing the Number of Clusters
Useful tools:
- Elbow method.
- Silhouette score.
- BIC or AIC for GMMs.
- Stability checks.
- Domain interpretability.
The best number of clusters is not only a metric decision. It should also be useful for the downstream task.
Closing
K-means is a fast hard-clustering baseline. GMMs are a probabilistic soft-clustering extension. Use both with scaling, stability checks, and careful interpretation.