Curse of Dimensionality in Clustering and How Model-Based Methods Cope

Definition. The curse of dimensionality refers to the phenomena that appear when the number of attributes $d$ becomes large: statistical, geometric and computational techniques that work in low dimensions break down.

1. What goes wrong in high dimensions

1. Distance concentration. For random points, the difference between the nearest and farthest neighbour shrinks relative to the distances themselves. If $d_{max}$ and $d_{min}$ are the farthest and nearest distances from a query point, then the relative contrast $R = (d_{max} - d_{min})/d_{min}$ tends to 0 as $d \to \infty$: all points become almost equidistant, so nearest-neighbour, k-means and density-based rules lose their discriminating power.
2. Sparsity of the space. The volume of a unit hypercube grows as $1^d$ exponentially, while data samples grow only linearly. To keep the same occupancy one would need exponentially many points, so observed data occupies a vanishing fraction of the space — clusters that look dense in 2-D become empty regions in high $d$.
3. Concentration at the boundary. In high dimensions almost all the volume of a hypercube lies in its corners and near its surface, so intuition built from 2-D/3-D pictures no longer applies.
4. Density estimation fails. Histograms and kernel estimates need samples exponential in $d$; DBSCAN's eps neighbourhoods become empty or contain all points at once.
5. Spurious correlations and overfitting. With many attributes, chance patterns look like structure, and models fit noise.
6. Cost. Every distance or similarity computation is $O(d)$, making mining slow as well as unreliable.

2. How model-based clustering helps

Aspect	Effect
Parametric assumption	A mixture of $K$ Gaussians in $d$ dimensions is described by only $O(K d^2)$ parameters, whereas a non-parametric density estimate would need exponentially many samples — the model imposes structure that the data cannot supply
Joint distribution	Covariance matrices model correlations between attributes, so redundant dimensions do not each add independent noise to the comparison
Latent structure	Extensions such as factor analysis and probabilistic PCA assume data lies near a low-dimensional subspace, explicitly reducing the effective dimension before clustering
Regularised estimation	Covariance shrinkage and tied/diagonal covariance options stop the $d(d+1)/2$ covariance terms from being estimated from too little data
Model selection	BIC penalises the parameter count, discouraging over-complex models that would simply memorise noise in high dimensions
Probabilistic output	Membership probabilities degrade gracefully, instead of forcing a meaningless nearest-centre assignment when all distances are nearly equal
Outlier handling	Heavy-tailed components (Student-$t$ mixtures) absorb anomalous points instead of pulling centroids around

3. Limitations to mention

Even mixture models struggle when $d$ is large relative to $n$; in practice they are combined with dimensionality reduction (PCA), feature selection or subspace clustering (e.g. PROCLUS, CLIQUE) before fitting.

Conclusion: dimensionality destroys clustering mainly through distance concentration and exponential sparsity; model-based methods respond by imposing a compact probabilistic form, exploiting attribute correlations and selecting complexity through BIC.