▶Squared distance
Every coordinate contributes a nonnegative squared difference.
K-Means in High Dimensions
When irrelevant coordinates overwhelm distance
K-means can fail even when one coordinate separates groups clearly. In many dimensions, accumulated noise dominates Euclidean distance and makes candidate neighbors look alike.
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The essential idea: distance concentration weakens the comparisons that distance-based clustering depends on.
One Concept · One Example
What to notice
- Only one coordinate may carry cluster information.
- Noise contributes to every Euclidean distance.
- Relative distance variation shrinks like one over square root of dimension.
The curse appears in the distance
Concentration
The standard deviation becomes small relative to the mean as dimension grows.
The same first two dimensions, different clustering
Keep the first two coordinates visible while increasing how many noisy dimensions k-means uses. Compare the data-generating groups with the resulting cluster labels.
In high dimensions, irrelevant features can make distances nearly indistinguishable and undermine k-means.