List of unsolved problems in mathematics

Why do random points in high dimensions seem to be evenly spaced?

Imagine throwing darts at a wall with many holes; no matter where you stand, you expect them to hit the holes evenly.

As you throw more darts into a high-dimensional space, they tend to spread out evenly, covering the space uniformly.

Example

If you throw 100 darts at a wall with 10 holes, you'll likely hit each hole around 10 times.

Remember this

Random points in high dimensions are nearly equidistant because they evenly cover the space.

Text adapted from Wikipedia, licensed under CC BY-SA 4.0.

Related concepts

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cosine similarity works better than Euclidean distance in high dimensions

Cosine similarity measures orientation, not magnitude, making it more robust to irrelevant dimensions in high-dimensional spaces

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Manifold hypothesis

High-dimensional data lies on lower-dimensional manifolds

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