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

random projection to O(log n/ε²) dimensions preserves pairwise distances within 1±ε

Can you shrink a 3D object into 2D without losing its shape?

Johnson–Lindenstrauss lemma

Can you fit more clothes in a drawer without folding them?

the Johnson-Lindenstrauss lemma says

Can we shrink big data without losing important details?

Mandelbrot set

Ever wondered why some shapes never look the same no matter how much you zoom in?

Random projection

Ever wondered how computers quickly understand vast amounts of text?

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

Swipe through more Machine Learning concepts