1 · Pick a regime2 · Raise D3 · Demand k neighbors4 · Watch distances concentrate
Prediction: If we insist that a neighborhood contain 10% of uniformly distributed data, should the required edge length get smaller or larger as dimension increases?
Reveal the slide connection
It gets dramatically larger. For a D-dimensional unit cube, a hypercube containing fraction f has edge length e = f^(1/D). At D = 10 and f = 0.1, e is about 0.79. The neighborhood is no longer very local.