The interactive film is in the book's Companion

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Random Directions Are Nearly Orthogonal

Chapter 13 · Mathematics for AI and Machine Learning · Advanced · 2:09

An interactive live film: random unit vectors on a sphere whose cosines with a fixed direction crowd onto the equator as the dimension grows (spread about one over the square root of d), and a Johnson-Lindenstrauss panel of squared-distance ratios after a random projection to k dimensions.

Narration

Take two random directions: Gaussian vectors scaled to unit length. Fix the red one, v, pointing up; by symmetry its direction does not matter. Each blue dot is a random unit vector u, at the height of its cosine with v.

In three dimensions the cosines spread from minus one to one, and the histogram on the right is flat. On the green equator, u is orthogonal to v.

The rest of the narration is in the book's Companion.

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