random-projection-orthogonal-beta-distribution

Summary

When a unit-length vector is projected onto a random lower-dimensional subspace, the expected squared length of the result is exactly the fraction of dimensions kept (k out of d), and the full spread of possible outcomes follows a known Beta distribution. This gives the system a precise statistical benchmark for evaluating how much geometric structure random projections preserve or destroy, which underpins reasoning about dimensionality reduction and compressed representations.