  This work constructs Jonson-Lindenstrauss embeddings with best accuracy, as
measured by variance, mean-squared error and exponential concentration of the
length distortion. Lower bounds for any data and embedding dimensions are
determined, and accompanied by matching and efficiently samplable constructions
(built on orthogonal matrices). Novel techniques: a unit sphere
parametrization, the use of singular-value latent variables and Schur-convexity
are of independent interest.
