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Random Embeddings with Optimal Accuracy

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.