Random Embeddings with Optimal Accuracy

TL;DR

This paper develops optimal Johnson-Lindenstrauss embeddings with minimal variance and error, providing theoretical bounds and efficient constructions based on orthogonal matrices, introducing new mathematical techniques.

Contribution

It introduces the first optimal embeddings with provable bounds and efficient sampling methods, advancing the theoretical understanding of dimensionality reduction.

Findings

Achieves minimal variance and mean-squared error in embeddings

Provides matching lower bounds for data and embedding dimensions

Introduces novel techniques like sphere parametrization and Schur-convexity

Abstract

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.