Johnson-Lindenstrauss Transforms with Best Confidence
Proceedings of Thirty Fourth Conference on Learning Theory, PMLR 134:3989-4007, 2021.
The seminal result of Johnson and Lindenstrauss on random embeddings has been intensively studied in applied and theoretical computer science. Despite that vast body of literature, we still lack of complete understanding of statistical properties of random projections; a particularly intriguing question is: why are the theoretical bounds that far behind the empirically observed performance? Motivated by this question, this work develops Johnson-Lindenstrauss distributions with optimal, data-oblivious, statistical confidence bounds. These bounds are numerically best possible, for any given data dimension, embedding dimension, and distortion tolerance. They improve upon prior works in terms of statistical accuracy, as well as exactly determine the no-go regimes for data-oblivious approaches. Furthermore, the projection matrices are efficiently samplable. The construction relies on orthogonal matrices, and the proof uses certain elegant properties of the unit sphere. In particular, the following techniques introduced in this work are of independent interest: a) a compact expression for the projection distortion in terms of singular eigenvalues of the projection matrix, b) a parametrization linking the unit sphere and the Dirichlet distribution and c) anti-concentration bounds for the Dirichlet distribution. Besides the technical contribution, the paper presents applications and numerical evaluation along with working implementation in Python (shared as a GitHub repository).