Nonlinear estimators and tail bounds for dimension reduction in l ₁ using cauchy random projections

For dimension reduction in l₁, one can multiply a data matrix A ε ℝ^n×D by R ε ℝ^D×k (k ≪ D) whose entries are i.i.d. samples of Cauchy. The impossibility result says one can not recover the pairwise l₁ distances in A from B = AR ε^ℝn×k, using linear estimators. However, nonlinear estimators are still useful for certain applications in data stream computations, information retrieval, learning, and data mining. We propose three types of nonlinear estimators: the bias-corrected sample median estimator, the bias-corrected geometric mean estimator, and the bias-corrected maximum likelihood estimator. We derive tail bounds for the geometric mean estimator and establish that k = O (&logn/ε²) suffices with the constants explicitly given. Asymptotically (as k → ∞), both the sample median estimator and the geometric mean estimator are about 80% efficient compared to the maximum likelihood estimator (MLE). We analyze the moments of the MLE and propose approximating the distribution of the MLE by an inverse Gaussian.