Nonlinear Estimators and Tail Bounds for Dimension Reduction in l_1 Using Cauchy Random Projections

摘要

For dimension reduction in l_1, one can multiply a data matrix A ∈ R~(n×D) by R ∈ R~(D×k) (k ＜＜ D) whose entries are I.I.d. samples of Cauchy. The impossibility result says one can not recover the pairwise l_1 distances in A from B = AR ∈ R~(n×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 (log n/∈~2) 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.