Very Sparse Stable Random Projections, Estimators and Tail Bounds for Stable Random Projections

Abstract: This paper will focus on three different aspects in improving the current practice of stable random projections. Firstly, we propose {\em very sparse stable random projections} to significantly reduce the processing and storage cost, by replacing the $\alpha$-stable distribution with a mixture of a symmetric $\alpha$-Pareto distribution (with probability $\beta$, $0<\beta\leq1$) and a point mass at the origin (with a probability $1-\beta$). This leads to a significant $\frac{1}{\beta}$-fold speedup for small $\beta$. Secondly, we provide an improved estimator for recovering the original $l_\alpha$ norms from the projected data. The standard estimator is based on the (absolute) sample median, while we suggest using the geometric mean. The geometric mean estimator we propose is strictly unbiased and is easier to study. Moreover, the geometric mean estimator is more accurate, especially non-asymptotically. Thirdly, we provide an adequate answer to the basic question of how many projections (samples) are needed for achieving some pre-specified level of accuracy. \cite{Proc:Indyk_FOCS00,Article:Indyk_TKDE03} did not provide a criterion that can be used in practice. The geometric mean estimator we propose allows us to derive sharp tail bounds which can be expressed in exponential forms with constants explicitly given.