  The method of stable random projections is a tool for efficiently computing
the $l_\alpha$ distances using low memory, where $0<\alpha \leq 2$ is a tuning
parameter. The method boils down to a statistical estimation task and various
estimators have been proposed, based on the geometric mean, the harmonic mean,
and the fractional power etc.
  This study proposes the optimal quantile estimator, whose main operation is
selecting, which is considerably less expensive than taking fractional power,
the main operation in previous estimators. Our experiments report that the
optimal quantile estimator is nearly one order of magnitude more
computationally efficient than previous estimators. For large-scale learning
tasks in which storing and computing pairwise distances is a serious
bottleneck, this estimator should be desirable.
  In addition to its computational advantages, the optimal quantile estimator
exhibits nice theoretical properties. It is more accurate than previous
estimators when $\alpha>1$. We derive its theoretical error bounds and
establish the explicit (i.e., no hidden constants) sample complexity bound.
