  A classical result of Johnson and Lindenstrauss states that a set of $n$ high
dimensional data points can be projected down to $O(\log n/\epsilon^2)$
dimensions such that the square of their pairwise distances is preserved up to
a small distortion $\epsilon\in(0,1)$. It has been proved that the JL lemma is
optimal for the general case, therefore, improvements can only be explored for
special cases. This work aims to improve the $\epsilon^{-2}$ dependency based
on techniques inspired by the Hutch++ Algorithm, which reduces $\epsilon^{-2}$
to $\epsilon^{-1}$ for the related problem of implicit matrix trace estimation.
We first present an algorithm to estimate the Euclidean lengths of the rows of
a matrix. We prove for it element-wise probabilistic bounds that are at least
as good as standard JL approximations in the worst-case, but are asymptotically
better for matrices with decaying spectrum. Moreover, for any matrix,
regardless of its spectrum, the algorithm achieves $\epsilon$-accuracy for the
total, Frobenius norm-wise relative error using only $O(\epsilon^{-1})$
queries. This is a quadratic improvement over the norm-wise error of standard
JL approximations. We also show how these results can be extended to estimate
(i) the Euclidean distances between data points and (ii) the statistical
leverage scores of tall-and-skinny data matrices, which are ubiquitous for many
applications, with analogous theoretical improvements. Proof-of-concept
numerical experiments are presented to validate the theoretical analysis.
