# Optimal bounds for Johnson-Lindenstrauss transforms and streaming problems with subconstant error

## Abstract

The Johnson-Lindenstrauss transform is a dimensionality reduction technique with a wide range of applications to theoretical computer science. It is specified by a distribution over projection matrices from ℝn →ℝk where k ≤ n and states that k = O(ε -2 log 1/δ) dimensions suffice to approximate the norm of any fixed vector in ℝn to within a factor of 1±ε with probability at least 1-δ. In this article, we show that this bound on k is optimal up to a constant factor, improving upon a previous Ω((ε -2 log 1/δ)/ log(1/ε)) dimension bound of Alon. Our techniques are based on lower bounding the information cost of a novel one-way communication game and yield the first space lower bounds in a data stream model that depend on the error probability δ. For many streaming problems, the most naïve way of achieving error probability δ is to first achieve constant probability, then take the median of O(log 1/δ) independent repetitions. Our techniques show that for a wide range of problems, this is in fact optimal! As an example, we show that estimating the ℓp- distance for any p ε [0, 2] requires Ω(ε-2 log n log 1/δ) space, even for vectors in {0, 1}n. This is optimal in all parameters and closes a long line of work on this problem. We also show the number of distinct elements requires Ω(ε-2 log 1/δ + log n) space, which is optimal if ε-2 = Ω(log n). We also improve previous lower bounds for entropy in the strict turnstile and general turnstile models by a multiplicative factor of Ω(log 1/δ). Finally, we give an application to one-way communication complexity under product distributions, showing that, unlike the case of constant δ, the VC-dimension does not characterize the complexity when δ = o(1). © 2013 ACM.