NDLI: Optimal bounds for Johnson-Lindenstrauss transforms and streaming problems with sub-constant error

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Optimal bounds for Johnson-Lindenstrauss transforms and streaming problems with sub-constant error

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Optimal bounds for Johnson-Lindenstrauss transforms and streaming problems with sub-constant error

Content Provider	ACM Digital Library
Author	Woodruff, David Jayram, T. S.
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 $R^{n}$ → $R^{k}$ where k ≪ d and states that k = $O(ε^{™2}$ log 1/Δ) dimensions suffice to approximate the norm of any fixed vector in $R^{d}$ to within a factor of 1 ± ε with probability at least 1 ™ Δ. In this paper 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 $l_{p}-distance$ for any p ∈ [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 in the case of constant Δ, the VC-dimension does not characterize the complexity when Δ = o(1).
Starting Page	1
Ending Page	10
Page Count	10
File Format	PDF
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2011-01-23
Access Restriction	Subscribed
Content Type	Text
Resource Type	Article

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