Emergent Mind
The Johnson-Lindenstrauss lemma is optimal for linear dimensionality reduction
(1411.2404)
Published Nov 10, 2014
in
cs.IT
,
cs.CG
,
cs.DS
,
math.FA
,
and
math.IT
Abstract
For any $n>1$ and $0<\varepsilon<1/2$, we show the existence of an $n{O(1)}$-point subset $X$ of $\mathbb{R}n$ such that any linear map from $(X,\ell2)$ to $\ell2m$ with distortion at most $1+\varepsilon$ must have $m = \Omega(\min{n, \varepsilon{-2}\log n})$. Our lower bound matches the upper bounds provided by the identity matrix and the Johnson-Lindenstrauss lemma, improving the previous lower bound of Alon by a $\log(1/\varepsilon)$ factor.
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