This is an announcement for the paper "Toward a unified theory of sparse dimensionality reduction in Euclidean space" by Jean Bourgain and Jelani Nelson.

Abstract: Let $\Phi\in\mathbb{R}^{m\times n}$ be a sparse Johnson-Lindenstrauss transform [Kane, Nelson, SODA 2012] with $s$ non-zeroes per column. For $T$ a subset of the unit sphere, $\varepsilon\in(0,1/2)$ given, we study settings for $m,s$ required to ensure $$ \mathop{\mathbb{E}}_\Phi \sup_{x\in T} \left||\Phi x|_2^2 - 1 \right| < \varepsilon , $$ i.e. so that $\Phi$ preserves the norm of every $x\in T$ simultaneously and multiplicatively up to $1+\varepsilon$. In particular, our most general theorem shows that it suffices to set $m = \tilde{\Omega}(\gamma_2^2(T) + 1)$ and $s = \tilde{\Omega}(1)$ as long as $s,m$ additionally satisfy a certain tradeoff condition that is governed by the geometry of $T$ (and as we show for several examples of interest, is easy to verify). Here $\gamma_2$ is Talagrand's functional, and we write $f = \tilde{\Omega}(g)$ to mean $f \ge Cg (\varepsilon^{-1}\log n)^c$ for some constants $C,c>0$. Our result can be seen as an extension to sparse $\Phi$ of works of [Klartag, Mendelson, J. Funct. Anal. 2005], [Gordon, GAFA 1988], and [Mendelson, Pajor, Tomczak-Jaegermann, GAFA 2007], which were concerned with dense $\Phi$ having i.i.d. (sub)gaussian entries. Our work introduces a theory that qualitatively unifies several results related to the Johnson-Lindenstrauss lemma, subspace embeddings, and Fourier-based methods for obtaining matrices satisfying the restricted isometry property.

Archive classification: cs.DS cs.CG cs.IT math.FA math.IT math.PR

Submitted from: minilek@seas.harvard.edu

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1311.2542

or

http://arXiv.org/abs/1311.2542