%0 Journal Article
%T Small width, low distortions: quasi-isometric embeddings with quantized sub-Gaussian random projections
%A Laurent Jacques
%J Mathematics
%D 2015
%I arXiv
%X Under which conditions a subset $\mathcal K$ of $\mathbb R^N$ can be embedded in another one of $\delta \mathbb Z^M$ for some resolution $\delta>0$? We address this general question through the specific use of a quantized random linear mapping ${\bf A}:\mathbb R^N \to \delta \mathbb Z^M$ combining a linear projection of $\mathbb R^N$ in $\mathbb R^M$ associated to a random matrix $\boldsymbol \Phi \in \mathbb R^{M\times N}$ with a uniform scalar (dithered) quantization $\mathcal Q$ of $\mathbb R^M$ in $\delta\mathbb Z^M$. The targeted embedding relates the $\ell_2$-distance of any pair of vectors in $\mathcal K$ with the $\ell_1$-distance of their respective mappings in $\delta \mathbb Z^M$, allowing for both multiplicative and additive distortions between these two quantities, i.e., describing a $\ell_2/\ell_1$-quasi-isometric embedding. We show that the sought conditions depend on the Gaussian mean width $w(\mathcal K)$ of the subset $\mathcal K$. In particular, given a symmetric sub-Gaussian distribution $\varphi$ and a precision $\epsilon > 0$, if $M \geq C \epsilon^{-5} w(\mathcal K)^2$ and if the sensing matrix $\boldsymbol \Phi$ has entries i.i.d. as $\varphi$, then, with high probability, the mapping $\bf A$ provides a $\ell_2/\ell_1$-quasi-isometry between $\mathcal K$ and its image in $\delta \mathbb Z^M$. Moreover, in this embedding, the additive distortion is of order $\delta\epsilon$ while the multiplicative one grows with $\epsilon$. For non-Gaussian random $\boldsymbol \Phi$, the multiplicative error is also impacted by the sparsity of the vectors difference, i.e., being smaller for "not too sparse" difference. When $\mathcal K$ is the set of bounded $K$-sparse vectors in any orthonormal basis, then only $M \geq C \epsilon^{-2} \log(c N/K\epsilon^{3/2})$ measurements suffice. Remark: all values $C,c>0$ above only depend on $\delta$ and on the distribution $\varphi$.
%U http://arxiv.org/abs/1504.06170v2