A unified framework for linear dimensionality reduction in L1

Abstract: For a family of interpolation norms $| \cdot |<em>{1,2,s}$ on $\mathbb{R}<sup>n$, we provide a distribution over random matrices $\Phi_s \in \mathbb{R}^{m \times n}$ parametrized by sparsity level $s$ such that for a fixed set $X$ of $K$ points in $\mathbb{R}<sup>n$, if $m \geq C s \log(K)$ then with high probability, $$\frac{1}{2} | x |</em>{1,2,s} \leq | \Phi_s (x) |<em>1 \leq 2 | x|</em>{1,2,s}$$ for all $x\in X$. Several existing results in the literature reduce to special cases of this result at different values of $s$: for $s=n$, $| x|<em>{1,2,n} \equiv | x |</em>{1}$ and we recover that dimension reducing linear maps can preserve the $\ell_1$-norm up to a distortion proportional to the dimension reduction factor, which is known to be the best possible such result. For $s=1$, $|x |<em>{1,2,1} \equiv | x |</em>{2}$, and we recover an $\ell_2 / \ell_1$ variant of the Johnson-Lindenstrauss Lemma for Gaussian random matrices. Finally, if $x$ is $s$-sparse, then $| x |_{1,2,s} = | x |_1$ and we recover that $s$-sparse vectors in $\ell_1<sup>n$ embed into $\ell_1^{{\mathcal{O}(s} \log(n))}$ via sparse random matrix constructions.