Almost Euclidean sections of the N-dimensional cross-polytope using O(N) random bits

On the isotropic constant of random polytopes with vertices on an $\ell_p$-sphere

The symmetric convex hull of random points that are independent and distributed according to the cone probability measure on the $\ell_p$-unit sphere of $\mathbb R^n$ for some $1\leq p < \infty$ is considered. We prove that these random polytopes have uniformly absolutely bounded isotropic constants with overwhelming probability. This generalizes the result for the Euclidean sphere ($p=2$) obtained by D. Alonso-Gutiérrez. The proof requires several different tools including a probabilistic representation of the cone measure due to G. Schechtman and J. Zinn and moment estimates for sums of inde…

Almost-Euclidean subspaces of $\ell_1^N$ via tensor products: a simple approach to randomness reduction

It has been known since 1970's that the N-dimensional $\ell_1$-space contains nearly Euclidean subspaces whose dimension is $Ω(N)$. However, proofs of existence of such subspaces were probabilistic, hence non-constructive, which made the results not-quite-suitable for subsequently discovered applications to high-dimensional nearest neighbor search, error-correcting codes over the reals, compressive sensing and other computational problems. In this paper we present a "low-tech" scheme which, for any $a > 0$, allows to exhibit nearly Euclidean $Ω(N)$-dimensional subspaces of $\ell_1^N$ while usi…

Almost Euclidean subspaces of \ell_1^N via expander codes

We give an explicit (in particular, deterministic polynomial time) construction of subspaces X of R^N of dimension (1-o(1))N such that for every element x in X, |x|_1 and N^{1/2} |x|_2 are equivalent up to a factor of (log N)^{log log log N}. If we are allowed to use N^{o(1)} random bits, this factor can be improved to poly(log N). Our construction makes use of unbalanced bipartite graphs to impose local linear constraints on vectors in the subspace, and our analysis relies on expansion properties of the graph. This is inspired by similar constructions of error-correcting codes.

对交叉多面体的Banach-Mazur距离的改进下界

An improved lower bound on the Banach--Mazur distance to the cross-polytope

该研究改进了对交叉多面体的Banach-Mazur距离的下界，通过处理“许多小系数”和“少数小系数”两种情况，得出了指数为4/7的改进结果。