基于扩展码的\ell_1^N的几乎欧氏子空间

膨胀图是 $\ell^p$ 粗 Baum-Connes 猜想的反例

Expanders are counterexamples to the $\ell^p$ coarse Baum-Connes conjecture

本文通过构造来自剩余有限双曲群的膨胀图，证明了 $\ell^p$ 粗 Baum-Connes 装配映射在 $1<p<\infty$ 时不是满射，从而给出该猜想的反例。

Expanders with respect to Hadamard spaces and random graphs

It is shown that there exists a sequence of 3-regular graphs $\{G_n\}_{n=1}^\infty$ and a Hadamard space $X$ such that $\{G_n\}_{n=1}^\infty$ forms an expander sequence with respect to $X$, yet random regular graphs are not expanders with respect to $X$. This answers a question of \cite{NS11}. $\{G_n\}_{n=1}^\infty$ are also shown to be expanders with respect to random regular graphs, yielding a deterministic sublinear time constant factor approximation algorithm for computing the average squared distance in subsets of a random graph. The proof uses the Euclidean cone over a random graph, an a…

Euclidean preferences in the plane under $\ell_1$, $\ell_2$ and $\ell_\infty$ norms

We present various results about Euclidean preferences in the plane under $\ell_1$, $\ell_2$ and $\ell_{\infty}$ norms. When there are four candidates, we show that the maximal size (in terms of the number of pairwise distinct preferences) of Euclidean preference profiles in the plane under norm $\ell_1$ or $\ell_{\infty}$ is 19. Whatever the number of candidates, we prove that at most four distinct candidates can be ranked in last position of a two-dimensional Euclidean preference profile under norm $\ell_1$ or $\ell_\infty$, which generalizes the case of one-dimensional Euclidean preferences…

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…

$\ell_p$-球的欧几里得截面的半径

Radii of Euclidean sections of $\ell_p$-balls

研究$\ell_p$-球的欧几里得截面半径，探讨其在理论计算机科学中的应用及开放问题

Equilateral sets in the $\ell_1$ sum of Euclidean spaces

Let $E^n$ denote the (real) $n$-dimensional Euclidean space. It is not known whether an equilateral set in the $\ell_1$ sum of $E^a$ and $E^b$, denoted here as $E^a \oplus_1 E^b$, has maximum size at least $\dim(E^a \oplus_1 E^b) + 1 = a + b + 1$ for all pairs of $a$ and $b$. We show, via some explicit constructions of equilateral sets, that this holds for all $a \leqslant 27$, as well as some other instances.