Metric structures in L_1: Dimension, snowflakes, and average distortion

Embedding snowflakes of Carnot groups into bounded dimensional Euclidean spaces with optimal distortion

We show that for any Carnot group $G$ there exists a natural number $D_G$ such that for any $0<\varepsilon<1/2$ the metric space $(G,d_G^{1-\varepsilon})$ admits a bi-Lipschitz embedding into $\mathbb{R}^{D_G}$ with distortion $O_G(\varepsilon^{-1/2})$. This is done by building on the approach of T. Tao (2021), who established the above assertion when $G$ is the Heisenberg group using a new variant of the Nash--Moser iteration scheme combined with a new extension theorem for orthonormal vector fields. Beyond the need to overcome several technical issues that arise in the more general setting o…

欧几里得雪花嵌入的阈值现象与降维的不可能性

A threshold phenomenon for embeddings of Euclidean snowflakes and impossibility of dimension reduction

该研究证明了欧几里得雪花嵌入存在阈值现象，当p超过阈值2/θ时，ℓₚ中点集的低失真嵌入需要超线性增长的维度，表明Johnson-Lindenstrauss降维引理对ℓₚ不成立

Snowflake universality of Wasserstein spaces

For $p\in (1,\infty)$ let $\mathscr{P}_p(\mathbb{R}^3)$ denote the metric space of all $p$-integrable Borel probability measures on $\mathbb{R}^3$, equipped with the Wasserstein $p$ metric $\mathsf{W}_p$. We prove that for every $\varepsilon>0$, every $θ\in (0,1/p]$ and every finite metric space $(X,d_X)$, the metric space $(X,d_{X}^θ)$ embeds into $\mathscr{P}_p(\mathbb{R}^3)$ with distortion at most $1+\varepsilon$. We show that this is sharp when $p\in (1,2]$ in the sense that the exponent $1/p$ cannot be replaced by any larger number. In fact, for arbitrarily large $n\in \mathbb{N}$ there …

An average John theorem

We prove that the $\frac12$-snowflake of a finite-dimensional normed space $(X,\|\cdot\|_X)$ embeds into a Hilbert space with quadratic average distortion $$O\Big(\sqrt{\log \mathrm{dim}(X)}\Big).$$ We deduce from this (optimal) statement that if an $n$-vertex expander embeds with average distortion $D\geqslant 1$ into $(X,\|\cdot\|_X)$, then necessarily $\mathrm{dim}(X)\geqslant n^{Ω(1/D)}$, which is sharp by the work of Johnson, Lindenstrauss and Schechtman (1987). This improves over the previously best-known bound $\mathrm{dim}(X)\gtrsim (\log n)^2/D^2$ of Linial, London and Rabinovich (199…

Embedding the Heisenberg group into a bounded dimensional Euclidean space with optimal distortion

Let $H := \begin{pmatrix} 1 & {\mathbf R} & {\mathbf R} \\ 0 & 1 &{\mathbf R} \\ 0 & 0 & 1 \end{pmatrix}$ denote the Heisenberg group with the usual Carnot-Carathéodory metric $d$. It is known (since the work of Pansu and Semmes) that the metric space $(H,d)$ cannot be embedded in a bilipchitz fashion into a Hilbert space; however, from a general theorem of Assouad, for any $0 < \varepsilon < 1$, the snowflaked metric space $(H,d^{1-\varepsilon})$ embeds into an infinite-dimensional Hilbert space with distortion $O( \varepsilon^{-1/2} )$. This distortion bound was shown by Austin, Naor, and Te…

A Nonlinear Approach to Dimension Reduction

The $l_2$ flattening lemma of Johnson and Lindenstrauss [JL84] is a powerful tool for dimension reduction. It has been conjectured that the target dimension bounds can be refined and bounded in terms of the intrinsic dimensionality of the data set (for example, the doubling dimension). One such problem was proposed by Lang and Plaut [LP01] (see also [GKL03,MatousekProblems07,ABN08,CGT10]), and is still open. We prove another result in this line of work: The snowflake metric $d^{1/2}$ of a doubling set $S \subset l_2$ embeds with constant distortion into $l_2^D$, for dimension $D$ that depends …