发表机构
Princeton（普林斯顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
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
AI 中文摘要
固定0<θ≤1，我们证明：若1≤p≤2/θ，则ℓ₂ᵏ的θ-雪花（即配备度量((x,y)∈ℝᵏ×ℝᵏ)→‖x−y‖₂^θ的ℝᵏ）可以O(1)的失真嵌入到ℓₚᵐ中，其中整数m≲_{p,θ}k，当k→∞时该结果是最优的，可通过维度比较看出。然而，当p大于阈值2/θ时，行为发生变化：若欧几里得球面S^{k−1}的一个(1/√k)-稠密子集以O(1)的失真嵌入到ℓₚᵐ中，则必有m≳_{p,θ}(k/logk)^{pθ/2}，当pθ/2>1时，该值随k超线性增长，且当k→∞时，该维度界在低阶因子范围内是最优的。我们由此推论：若2<p<∞，则ℓₚ中存在任意大的n点子集，其性质为：若它们以O(1)的失真嵌入到ℓₚᵐ中，则必有m≳_p((logn)/(loglogn)²)^{p/2}，这表明Johnson-Lindenstrauss降维引理的结论对ℓₚ不成立
英文摘要
Fix $0<θ\leqslant 1$. We prove that if $1\leqslant p \leqslant 2/θ$, then the $θ$-snowflake of $\ell_2^k$, namely, $\mathbb{R}^k$ equipped with the metric $((x,y)\in \mathbb{R}^k\times \mathbb{R}^k)\mapsto \|x-y\|_2^θ$, embeds with distortion $O(1)$ into $\ell_p^m$ for some integer $m\lesssim_{p,θ}k$, which is optimal as $k\to \infty$, as seen by comparing dimensions. However, for $p$ larger than the sharp threshold $2/θ$ the following change in behavior occurs: If a $(1/\sqrt{k})$-dense subset of the Euclidean sphere $S^{k-1}$ embeds into $\ell_p^m$ with distortion $O(1)$, then necessarily $m\gtrsim_{p,θ}( k/\log k)^{pθ/2}$, which grows super-linearly in $k$ as $pθ/2>1$, and this dimension bound is optimal as $k\to \infty$ up to lower order factors. We deduce from this statement that if $2<p<\infty$, then there exist arbitrarily large $n$-point subsets of $\ell_p$ with the property that if they embed with distortion $O(1)$ into $\ell_p^m$, then necessarily $m\gtrsim_p ((\log n)/(\log\log n)^2)^{p/2}$, thus demonstrating that the statement of the Johnson--Lindenstrauss dimension reduction lemma fails to hold for $\ell_p$
