AI 中文总结
该研究针对$1\leq p<2$且$p\notin2\mathbb{Z}$的情况，建立了$\ell_p$子空间嵌入的近最优下界，改进了Li等人的此前结果，核心技术思路源自ChatGPT 5.6 Sol。
AI 中文摘要
对于$d\geq2$、$p\geq1$和$\epsilon>0$，设$N_p(d,\epsilon)$为满足对任意整数$n$与所有$A\in\mathbb{R}^{n\times d}$，存在矩阵$\Phi\in\mathbb{R}^{N\times n}$，使得对所有$x\in\mathbb{R}^d$都有$(1-\epsilon)\lVert Ax\rVert_p\leq\lVert\Phi A x\rVert_p\leq(1+\epsilon)\lVert Ax\rVert_p$的最小整数$N$。对所有满足$p\geq1$且$p\notin2\mathbb{Z}$的常数，当$d\gtrsim_p\log(1/\epsilon)$时，建立了下界$N_p(d,\epsilon)\gtrsim_p\frac{d}{\epsilon^2\operatorname{polylog}(d/\epsilon)}$，该结果改进了Li等人（SICOMP 2021）提出的此前下界$\Omega(1/(\epsilon^2\operatorname{polylog}(1/\epsilon)))$，且在$1\leq p<2$时的下界在对数因子范围内是最优的，核心技术思路源自ChatGPT 5.6 Sol。
英文摘要
For $d \geq 2$, $p \geq 1$ and $ε> 0$, let $N_p(d,ε)$ be the smallest integer $N$ such that for every integer $n$ and every $A\in\mathbb{R}^{n\times d}$, there exists a matrix $Φ\in\mathbb{R}^{N\times n}$ satisfying $(1-ε)\lVert Ax\rVert_p\leq \lVertΦA x\rVert_p\leq (1+ε)\lVert Ax\rVert_p$ for all $x\in\mathbb{R}^d$. For every constant $p\geq 1$ with $p\not\in 2\mathbb{Z}$, when $d\gtrsim_p \log(1/ε)$, the bound \[ N_p(d,ε) \gtrsim_{p} \frac{d}{ε^2 \operatorname{polylog}(d/ε)} \] is established. This improves the previous lower bound $Ω(1/(ε^2\operatorname{polylog}(1/ε)))$ due to Li et al. (SICOMP 2021) and is optimal up to logarithmic factors for $1\leq p<2$. The central technical idea originated from ChatGPT 5.6 Sol.
