AI 中文总结
该研究针对满足包含关系的凸体，推导了ℓ₀等周系数的改进下界，应用于坐标命中跑算法的混合时间界，还给出互补上界，上下界存在n倍差距。
AI 中文摘要
我们研究ℝⁿ（n≥2）中凸体K的ℓ₀等周性。对于K的Borel集S，令∂₀ᴷS为K\backslashS中可通过改变至多一个坐标（即S的ℓ₀边界）从S到达的点集。假设对于某个无条件凸体Q⊂ℝⁿ、正数r和R，以及可能不同的中心x₀、y₀，满足x₀+rQ⊂K⊂y₀+RQ。记s=vol(S)/vol(K)，我们证明当0<s≤1/2时，vol(∂₀ᴷS)/vol(S)≥(cr)/(nR)min{1,log(e/s)/n}，其中c>0为绝对常数。由此可得，相关的ℓ₀等周系数至少为cr/(n²R)。此前仅已知ℓ₂和ℓ∞正则性的直接下界，而我们的下界直接适用于任意Q正则性（Q为无条件凸体）。与ℓ₂和ℓ∞正则性相比，我们的下界结果在任意s下均比此前最佳已知下界提升了n倍。作为应用，我们给出了坐标命中跑算法（Coordinate Hit and Run walk，CHAR）的改进混合时间界。我们的下界证明基于对规范路径方法的改进，应用于凸体上的连续汉明图；规范路径的构造可视为对从S到S^c的某些质量传输映射进行适当的坐标离散化。我们还给出了任意Q正则性的互补上界，上下界之间整体存在n倍的差距。
英文摘要
We study $\ell_0$ isoperimetry for a convex body $K\subset \mathbb{R}^n$, $n\ge2$. For a Borel set $S\subset K$, let $\partial_0^K S$ be the set of points in $K \setminus S$ that can be reached from $S$ by changing at most one coordinate (i.e. the $\ell_0$ boundary of $S$). Suppose that, for some unconditional convex body $Q \subset \mathbb{R}^n$, numbers $r,R>0$, and possibly different centers $x_0,y_0$, \[ x_0+rQ \subset K\subset y_0+RQ. \] Writing $s=\text{vol}(S)/\text{vol}(K)$, we prove that whenever $0<s \le 1/2$, \[ \frac{\text{vol}(\partial_0^K S)}{\text{vol}(S)} \ge \frac{cr}{nR} \min\left\{1,\frac{\log(e/s)}{n}\right\}, \] where $c > 0$ is an absolute constant. Consequently, the associated $\ell_0$-isoperimetric coefficient is at least $cr/(n^2R)$. Previous direct lower bounds were only known for $\ell_2$ and $\ell_\infty$ regularity whereas our lower bound holds directly for any $Q$-regularity, where $Q$ is an unconditional convex body. Compared to $\ell_2$ and $\ell_\infty$ regularity, our lower bound result improves upon the previously best known lower bounds, for any $s$, by a factor of $n$. As an application of our result, we give improved mixing time bounds for the Coordinate Hit and Run walk (CHAR). Our proof of the lower bound is based on a modification of the method of canonical paths applied to a continuous Hamming graph over our convex body. Our construction of canonical paths can be viewed as a suitable coordinate discretization of certain mass transport maps from $S$ to $S^c$. We also give complementary upper-bounds for any $Q$-regularity, with an overall factor of $n$ gap between the two.
Comments36 pages, 2 figures
