改进了加权Dikin随机游走的混合时间分析，实现d²级采样效率。
On two proofs of $d^2$ mixing of weighted Dikin walks
- 通过控制高概率区域的接受率，建立总变差混合上界。
- 对多面体采样得$ ilde{O}(d^2)$混合时间，对截断PSD锥得$ ilde{O}(d^4)$。
- 首次在χ²散度下实现$ ilde{O}(d^2)$，适合优化与采样研究者。
本文研究在多面体和截断正定锥上采样指数分布时加权Dikin随机游走的混合时间。首个结果在强自协调性、$arν$-对称性和局部度量混合迹正则性条件下，给出总变差混合上界。核心思路是控制高概率区域的Metropolis-Hastings接受率而非所有点。将该框架应用于Lee-Sidford、Lewis权重及John度量，得到多面体采样的$ ilde{O}(d^2)$混合时间；应用于混合势函数，得到截断正定锥采样的$ ilde{O}(d^4)$混合时间。第二个结果引入新的四阶自举条件，获得更强的χ²散度保证和逐点接受率控制。对适当缩放的Lee-Sidford度量，χ²散度下的混合时间提升至$ ilde{O}(d^2)$，优于此前$ ilde{O}(d^{9/4})$的界限。
原文摘要 · Abstract (English)
We study the mixing time of weighted Dikin walks for sampling from exponential distributions on polytopes and truncated positive-semidefinite (PSD) cones. Our first result gives a general total-variation mixing bound under strong self-concordance, $\barν$-symmetry, and mixed-trace regularity on the local metric. The key idea is to control the Metropolis--Hastings acceptance probability on a high-probability region rather than at every point. Applying this framework to the Lee--Sidford, Lewis-weight, and John metrics yields an $\widetilde O(d^2)$ mixing bound for sampling from polytopes, while applying it to a hybrid barrier yields an $\widetilde O(d^4)$ mixing bound for sampling from truncated PSD cones. Our second result establishes stronger $χ^2$-divergence guarantees and pointwise acceptance control using a new fourth-order bootstrap condition. For a suitably scaled Lee--Sidford metric, this yields an $\widetilde O(d^2)$ mixing bound in $χ^2$-divergence, improving on the previous $\widetilde O(d^{9/4})$ bound.
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