大都会调整迪金游走的通用框架：多面体上的维度平方混合与谱面体上的对数行列式游走
A General Framework for Metropolis-Adjusted Dikin Walks: Dimension-Square Mixing on Polytopes and Log-Det Walks on Spectrahedra
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中文总结 AI 辅助
该研究提出大都会调整迪金游走的通用框架，推导多面体和谱面体上的混合步数界，通过提议比较和TensorSRHT构造实现精确算术实现，为相关采样问题提供理论与方法支撑。
中文摘要 AI 辅助
我们通过同时保留提议行列式和逆二次型来分析精确度量的大都会调整迪金游走。它们的主导非中心化项在完全对数接受率中抵消，留下可通过二阶工具控制的中心化波动。对于由n个不等式和凸L-利普希茨势构成的多面体，这为正则化Lee--Sidford游走产生了暖启动混合，所需步数为\\(\widetilde O((d^{2}+dL^{2}R^{2})\log(w/\delta))\\)。对于带有n×n块的谱面体，对数行列式游走的混合步数为\\(\widetilde O((\psi^\star nd+dL^{2}R^{2})\log(w/\delta))\\)，其中\\(\psi^\star\\)衡量矩阵杠杆率。两种分析共享接受到混合的归约。提议比较论证将多面体界转移到由高精度刘易斯权重计算的适当填充的O(1/d)精确度量。对于谱面体，给定\\(\widehat\psi\ge\psi^\star\\)，直接或双种子TensorSRHT构造给出精确算术实现，混合界中\\(\psi^\star\\)被替换为\\(\widehat\psi\\)。
英文摘要
We analyze exact-metric, Metropolis-adjusted Dikin walks by keeping the proposal determinant and reverse quadratic form together. Their leading uncentered terms cancel in the complete logarithmic acceptance ratio, leaving centered fluctuations that can be controlled with second-order tools. For a polytope given by $n$ inequalities and a convex $L$-Lipschitz potential, this yields warm-start mixing in $\widetilde O((d^{2}+dL^{2}R^{2})\log(w/δ))$ steps for the regularized Lee--Sidford walk. For a spectrahedron with $n\times n$ blocks, the log-det walk mixes in $\widetilde O((ψ^\star nd+dL^{2}R^{2})\log(w/δ))$ steps, where $ψ^\star$ measures matrix leverage. The two analyses share an acceptance-to-mixing reduction. A proposal-comparison argument transfers the polytope bound to an appropriately padded $O(1/d)$-accurate metric computed from high-precision Lewis weights. For spectrahedra, given $\widehatψ\geψ^\star$, a direct-or-two-seed TensorSRHT construction gives an exact-arithmetic implementation with $ψ^\star$ replaced by $\widehatψ$ in the mixing bound.
