利用多尺度分解从冷启动凸集采样
Sampling from convex sets with a cold start using multiscale decompositions
AI总结：
本文提出基于多尺度分解的随机游走族，证明从冷启动（初始密度可指数大）出发在凸体上混合时间仍为多项式，并推论坐标命中并运行游走也快速混合。
摘要：
在凸体 $K\subseteq\mathbb{R}^n$ 中运行随机游走是从该体中近似均匀采样的标准方法。要求是从合适的初始分布出发，游走的分布经过多项式步数（在 $n$ 和纵横比 $R/r$ 上，即当 $rB_2 \subseteq K \subseteq RB_{2}$ 时）后接近 $K$ 上的均匀分布 $π_K$。此类游走快速混合的证明通常要求初始分布相对于 $π_K$ 的概率密度 $η_0$ 至多为 $\mathrm{poly}(n)$：这被称为“暖启动”。实现暖启动通常需要在开始随机游走之前进行非平凡的预处理。这促使我们证明从“冷启动”出发的快速混合，其中 $η_0$ 可以高达 $\exp(\mathrm{poly}(n))$。与暖启动不同，冷启动通常很容易实现。然而，随机游走从冷启动不一定能快速混合：一个例子是众所周知的“球游走”。另一方面，Lovász 和 Vempala 证明了“命中并运行”随机游走从冷启动快速混合。对于相关的坐标命中并运行（CHR）游走，在计算实验中已被证明有前景，其从暖启动的快速混合最近才被证明，但从冷启动的快速混合问题仍然开放。我们构造了一族随机游走，灵感来自将 $\mathbb{R}^n$ 的子集分解为可数个轴对齐的二进立方体的经典分解。我们证明即使从冷启动出发，这些游走的混合时间也受 $n$ 和纵横比的多项式界限。我们的主要技术成分是 $K$ 的等周不等式，针对一种放大 $K$ 边界附近点之间距离的度量。作为推论，我们证明了 CHR 游走也从冷启动和从不太靠近 $K$ 边界的点快速混合。
英文摘要：
Running a random walk in a convex body $K\subseteq\mathbb{R}^n$ is a standard approach to sample approximately uniformly from the body. The requirement is that from a suitable initial distribution, the distribution of the walk comes close to the uniform distribution $π_K$ on $K$ after a number of steps polynomial in $n$ and the aspect ratio $R/r$ (i.e., when $rB_2 \subseteq K \subseteq RB_{2}$). Proofs of rapid mixing of such walks often require the probability density $η_0$ of the initial distribution with respect to $π_K$ to be at most $\mathrm{poly}(n)$: this is called a "warm start". Achieving a warm start often requires non-trivial pre-processing before starting the random walk. This motivates proving rapid mixing from a "cold start", wherein $η_0$ can be as high as $\exp(\mathrm{poly}(n))$. Unlike warm starts, a cold start is usually trivial to achieve. However, a random walk need not mix rapidly from a cold start: an example being the well-known "ball walk". On the other hand, Lovász and Vempala proved that the "hit-and-run" random walk mixes rapidly from a cold start. For the related coordinate hit-and-run (CHR) walk, which has been found to be promising in computational experiments, rapid mixing from a warm start was proved only recently but the question of rapid mixing from a cold start remained open. We construct a family of random walks inspired by classical decompositions of subsets of $\mathbb{R}^n$ into countably many axis-aligned dyadic cubes. We show that even with a cold start, the mixing times of these walks are bounded by a polynomial in $n$ and the aspect ratio. Our main technical ingredient is an isoperimetric inequality for $K$ for a metric that magnifies distances between points close to the boundary of $K$. As a corollary, we show that the CHR walk also mixes rapidly both from a cold start and from a point not too close to the boundary of $K$.
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