论文标题
高维域的离散化
Discretization on high-dimensional domains
论文作者
论文摘要
令$μ$为紧凑的路径连接度量空间$（x，ρ）$的borel概率度量，其中存在常数$ c，β> 1 $，以便每个开放球$ b \ geq c r^β$ for a radius $ r> r> r> r> r> r> r> 0 $。对于一类LipsChitz的功能$φ：[0，\ infty）\至r $，将有限维度的连续功能子空间分别为r $，我们在某些温和条件下，在公制$ρ$的某些温和条件下，对于每个正面整数$ n \ geq 2 $，以及每个$ g fives $ g \ in l lim g \ in l l^y l^y l^fty（x） $ \ | g \ | _ \ infty = 1 $，存在点$ y_1，\ ldots，y_ {n} \ in x $和实数$λ_1，\ ldots，λ_{n} $，因此对于任何$ x \ in x $， \ begin {align*} ＆\ left | \int_xφ（ρ（x，y））g（y）\，dμ（y） - \ sum_ {j = = 1}^{n}λ_jφ（ρ（x，y_j））\ right | \ leq c n^{ - \ frac {1} {2} - \ frac {3} {2β}}} \ sqrt {\ log n}， \ end {align*} 其中常数$ c> 0 $独立于$ n $和$ g $。如果$ x $是$ r^{d+1} $的单位球$ s^d $，则具有USUSAL GEODESIC距离时，我们还证明，这里的常数$ c $在这里独立于尺寸$ d $。我们的估计值比从标准蒙特卡洛方法获得的估计值更好，该方法通常产生较弱的上限$ n^{ - \ frac12} \ sqrt {\ log n} $。
Let $μ$ be a Borel probability measure on a compact path-connected metric space $(X, ρ)$ for which there exist constants $c,β>1$ such that $μ(B) \geq c r^β$ for every open ball $B\subset X$ of radius $r>0$. For a class of Lipschitz functions $Φ:[0,\infty)\to R$ that piecewisely lie in a finite-dimensional subspace of continuous functions, we prove under certain mild conditions on the metric $ρ$ and the measure $μ$ that for each positive integer $N\geq 2$, and each $g\in L^\infty(X, dμ)$ with $\|g\|_\infty=1$, there exist points $y_1, \ldots, y_{ N}\in X$ and real numbers $λ_1, \ldots, λ_{ N}$ such that for any $x\in X$, \begin{align*} & \left| \int_X Φ(ρ(x, y)) g(y) \,d μ(y) - \sum_{j = 1}^{ N} λ_j Φ(ρ(x, y_j)) \right| \leq C N^{- \frac{1}{2} - \frac{3}{2β}} \sqrt{\log N}, \end{align*} where the constant $C>0$ is independent of $N$ and $g$. In the case when $X$ is the unit sphere $S^d$ of $R^{d+1}$ with the ususal geodesic distance, we also prove that the constant $C$ here is independent of the dimension $d$. Our estimates are better than those obtained from the standard Monte Carlo methods, which typically yield a weaker upper bound $N^{-\frac12}\sqrt{\log N}$.
