论文标题
通过随机抽样近似$ l_p $单位球
Approximating $L_p$ unit balls via random sampling
论文作者
论文摘要
令$ x $为$ r^d $中的同型随机向量，可以满足s^{d-1} $中的每个$ v \，$ \ | <x，v> \ | _ {l_q} \ leq l \ | <| <| <x，v>> \ | <x，v> \ | _ {l_p} $ for $ q Q \ geq qeq 2p $。我们表明，以$ 0 <\ varepsilon <1 $，一组$ n = c（p，q，q，\ varepsilon）d $随机点，根据$ x $独立选择，可用于构建$ 1 \ pm \ varepsilon $ your the $ l_p $ ball of $ r r^d $ x $ x $ x $ x $ x $ by x $。此外，$ c（p，q，\ varepsilon）\ leq c^p \ varepsilon^{ - 2} \ log（2/\ varepsilon）$;当$ q = 2p $以概率至少$ 1-2 \ exp（-cn \ varepsilon^2/\ log^2（2/\ varepsilon））$时，并且如果$ q $要大于$ p $ - 例如，$ q = 4p $，则至少在$ 1-2 \ 2 var（-cn）中实现了近似值，而$ q $至少要大于$ q = 4p $ c（-ccn），则至少$ q $要大于$ p $ - q = 4p \ c。 特别是，当$ x $是一个log-concove随机向量时，此估计值改善了先前的最先进的---- $ n = c^\ prime（p，\ varepsilon）d^{p/2} \ log d $ d $随机点足够，并且近似值具有恒定的可能性。
Let $X$ be an isotropic random vector in $R^d$ that satisfies that for every $v \in S^{d-1}$, $\|<X,v>\|_{L_q} \leq L \|<X,v>\|_{L_p}$ for some $q \geq 2p$. We show that for $0<\varepsilon<1$, a set of $N = c(p,q,\varepsilon) d$ random points, selected independently according to $X$, can be used to construct a $1 \pm \varepsilon$ approximation of the $L_p$ unit ball endowed on $R^d$ by $X$. Moreover, $c(p,q,\varepsilon) \leq c^p \varepsilon^{-2}\log(2/\varepsilon)$; when $q=2p$ the approximation is achieved with probability at least $1-2\exp(-cN \varepsilon^2/\log^2(2/\varepsilon))$ and if $q$ is much larger than $p$---say, $q=4p$, the approximation is achieved with probability at least $1-2\exp(-cN \varepsilon^2)$. In particular, when $X$ is a log-concave random vector, this estimate improves the previous state-of-the-art---that $N=c^\prime(p,\varepsilon) d^{p/2}\log d$ random points are enough, and that the approximation is valid with constant probability.
