割范数与无界核的采样引理
The cut norm and Sampling Lemmas for unbounded kernels
AI总结：
本文推广有界核的采样引理至无界核，证明其随机样本的割范数以高概率接近原核，并给出样本在割度量下收敛的充分条件。
摘要：
将Borgs、Chayes、Lovász、Sós和Vesztergombi（2008）的有界核结果推广，我们针对割范数证明了无界核的两个采样引理。一方面，我们证明给定一个（对称）核$U\in L^p([0,1]^2)$，其中$3<p<\infty$，$U$的随机$k$-样本的割范数以高概率落在$U$的割范数的$O(k^{-\frac14+\frac{1}{4p}})$范围内。样本的割范数有强烈偏向于大于原始值的趋势，这使我们能够实际获得更强的关于其可能小多少的高概率界，阶为$O(k^{-\frac 12+\frac1p+\varepsilon})$（此处对任意$p>2$成立）。这些结果随后部分扩展到向量值核的情形。另一方面，我们证明以高概率，$k$-样本在割度量下也接近$U$，尽管界较弱，阶为$O((\ln k)^{-\frac12+\frac1{2p}})$（对任意适当的$p>2$）。作为推论，我们得到只要$U\in L^p$且$p>4$，当$k\to\infty$时，$k$-样本在割度量下几乎必然收敛到$U$。
英文摘要：
Generalizing the bounded kernel results of Borgs, Chayes, Lovász, Sós and Vesztergombi (2008), we prove two Sampling Lemmas for unbounded kernels with respect to the cut norm. On the one hand, we show that given a (symmetric) kernel $U\in L^p([0,1]^2)$ for some $3<p<\infty$, the cut norm of a random $k$-sample of $U$ is with high probability within $O(k^{-\frac14+\frac{1}{4p}})$ of the cut norm of $U$. The cut norm of the sample has a strong bias to being larger than the original, allowing us to actually obtain a stronger high probability bound of order $O(k^{-\frac 12+\frac1p+\varepsilon})$ for how much smaller it can be (for any $p>2$ here). These results are then partially extended to the case of vector valued kernels. On the other hand, we show that with high probability, the $k$-samples are also close to $U$ in the cut metric, albeit with a weaker bound of order $O((\ln k)^{-\frac12+\frac1{2p}})$ (for any appropriate $p>2$). As a corollary, we obtain that whenever $U\in L^p$ with $p>4$, the $k$-samples converge almost surely to $U$ in the cut metric as $k\to\infty$.
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