We study local filters for the Lipschitz property of real-valued functions $f: V \to [0,r]$, where the Lipschitz property is defined with respect to an arbitrary undirected graph $G=(V,E)$. We give nearly optimal local Lipschitz filters both with respect to $\ell_1$ distance and $\ell_0$ distance. Previous work only considered unbounded-range functions over $[n]^d$. Jha and Raskhodnikova (SICOMP `13) gave an algorithm for such functions with lookup complexity exponential in $d$, which Awasthi et al.\ (ACM Trans. Comput. Theory) showed was necessary in this setting. By considering the natural class of functions whose range is bounded in $[0,r]$, we circumvent this lower bound and achieve running time $(d^r\log n)^{O(\log r)}$ for the $\ell_1$-respecting filter and $d^{O(r)}\text{polylog }n$ for the $\ell_0$-respecting filter for functions over $[n]^d$. Furthermore, we show that our algorithms are nearly optimal in terms of the dependence on $r$ for the domain $\{0,1\}^d$, an important special case of the domain $[n]^d$. In addition, our lower bound resolves an open question of Awasthi et al., removing one of the conditions necessary for their lower bound for general range. We prove our lower bound via a reduction from distribution-free Lipschitz testing. Finally, we provide two applications of our local filters. First, they can be used in conjunction with the Laplace mechanism for differential privacy to provide filter mechanisms for privately releasing outputs of black box functions even in the presence of malicious clients. Second, we use them to obtain the first tolerant testers for the Lipschitz property.

翻译：我们研究实值函数 $f: V \to [0,r]$ 关于Lipschitz性质的局部滤波器，其中Lipschitz性质是针对任意无向图 $G=(V,E)$ 定义的。我们给出了在 $\ell_1$ 距离和 $\ell_0$ 距离下近乎最优的局部Lipschitz滤波器。先前的研究仅考虑了定义在 $[n]^d$ 上的无界值函数。Jha和Raskhodnikova (SICOMP `13) 针对此类函数提出了一种查找复杂度随 $d$ 呈指数增长的算法，而Awasthi等人 (ACM Trans. Comput. Theory) 证明在此设置下该复杂度是必要的。通过考虑值域有界于 $[0,r]$ 的自然函数类，我们规避了这一下界，并针对定义在 $[n]^d$ 上的函数实现了 $\ell_1$ 尊重滤波器的运行时间为 $(d^r\log n)^{O(\log r)}$，以及 $\ell_0$ 尊重滤波器的运行时间为 $d^{O(r)}\text{polylog }n$。此外，我们证明在定义域 $\{0,1\}^d$（即 $[n]^d$ 的重要特例）上，我们的算法在依赖参数 $r$ 的意义下是近乎最优的。同时，我们的下界解决了Awasthi等人提出的一个开放问题，去除了其针对一般值域下界所需的某个条件。我们通过从分布无关的Lipschitz测试进行归约来证明该下界。最后，我们给出了局部滤波器的两个应用。其一，它们可与差分隐私中的拉普拉斯机制结合，为黑盒函数在恶意客户端存在时的隐私输出提供滤波机制。其二，我们利用它们获得了首个针对Lipschitz性质的容错测试器。