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. We demonstrate that important applications of local Lipschitz filters can be accomplished with filters for functions with bounded-range. For functions $f: [n]^d\to [0,r]$, we circumvent the 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. Our local filters provide a novel Lipschitz extension that can be implemented locally. Furthermore, we show that our algorithms have nearly optimal dependence on $r$ for the domain $\{0,1\}^d$. 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 and a new technique for proving hardness for {\em adaptive} algorithms. We provide two applications of our local filters to arbitrary real-valued functions. In the first application, we use them in conjunction with the Laplace mechanism for differential privacy and noisy binary search to provide mechanisms for privately releasing outputs of black-box functions, even in the presence of malicious clients. In the second application, we use our local filters to obtain the first nontrivial tolerant tester 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性质是基于任意无向图$G=(V,E)$定义的。我们分别给出了关于$\ell_1$-距离和$\ell_0$-距离的近乎最优的局部Lipschitz滤波器。以往的工作仅考虑定义在$[n]^d$上的无界值函数。Jha和Raskhodnikova (SICOMP `13) 针对此类函数提出了一种查询复杂度在$d$上呈指数增长的算法，而Awasthi等人 (ACM Trans. Comput. Theory) 证明了在此设定下该复杂度是必要的。我们证明，局部Lipschitz滤波器的重要应用可通过针对有界值函数的滤波器实现。对于函数$f: [n]^d\to [0,r]$，我们突破了下界限制，实现了分别在$\ell_1$-保持滤波器中达到$(d^r\log n)^{O(\log r)}$的运行时间，以及在$\ell_0$-保持滤波器中达到$d^{O(r)}\text{polylog } n$的运行时间。我们的局部滤波器提供了一种可在局部实现的新型Lipschitz延拓方法。此外，我们证明在定义域$\{0,1\}^d$上，算法对参数$r$的依赖性近乎最优。同时，我们的下界结果解决了Awasthi等人提出的一个开放问题，去除了其关于一般值域下界证明中需要的条件之一。我们通过从无分布Lipschitz测试的归约和一种证明自适应算法难解性的新技术来证明下界。我们将局部滤波器应用于两类任意实值函数问题。第一个应用中，我们将其与差分隐私的拉普拉斯机制和噪声二分搜索结合，在存在恶意客户端的情况下，为黑盒函数的输出提供隐私保护发布机制。第二个应用中，我们利用局部滤波器首次获得了关于Lipschitz性质的非平凡容错测试器。