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 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：我们研究了实值函数$f: V \to [0,r]$关于Lipschitz性质的局部滤波器，其中Lipschitz性质是相对于任意无向图$G=(V,E)$定义的。我们针对$\ell_1$距离和$\ell_0$距离分别给出了近乎最优的局部Lipschitz滤波器。以往的研究仅考虑了定义域为$[n]^d$的无界范围函数。Jha和Raskhodnikova（SICOMP `13）针对此类函数提出了一种算法，其查询复杂度随$d$呈指数增长，而Awasthi等人（ACM Trans. Comput. Theory）证明在此设定下这种复杂度是必要的。我们证明，局部Lipschitz滤波器的重要应用可以通过针对有界范围函数的滤波器来实现。对于函数$f: [n]^d\to [0,r]$，我们规避了下界限制，并在$\ell_1$保持滤波器上实现了运行时间$(d^r\log n)^{O(\log r)}$，在$\ell_0$保持滤波器上实现了$d^{O(r)}\text{polylog } n$。我们的局部滤波器提供了一种可在局部实现的新型Lipschitz拓展。此外，我们证明对于定义域$\{0,1\}^d$，我们的算法在参数$r$上具有近乎最优的依赖关系。同时，我们的下界解决了Awasthi等人提出的一个开放问题，移除了其关于一般范围下界所需的某个条件。我们通过从分布无关Lipschitz测试的归约以及一种证明自适应算法困难性的新技术来证明下界。我们提供了局部滤波器在任意实值函数上的两个应用。在第一个应用中，我们将这些滤波器与用于差分隐私的拉普拉斯机制及噪声二分搜索相结合，从而在存在恶意客户端的情况下，为黑箱函数的隐私保护输出提供机制。在第二个应用中，我们利用局部滤波器首次获得了针对Lipschitz性质的非平凡容忍测试器。