Local Lipschitz Filters for Bounded-Range Functions with Applications to Arbitrary Real-Valued Functions

TL;DR

This paper introduces nearly optimal local Lipschitz filters for bounded-range functions on arbitrary graphs, enabling efficient Lipschitz extension, private data release, and tolerant property testing with improved theoretical guarantees.

Contribution

It develops nearly optimal local Lipschitz filters for bounded-range functions on arbitrary graphs, overcoming previous lower bounds and enabling new applications.

Findings

Achieved running times $(d^r ext{log} n)^{O( ext{log} r)}$ and $d^{O(r)} ext{polylog } n$ for Lipschitz filters.

Provided nearly optimal dependence on $r$ for functions on $ ext{domain} extstyle o extstyle ext{range}=[0,r]$.

Resolved an open lower bound problem for general range functions, using a reduction from distribution-free Lipschitz testing.

Abstract

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.