Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2307.02193

Directed Poincar\'e Inequalities and $L^1$ Monotonicity Testing of Lipschitz Functions

Renato Ferreira Pinto Jr
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the connection between directed isoperimetric inequalities and monotonicity testing. In recent years, this connection has unlocked breakthroughs for testing monotonicity of functions defined on discrete domains. Inspired the rich history of isoperimetric inequalities in continuous settings, we propose that studying the relationship between directed isoperimetry and monotonicity in such settings is essential for understanding the full scope of this connection. Hence, we ask whether directed isoperimetric inequalities hold for functions $f : [0,1]^n \to \mathbb{R}$, and whether this question has implications for monotonicity testing. We answer both questions affirmatively. For Lipschitz functions $f : [0,1]^n \to \mathbb{R}$, we show the inequality $d^{\mathsf{mono}}_1(f) \lesssim \mathbb{E}\left[\|\nabla^- f\|_1\right]$, which upper bounds the $L^1$ distance to monotonicity of $f$ by a measure of its "directed gradient". A key ingredient in our proof is the monotone rearrangement of $f$, which generalizes the classical "sorting operator" to continuous settings. We use this inequality to give an $L^1$ monotonicity tester for Lipschitz functions $f : [0,1]^n \to \mathbb{R}$, and this framework also implies similar results for testing real-valued functions on the hypergrid.

Explore related subjects

cs.DS

Keep this discovery

BibTeXRIS

Renato Ferreira Pinto Jr. 2023-07-05. Directed Poincar\'e Inequalities and $L^1$ Monotonicity Testing of Lipschitz Functions. https://arxiv.org/abs/2307.02193

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗
arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗