DROPS

Document

RANDOM

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.42

On the Spectral Expansion of Monotone Subsets of the Hypercube

Authors: Yumou Fei and Renato Ferreira Pinto Jr.

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract

We study the spectral gap of subgraphs of the hypercube induced by monotone subsets of vertices. For a monotone subset A ⊆ {0,1}ⁿ of density μ(A), the previous best lower bound on the spectral gap, due to Cohen [Cohen, 2016], was γ ≳ μ(A)/n², improving upon the earlier bound γ ≳ μ(A)²/n² established by Ding and Mossel [Ding and Mossel, 2014]. In this paper, we prove the optimal lower bound γ ≳ μ(A)/n. As a corollary, we improve the mixing time upper bound of the random walk on constant-density monotone sets from O(n³), as shown by Ding and Mossel, to O(n²). Along the way, we develop two new inequalities that may be of independent interest: (1) a directed L²-Poincaré inequality on the hypercube, and (2) an "approximate" FKG inequality for monotone sets.

Cite as

Yumou Fei and Renato Ferreira Pinto Jr.. On the Spectral Expansion of Monotone Subsets of the Hypercube. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 42:1-42:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

Copy BibTex To Clipboard

@InProceedings{fei_et_al:LIPIcs.APPROX/RANDOM.2025.42,
  author =	{Fei, Yumou and Ferreira Pinto Jr., Renato},
  title =	{{On the Spectral Expansion of Monotone Subsets of the Hypercube}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{42:1--42:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.42},
  URN =		{urn:nbn:de:0030-drops-244081},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.42},
  annote =	{Keywords: Random walks, mixing time, FKG inequality, Poincar\'{e} inequality, directed isoperimetry}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2024.47

Distribution Testing with a Confused Collector

Authors: Renato Ferreira Pinto Jr. and Nathaniel Harms

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract

We are interested in testing properties of distributions with systematically mislabeled samples. Our goal is to make decisions about unknown probability distributions, using a sample that has been collected by a confused collector, such as a machine-learning classifier that has not learned to distinguish all elements of the domain. The confused collector holds an unknown clustering of the domain and an input distribution μ, and provides two oracles: a sample oracle which produces a sample from μ that has been labeled according to the clustering; and a label-query oracle which returns the label of a query point x according to the clustering. Our first set of results shows that identity, uniformity, and equivalence of distributions can be tested efficiently, under the earth-mover distance, with remarkably weak conditions on the confused collector, even when the unknown clustering is adversarial. This requires defining a variant of the distribution testing task (inspired by the recent testable learning framework of Rubinfeld & Vasilyan), where the algorithm should test a joint property of the distribution and its clustering. As an example, we get efficient testers when the distribution tester is allowed to reject if it detects that the confused collector clustering is "far" from being a decision tree. The second set of results shows that we can sometimes do significantly better when the clustering is random instead of adversarial. For certain one-dimensional random clusterings, we show that uniformity can be tested under the TV distance using Õ((√n)/(ρ^{3/2} ε²)) samples and zero queries, where ρ ∈ (0,1] controls the "resolution" of the clustering. We improve this to O((√n)/(ρ ε²)) when queries are allowed.

Cite as

Renato Ferreira Pinto Jr. and Nathaniel Harms. Distribution Testing with a Confused Collector. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 47:1-47:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{ferreirapintojr._et_al:LIPIcs.ITCS.2024.47,
  author =	{Ferreira Pinto Jr., Renato and Harms, Nathaniel},
  title =	{{Distribution Testing with a Confused Collector}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{47:1--47:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.47},
  URN =		{urn:nbn:de:0030-drops-195755},
  doi =		{10.4230/LIPIcs.ITCS.2024.47},
  annote =	{Keywords: Distribution testing, property testing, uniformity testing, identity testing, earth-mover distance, sublinear algorithms}
}

Document

RANDOM

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.61

Directed Poincaré Inequalities and L¹ Monotonicity Testing of Lipschitz Functions

Authors: Renato Ferreira Pinto Jr.

Published in: LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

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 → R, and whether this question has implications for monotonicity testing. We answer both questions affirmatively. For Lipschitz functions f:[0,1]ⁿ → ℝ, we show the inequality d^mono₁(f) ≲ 𝔼 [‖∇^- f‖₁], which upper bounds the L¹ 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 L¹ monotonicity tester for Lipschitz functions f:[0,1]ⁿ → ℝ, and this framework also implies similar results for testing real-valued functions on the hypergrid.

Cite as

Renato Ferreira Pinto Jr.. Directed Poincaré Inequalities and L¹ Monotonicity Testing of Lipschitz Functions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 61:1-61:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{ferreirapintojr.:LIPIcs.APPROX/RANDOM.2023.61,
  author =	{Ferreira Pinto Jr., Renato},
  title =	{{Directed Poincar\'{e} Inequalities and L¹ Monotonicity Testing of Lipschitz Functions}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{61:1--61:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.61},
  URN =		{urn:nbn:de:0030-drops-188867},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.61},
  annote =	{Keywords: Monotonicity testing, property testing, isoperimetric inequalities, Poincar\'{e} inequalities}
}

Search Results

Documents authored by Ferreira Pinto Jr., Renato

On the Spectral Expansion of Monotone Subsets of the Hypercube

Abstract

Cite as

Distribution Testing with a Confused Collector

Abstract

Cite as

Directed Poincaré Inequalities and L¹ Monotonicity Testing of Lipschitz Functions

Abstract

Cite as

Thanks for your feedback!

Could not send message