4 Search Results for "Ivanov, Peter"

Document
DOI: 10.4230/LIPIcs.CCC.2023.3
On Correlation Bounds Against Polynomials

Authors: Peter Ivanov, Liam Pavlovic, and Emanuele Viola

Published in: LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

Abstract
We study the fundamental challenge of exhibiting explicit functions that have small correlation with low-degree polynomials over 𝔽₂. Our main contributions include: 1) In STOC 2020, CHHLZ introduced a new technique to prove correlation bounds. Using their technique they established new correlation bounds for low-degree polynomials. They conjectured that their technique generalizes to higher degree polynomials as well. We give a counterexample to their conjecture, in fact ruling out weaker parameters and showing what they prove is essentially the best possible. 2) We propose a new approach for proving correlation bounds with the central "mod functions," consisting of two steps: (I) the polynomials that maximize correlation are symmetric and (II) symmetric polynomials have small correlation. Contrary to related results in the literature, we conjecture that (I) is true. We argue this approach is not affected by existing "barrier results." 3) We prove our conjecture for quadratic polynomials. Specifically, we determine the maximum possible correlation between quadratic polynomials modulo 2 and the functions (x_1,… ,x_n) → z^{∑ x_i} for any z on the complex unit circle, and show that it is achieved by symmetric polynomials. To obtain our results we develop a new proof technique: we express correlation in terms of directional derivatives and analyze it by slowly restricting the direction. 4) We make partial progress on the conjecture for cubic polynomials, in particular proving tight correlation bounds for cubic polynomials whose degree-3 part is symmetric.
Cite as

Peter Ivanov, Liam Pavlovic, and Emanuele Viola. On Correlation Bounds Against Polynomials. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 3:1-3:35, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{ivanov_et_al:LIPIcs.CCC.2023.3,
  author =	{Ivanov, Peter and Pavlovic, Liam and Viola, Emanuele},
  title =	{{On Correlation Bounds Against Polynomials}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{3:1--3:35},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-282-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{264},
  editor =	{Ta-Shma, Amnon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.3},
  URN =		{urn:nbn:de:0030-drops-182734},
  doi =		{10.4230/LIPIcs.CCC.2023.3},
  annote =	{Keywords: Correlation bounds, Polynomials}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2022.9
Affine Extractors and AC0-Parity

Authors: Xuangui Huang, Peter Ivanov, and Emanuele Viola

Published in: LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

Abstract
We study a simple and general template for constructing affine extractors by composing a linear transformation with resilient functions. Using this we show that good affine extractors can be computed by non-explicit circuits of various types, including AC0-Xor circuits: AC0 circuits with a layer of parity gates at the input. We also show that one-sided extractors can be computed by small DNF-Xor circuits, and separate these circuits from other well-studied classes. As a further motivation for studying DNF-Xor circuits we show that if they can approximate inner product then small AC0-Xor circuits can compute it exactly - a long-standing open problem.
Cite as

Xuangui Huang, Peter Ivanov, and Emanuele Viola. Affine Extractors and AC0-Parity. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{huang_et_al:LIPIcs.APPROX/RANDOM.2022.9,
  author =	{Huang, Xuangui and Ivanov, Peter and Viola, Emanuele},
  title =	{{Affine Extractors and AC0-Parity}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{9:1--9:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-249-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{245},
  editor =	{Chakrabarti, Amit and Swamy, Chaitanya},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.9},
  URN =		{urn:nbn:de:0030-drops-171313},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.9},
  annote =	{Keywords: affine extractor, resilient function, constant-depth circuit, parity gate, inner product}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.80
Mixing in Non-Quasirandom Groups

Authors: W. T. Gowers and Emanuele Viola

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
We initiate a systematic study of mixing in non-quasirandom groups. Let A and B be two independent, high-entropy distributions over a group G. We show that the product distribution AB is statistically close to the distribution F(AB) for several choices of G and F, including: 1) G is the affine group of 2x2 matrices, and F sets the top-right matrix entry to a uniform value, 2) G is the lamplighter group, that is the wreath product of ℤ₂ and ℤ_{n}, and F is multiplication by a certain subgroup, 3) G is Hⁿ where H is non-abelian, and F selects a uniform coordinate and takes a uniform conjugate of it. The obtained bounds for (1) and (2) are tight. This work is motivated by and applied to problems in communication complexity. We consider the 3-party communication problem of deciding if the product of three group elements multiplies to the identity. We prove lower bounds for the groups above, which are tight for the affine and the lamplighter groups.
Cite as

W. T. Gowers and Emanuele Viola. Mixing in Non-Quasirandom Groups. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 80:1-80:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{gowers_et_al:LIPIcs.ITCS.2022.80,
  author =	{Gowers, W. T. and Viola, Emanuele},
  title =	{{Mixing in Non-Quasirandom Groups}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{80:1--80:9},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.80},
  URN =		{urn:nbn:de:0030-drops-156761},
  doi =		{10.4230/LIPIcs.ITCS.2022.80},
  annote =	{Keywords: Groups, representation theory, mixing, communication complexity, quasi-random}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.53
Fourier Growth of Structured 𝔽₂-Polynomials and Applications

Authors: Jarosław Błasiok, Peter Ivanov, Yaonan Jin, Chin Ho Lee, Rocco A. Servedio, and Emanuele Viola

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract
We analyze the Fourier growth, i.e. the L₁ Fourier weight at level k (denoted L_{1,k}), of various well-studied classes of "structured" m F₂-polynomials. This study is motivated by applications in pseudorandomness, in particular recent results and conjectures due to [Chattopadhyay et al., 2019; Chattopadhyay et al., 2019; Eshan Chattopadhyay et al., 2020] which show that upper bounds on Fourier growth (even at level k = 2) give unconditional pseudorandom generators. Our main structural results on Fourier growth are as follows: - We show that any symmetric degree-d m F₂-polynomial p has L_{1,k}(p) ≤ Pr [p = 1] ⋅ O(d)^k. This quadratically strengthens an earlier bound that was implicit in [Omer Reingold et al., 2013]. - We show that any read-Δ degree-d m F₂-polynomial p has L_{1,k}(p) ≤ Pr [p = 1] ⋅ (k Δ d)^{O(k)}. - We establish a composition theorem which gives L_{1,k} bounds on disjoint compositions of functions that are closed under restrictions and admit L_{1,k} bounds. Finally, we apply the above structural results to obtain new unconditional pseudorandom generators and new correlation bounds for various classes of m F₂-polynomials.
Cite as

Jarosław Błasiok, Peter Ivanov, Yaonan Jin, Chin Ho Lee, Rocco A. Servedio, and Emanuele Viola. Fourier Growth of Structured 𝔽₂-Polynomials and Applications. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 53:1-53:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{blasiok_et_al:LIPIcs.APPROX/RANDOM.2021.53,
  author =	{B{\l}asiok, Jaros{\l}aw and Ivanov, Peter and Jin, Yaonan and Lee, Chin Ho and Servedio, Rocco A. and Viola, Emanuele},
  title =	{{Fourier Growth of Structured \mathbb{F}₂-Polynomials and Applications}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{53:1--53:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.53},
  URN =		{urn:nbn:de:0030-drops-147462},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.53},
  annote =	{Keywords: Fourier analysis, Pseudorandomness, Fourier growth}
}
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