7 Search Results for "Linial, Nati"


Document
The Approximate Degree of Bipartite Perfect Matching

Authors: Gal Beniamini

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)


Abstract
The approximate degree of a Boolean function is the least degree of a real multilinear polynomial approximating it in the 𝓁_∞-norm over the Boolean hypercube. We show that the approximate degree of the Bipartite Perfect Matching function, which is the indicator over all bipartite graphs having a perfect matching of order n, is Θ̃(n^(3/2)). The upper bound is obtained by fully characterizing the unique multilinear polynomial representing the Boolean dual of the perfect matching function, over the reals. Crucially, we show that this polynomial has very small 𝓁₁-norm - only exponential in Θ(n log n). The lower bound follows by bounding the spectral sensitivity of the perfect matching function, which is the spectral radius of its cut-graph on the hypercube [Aaronson et al., 2021; Huang, 2019]. We show that the spectral sensitivity of perfect matching is exactly Θ(n^(3/2)).

Cite as

Gal Beniamini. The Approximate Degree of Bipartite Perfect Matching. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 1:1-1:26, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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@InProceedings{beniamini:LIPIcs.CCC.2022.1,
  author =	{Beniamini, Gal},
  title =	{{The Approximate Degree of Bipartite Perfect Matching}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{1:1--1:26},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.1},
  URN =		{urn:nbn:de:0030-drops-165634},
  doi =		{10.4230/LIPIcs.CCC.2022.1},
  annote =	{Keywords: Bipartite Perfect Matching, Boolean Functions, Approximate Degree}
}
Document
An Improved Protocol for the Exactly-N Problem

Authors: Nati Linial and Adi Shraibman

Published in: LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)


Abstract
In the 3-players exactly-N problem the players need to decide whether x+y+z = N for inputs x,y,z and fixed N. This is the first problem considered in the multiplayer Number On the Forehead (NOF) model. Even though this is such a basic problem, no progress has been made on it throughout the years. Only recently have explicit protocols been found for the first time, yet no improvement in complexity has been achieved to date. The present paper offers the first improved protocol for the exactly-N problem. This improved protocol has also interesting consequences in additive combinatorics. As we explain below, it yields a higher lower bound on the possible density of corner-free sets in [N]×[N].

Cite as

Nati Linial and Adi Shraibman. An Improved Protocol for the Exactly-N Problem. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 2:1-2:8, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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@InProceedings{linial_et_al:LIPIcs.CCC.2021.2,
  author =	{Linial, Nati and Shraibman, Adi},
  title =	{{An Improved Protocol for the Exactly-N Problem}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{2:1--2:8},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.2},
  URN =		{urn:nbn:de:0030-drops-142760},
  doi =		{10.4230/LIPIcs.CCC.2021.2},
  annote =	{Keywords: Communication complexity, Number-On-the-Forehead, Corner-free sets}
}
Document
On the Communication Complexity of High-Dimensional Permutations

Authors: Nati Linial, Toniann Pitassi, and Adi Shraibman

Published in: LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)


Abstract
We study the multiparty communication complexity of high dimensional permutations in the Number On the Forehead (NOF) model. This model is due to Chandra, Furst and Lipton (CFL) who also gave a nontrivial protocol for the Exactly-n problem where three players receive integer inputs and need to decide if their inputs sum to a given integer n. There is a considerable body of literature dealing with the same problem, where (N,+) is replaced by some other abelian group. Our work can be viewed as a far-reaching extension of this line of research. We show that the known lower bounds for that group-theoretic problem apply to all high dimensional permutations. We introduce new proof techniques that reveal new and unexpected connections between NOF communication complexity of permutations and a variety of well-known problems in combinatorics. We also give a direct algorithmic protocol for Exactly-n. In contrast, all previous constructions relied on large sets of integers without a 3-term arithmetic progression.

Cite as

Nati Linial, Toniann Pitassi, and Adi Shraibman. On the Communication Complexity of High-Dimensional Permutations. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 54:1-54:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


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@InProceedings{linial_et_al:LIPIcs.ITCS.2019.54,
  author =	{Linial, Nati and Pitassi, Toniann and Shraibman, Adi},
  title =	{{On the Communication Complexity of High-Dimensional Permutations}},
  booktitle =	{10th Innovations in Theoretical Computer Science Conference (ITCS 2019)},
  pages =	{54:1--54:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-095-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{124},
  editor =	{Blum, Avrim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.54},
  URN =		{urn:nbn:de:0030-drops-101470},
  doi =		{10.4230/LIPIcs.ITCS.2019.54},
  annote =	{Keywords: High dimensional permutations, Number On the Forehead model, Additive combinatorics}
}
Document
Weighted Polynomial Approximations: Limits for Learning and Pseudorandomness

Authors: Mark Bun and Thomas Steinke

Published in: LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)


Abstract
Low-degree polynomial approximations to the sign function underlie pseudorandom generators for halfspaces, as well as algorithms for agnostically learning halfspaces. We study the limits of these constructions by proving inapproximability results for the sign function. First, we investigate the derandomization of Chernoff-type concentration inequalities. Schmidt et al. (SIAM J. Discrete Math. 1995) showed that a tail bound of delta can be established for sums of Bernoulli random variables with only O(log(1/delta))-wise independence. We show that their results are tight up to constant factors. Secondly, the “polynomial regression” algorithm of Kalai et al. (SIAM J. Comput. 2008) shows that halfspaces can be efficiently learned with respect to log-concave distributions on R^n in the challenging agnostic learning model. The power of this algorithm relies on the fact that under log-concave distributions, halfspaces can be approximated arbitrarily well by low-degree polynomials. In contrast, we exhibit a large class of non-log-concave distributions under which polynomials of any degree cannot approximate the sign function to within arbitrarily low error.

Cite as

Mark Bun and Thomas Steinke. Weighted Polynomial Approximations: Limits for Learning and Pseudorandomness. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 625-644, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{bun_et_al:LIPIcs.APPROX-RANDOM.2015.625,
  author =	{Bun, Mark and Steinke, Thomas},
  title =	{{Weighted Polynomial Approximations: Limits for Learning and Pseudorandomness}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{625--644},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-89-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{40},
  editor =	{Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.625},
  URN =		{urn:nbn:de:0030-drops-53274},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.625},
  annote =	{Keywords: Polynomial Approximations, Pseudorandomness, Concentration, Learning Theory, Halfspaces}
}
Document
Combinatorial Discrepancy for Boxes via the gamma_2 Norm

Authors: Jirí Matoušek and Aleksandar Nikolov

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)


Abstract
The gamma_2 norm of a real m by n matrix A is the minimum number t such that the column vectors of A are contained in a 0-centered ellipsoid E that in turn is contained in the hypercube [-t, t]^m. This classical quantity is polynomial-time computable and was proved by the second author and Talwar to approximate the hereditary discrepancy: it bounds the hereditary discrepancy from above and from below, up to logarithmic factors. Here we provided a simplified proof of the upper bound and show that both the upper and the lower bound are asymptotically tight in the worst case. We then demonstrate on several examples the power of the gamma_2 norm as a tool for proving lower and upper bounds in discrepancy theory. Most notably, we prove a new lower bound of log(n)^(d-1) (up to constant factors) for the d-dimensional Tusnady problem, asking for the combinatorial discrepancy of an n-point set in d-dimensional space with respect to axis-parallel boxes. For d>2, this improves the previous best lower bound, which was of order approximately log(n)^((d-1)/2), and it comes close to the best known upper bound of O(log(n)^(d+1/2)), for which we also obtain a new, very simple proof. Applications to lower bounds for dynamic range searching and lower bounds in differential privacy are given.

Cite as

Jirí Matoušek and Aleksandar Nikolov. Combinatorial Discrepancy for Boxes via the gamma_2 Norm. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 1-15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{matousek_et_al:LIPIcs.SOCG.2015.1,
  author =	{Matou\v{s}ek, Jir{\'\i} and Nikolov, Aleksandar},
  title =	{{Combinatorial Discrepancy for Boxes via the gamma\underline2 Norm}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{1--15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.1},
  URN =		{urn:nbn:de:0030-drops-51091},
  doi =		{10.4230/LIPIcs.SOCG.2015.1},
  annote =	{Keywords: discrepancy theory, range counting, factorization norms}
}
Document
Embedding Hard Learning Problems Into Gaussian Space

Authors: Adam Klivans and Pravesh Kothari

Published in: LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)


Abstract
We give the first representation-independent hardness result for agnostically learning halfspaces with respect to the Gaussian distribution. We reduce from the problem of learning sparse parities with noise with respect to the uniform distribution on the hypercube (sparse LPN), a notoriously hard problem in theoretical computer science and show that any algorithm for agnostically learning halfspaces requires n^Omega(log(1/\epsilon)) time under the assumption that k-sparse LPN requires n^Omega(k) time, ruling out a polynomial time algorithm for the problem. As far as we are aware, this is the first representation-independent hardness result for supervised learning when the underlying distribution is restricted to be a Gaussian. We also show that the problem of agnostically learning sparse polynomials with respect to the Gaussian distribution in polynomial time is as hard as PAC learning DNFs on the uniform distribution in polynomial time. This complements the surprising result of Andoni et. al. 2013 who show that sparse polynomials are learnable under random Gaussian noise in polynomial time. Taken together, these results show the inherent difficulty of designing supervised learning algorithms in Euclidean space even in the presence of strong distributional assumptions. Our results use a novel embedding of random labeled examples from the uniform distribution on the Boolean hypercube into random labeled examples from the Gaussian distribution that allows us to relate the hardness of learning problems on two different domains and distributions.

Cite as

Adam Klivans and Pravesh Kothari. Embedding Hard Learning Problems Into Gaussian Space. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 793-809, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)


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@InProceedings{klivans_et_al:LIPIcs.APPROX-RANDOM.2014.793,
  author =	{Klivans, Adam and Kothari, Pravesh},
  title =	{{Embedding Hard Learning Problems Into Gaussian Space}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
  pages =	{793--809},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-74-3},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{28},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.793},
  URN =		{urn:nbn:de:0030-drops-47391},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.793},
  annote =	{Keywords: distribution-specific hardness of learning, gaussian space, halfspace-learning, agnostic learning}
}
Document
On the practically interesting instances of MAXCUT

Authors: Yonatan Bilu, Amit Daniely, Nati Linial, and Michael Saks

Published in: LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)


Abstract
For many optimization problems, the instances of practical interest often occupy just a tiny part of the algorithm's space of instances. Following (Y. Bilu and N. Linial, 2010), we apply this perspective to MAXCUT, viewed as a clustering problem. Using a variety of techniques, we investigate practically interesting instances of this problem. Specifically, we show how to solve in polynomial time distinguished, metric, expanding and dense instances of MAXCUT under mild stability assumptions. In particular, (1 + epsilon)-stability (which is optimal) suffices for metric and dense MAXCUT. We also show how to solve in polynomial time Omega(sqrt(n))-stable instances of MAXCUT, substantially improving the best previously known result.

Cite as

Yonatan Bilu, Amit Daniely, Nati Linial, and Michael Saks. On the practically interesting instances of MAXCUT. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 526-537, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)


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@InProceedings{bilu_et_al:LIPIcs.STACS.2013.526,
  author =	{Bilu, Yonatan and Daniely, Amit and Linial, Nati and Saks, Michael},
  title =	{{On the practically interesting instances of MAXCUT}},
  booktitle =	{30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)},
  pages =	{526--537},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-50-7},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{20},
  editor =	{Portier, Natacha and Wilke, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.526},
  URN =		{urn:nbn:de:0030-drops-39625},
  doi =		{10.4230/LIPIcs.STACS.2013.526},
  annote =	{Keywords: MAXCUT, Clustering, Hardness in practice, Stability, Non worst-case analysis}
}
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