Unambiguous DNFs and Alon-Saks-Seymour

Balodis, Kaspars; Ben-David, Shai; Göös, Mika; Jain, Siddhartha; Kothari, Robin

doi:10.1109/focs52979.2021.00020

Cited by 4 publications

(24 citation statements)

References 17 publications

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“…In a recent breakthrough, it was shown by Balodis, Ben-David, G ö ös, Jain, and Kothari [2] that the bound can be further improved to rank B (M) ≥ k Ω(log k) , which matches the upper bound in (1) up to log log k factors hidden in the Ω notation. Note that the result of [2] strengthens an earlier result of G ö ös, Pitassi, and Watson [15], who provided a near optimal separation between the binary rank of a 0, 1 matrix and the deterministic communication complexity of the problem associated with it. Interestingly, the above problem is closely related to a graph-theoretic problem proposed by Alon, Saks, and Seymour in 1991 (see [21]).…”

Section: Introductionmentioning

confidence: 62%

“…Theorem 1.1 can be viewed as a regular analogue of the aforementioned result of Balodis et al [2], showing that their near optimal separation between rank bin (M) and rank B (M) can also be attained by regular matrices M. Since every 0, 1 matrix M satisfies rank bin (M) ≥ rank B (M), Theorem 1.1 settles, in a strong form, the question of Pullman asked in [27,26] (and the variants of the question mentioned there) and confirms the conjecture of Hefner et al [18]. We remark that regular matrices M with rank B (M) larger than rank bin (M) can also be derived from [19] (see Section 1.2 for details).…”

Section: Our Contributionmentioning

confidence: 71%

“…These graphs were used there to derive the aforementioned separation between rank bin (M) and rank B (M) for 0, 1 matrices M (see [19,Section 4]). In a work of Bousquet, Lagoutte, and Thomassé [6], the two problems were shown to be essentially equivalent, allowing the authors of [2] to derive, for infinitely many integers k, the existence of a graph G satisfying bp(G) = k and χ(G) ≥ k Ω(log k) . As in the matrix setting, the gap is optimal up to log log k factors in the exponent.…”

Section: Introductionmentioning

confidence: 99%

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On the Binary and Boolean Rank of Regular Matrices

Haviv¹,

Parnas²

2022

Preprint

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A 0, 1 matrix is said to be regular if all of its rows and columns have the same number of ones. We prove that for infinitely many integers k, there exists a square regular 0, 1 matrix with binary rank k, such that the Boolean rank of its complement is k Ω(log k) . Equivalently, the ones in the matrix can be partitioned into k combinatorial rectangles, whereas the number of rectangles needed for any cover of its zeros is k Ω(log k) . This settles, in a strong form, a question of Pullman (Linear Algebra Appl. ,1988) and a conjecture of Hefner, Henson, Lundgren, and Maybee (Congr. Numer. ,1990). The result can be viewed as a regular analogue of a recent result of Balodis, Ben-David, G ö ös, Jain, and Kothari (FOCS, 2021), motivated by the clique vs. independent set problem in communication complexity and by the (disproved) Alon-Saks-Seymour conjecture in graph theory. As an application of the produced regular matrices, we obtain regular counterexamples to the Alon-Saks-Seymour conjecture and prove that for infinitely many integers k, there exists a regular graph with biclique partition number k and chromatic number k Ω(log k) .

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Section: Introductionmentioning

confidence: 62%

Section: Our Contributionmentioning

confidence: 71%

Section: Introductionmentioning

confidence: 99%

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On the Binary and Boolean Rank of Regular Matrices

Haviv¹,

Parnas²

2022

Preprint

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“…It might be interesting to note that our proof of it exploits a surprising connection with the theory of communication complexity. In particular, it hinges on the recent breakthroughs concerning the clique vs. independent set problem by Göös (2015); Ben- David, Hatami, and Tal (2017); Balodis, Ben-David, Göös, Jain, and Kothari (2021). We discuss this further in Section 2.4.3 below.…”

Section: Expressivitymentioning

confidence: 99%

“…Interestingly, its proof hinges on a recent breakthrough in communication complexity and its implications in graph theory: Göös (2015); Ben- David, Hatami, and Tal (2017); Balodis, Ben-David, Göös, Jain, and Kothari (2021). Despite the advantage that our proof of Theorem 11 is short and simple, it unfortunately provides only little insight on the structure of the concluded class H. In part, this is due to the complexity of the relevant result in graph theory, which is obtained by a series of reductions, some of which are unintuitive.…”

Section: Disambiguationsmentioning

confidence: 99%

A Theory of PAC Learnability of Partial Concept Classes

Alon¹,

Hanneke²,

Holzman³

et al. 2021

Preprint

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We extend the classical theory of PAC learning in a way which allows to model a rich variety of practical learning tasks where the data satisfy special properties that ease the learning process. For example, tasks where the distance of the data from the decision boundary is bounded away from zero, or tasks where the data lie on a lower dimensional surface. The basic and simple idea is to consider partial concepts: these are functions that can be undefined on certain parts of the space. When learning a partial concept, we assume that the source distribution is supported only on points where the partial concept is defined.This way, one can naturally express assumptions on the data such as lying on a lower dimensional surface, or that it satisfies margin conditions. In contrast, it is not at all clear that such assumptions can be expressed by the traditional PAC theory using learnable total concept classes, and in fact we exhibit easy-to-learn partial concept classes which provably cannot be captured by the traditional PAC theory. This also resolves, in a strong negative sense, a question posed by Attias, Kontorovich, and Mansour (2019).We characterize PAC learnability of partial concept classes and reveal an algorithmic landscape which is fundamentally different than the classical one. For example, in the classical PAC model, learning boils down to Empirical Risk Minimization (ERM). This basic principle follows from

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