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Characterizing Direct Product Testing via Coboundary Expansion
No full textSTOC ’24, June 24–28, 2024, Vancouver, BC, CanadaA d-dimensional simplicial complex X is said to support a direct product tester if any locally consistent function defined on its k-faces (where k≪ d) necessarily come from a function over its vertices. More precisely, a direct product tester has a distribution µ over pairs of k-faces (A,A′), and given query access to F: X(k)→{0,1}k it samples (A,A′)∼ µ and checks that F[A]|A∩ A′ = F[A′]|A∩ A′. The tester should have (1) the ”completeness property”, meaning that any assignment F which is a direct product assignment passes the test with probability 1, and (2) the ”soundness property”, meaning that if F passes the test with probability s, then F must be correlated with a direct product function. Dinur and Kaufman showed that a sufficiently good spectral expanding complex X admits a direct product tester in the ”high soundness” regime where s is close to 1. They asked whether there are high dimensional expanders that support direct product tests in the ”low soundness”, when s is close to 0. We give a characterization of high-dimensional expanders that support a direct product tester in the low soundness regime. We show that spectral expansion is insufficient, and the complex must additionally satisfy a variant of coboundary expansion, which we refer to as ”Unique-Games coboundary expanders”. Conversely, we show that this property is also sufficient to get direct product testers. This property can be seen as a high-dimensional generalization of the standard notion of coboundary expansion over non-Abelian groups for 2-dimensional complexes. It asserts that any locally consistent Unique-Games instance obtained using the low-level faces of the complex, must admit a good global solution
Rounding via Low Dimensional Embeddings
Full text linkA regular graph G = (V,E) is an (ε,γ) small-set expander if for any set of vertices of fractional size at most ε, at least γ of the edges that are adjacent to it go outside. In this paper, we give a unified approach to several known complexity-theoretic results on small-set expanders. In particular, we show:
1) Max-Cut: we show that if a regular graph G = (V,E) is an (ε,γ) small-set expander that contains a cut of fractional size at least 1-δ, then one can find in G a cut of fractional size at least 1-O(δ/(εγ⁶)) in polynomial time.
2) Improved spectral partitioning, Cheeger’s inequality and the parallel repetition theorem over small-set expanders. The general form of each one of these results involves square-root loss that comes from certain rounding procedure, and we show how this can be avoided over small set expanders. Our main idea is to project a high dimensional vector solution into a low-dimensional space while roughly maintaining ₂² distances, and then perform a pre-processing step using low-dimensional geometry and the properties of ₂² distances over it. This pre-processing leverages the small-set expansion property of the graph to transform a vector valued solution to a different vector valued solution with additional structural properties, which give rise to more efficient integral-solution rounding schemes
Characterizing Direct Product Testing via Coboundary Expansion
Full text linkA -dimensional simplicial complex is said to support a direct product
tester if any locally consistent function defined on its -faces (where ) necessarily come from a function over its vertices. More precisely, a
direct product tester has a distribution over pairs of -faces
, and given query access to it samples
and checks that . The
tester should have (1) the ``completeness property'', meaning that any
assignment which is a direct product assignment passes the test with
probability , and (2) the ``soundness property'', meaning that if passes
the test with probability , then must be correlated with a direct
product function.
Dinur and Kaufman showed that a sufficiently good spectral expanding complex
admits a direct product tester in the ``high soundness'' regime where
is close to . They asked whether there are high dimensional expanders that
support direct product tests in the ``low soundness'', when is close to
.
We give a characterization of high-dimensional expanders that support a
direct product tester in the low soundness regime. We show that spectral
expansion is insufficient, and the complex must additionally satisfy a variant
of coboundary expansion, which we refer to as \emph{Unique-Games coboundary
expanders}. Conversely, we show that this property is also sufficient to get
direct product testers. This property can be seen as a high-dimensional
generalization of the standard notion of coboundary expansion over non-Abelian
groups for 2-dimensional complexes. It asserts that any locally consistent
Unique-Games instance obtained using the low-level faces of the complex, must
admit a good global solution
Optimal Testing of Generalized Reed-Muller Codes in Fewer Queries
Full text linkA local tester for an error correcting code is a
tester that makes oracle queries to a given word and
decides to accept or reject the word . An optimal local tester is a local
tester that has the additional properties of completeness and optimal
soundness. By completeness, we mean that the tester must accept with
probability if . By optimal soundness, we mean that if the tester
accepts with probability at least (where is small),
then it must be the case that is -close to some codeword
in Hamming distance.
We show that Generalized Reed-Muller codes admit optimal testers with queries. Here, for a prime power , the Generalized Reed-Muller code, RM[n,q,d], consists of the
evaluations of all -variate degree polynomials over .
Previously, no tester achieving this query complexity was known, and the best
known testers due to Haramaty, Shpilka and Sudan(which is optimal) and due to
Ron-Zewi and Sudan(which was not known to be optimal) both required
queries. Our tester achieves query
complexity which is polynomially better than by a power of , which is
nearly the best query complexity possible for generalized Reed-Muller codes.
The tester we analyze is due to Ron-Zewi and Sudan, and we show that their
basic tester is in fact optimal. Our methods are more general and also allow us
to prove that a wide class of testers, which follow the form of the Ron-Zewi
and Sudan tester, are optimal. This result applies to testers for all
affine-invariant codes (which are not necessarily generalized Reed-Muller
codes).Comment: 42 pages, 8 page appendi
Solving Unique Games over Globally Hypercontractive Graphs
Full text linkWe study the complexity of affine Unique-Games (UG) over globally hypercontractive graphs, which are graphs that are not small set expanders but admit a useful and succinct characterization of all small sets that violate the small-set expansion property. This class of graphs includes the Johnson and Grassmann graphs, which have played a pivotal role in recent PCP constructions for UG, and their generalizations via high-dimensional expanders.
We show new rounding techniques for higher degree sum-of-squares (SoS) relaxations for worst-case optimization. In particular, our algorithm shows how to round "low-entropy" pseudodistributions, broadly extending the algorithmic framework of [Mitali Bafna et al., 2021]. At a high level, [Mitali Bafna et al., 2021] showed how to round pseudodistributions for problems where there is a "unique" good solution. We extend their framework by exhibiting a rounding for problems where there might be "few good solutions".
Our result suggests that UG is easy on globally hypercontractive graphs, and therefore highlights the importance of graphs that lack such a characterization in the context of PCP reductions for UG
Optimal Tiling of the Euclidean Space Using Permutation-Symmetric Bodies
Full text linkWhat is the least surface area of a permutation-symmetric body B whose ℤⁿ translations tile ℝⁿ? Since any such body must have volume 1, the isoperimetric inequality implies that its surface area must be at least Ω(√n). Remarkably, Kindler et al. showed that for general bodies B this is tight, i.e. that there is a tiling body of ℝⁿ whose surface area is O(√n).
In theoretical computer science, the tiling problem is intimately related to the study of parallel repetition theorems (which are an important component in PCPs), and more specifically in the question of whether a "strong version" of the parallel repetition theorem holds. Raz showed, using the odd cycle game, that strong parallel repetition fails in general, and subsequently these ideas were used in order to construct non-trivial tilings of ℝⁿ.
In this paper, motivated by the study of a symmetric parallel repetition, we consider the permutation-symmetric variant of the tiling problem in ℝⁿ. We show that any permutation-symmetric body that tiles ℝⁿ must have surface area at least Ω(n/√{log n}), and that this bound is tight, i.e. that there is a permutation-symmetric tiling body of ℝⁿ with surface area O(n/√{log n}). We also give matching bounds for the value of the symmetric parallel repetition of Raz’s odd cycle game.
Our result suggests that while strong parallel repetition fails in general, there may be important special cases where it still applies
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Full text linkWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
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