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Dor Minzer

Publications and source records attributed to Dor Minzer.

At least 37 records · Page 2Linked to original sources

arXiv2024

On Approximability of Satisfiable $k$-CSPs: VI

We prove local and global inverse theorems for general $3$-wise correlations over pairwise-connected distributions. Let $μ$ be a distribution over $Σ\times Γ\times Φ$ such that the supports of $μ_{xy}$, $μ_{xz}$, and $μ_{yz}$ are all connected, and let $f: Σ^n \to \mathbb{C}$, $g: Γ^n \to \mathbb{C}$, $h: Φ^n \to \mathbb{C}$ be $1$-bounded functions satisfying \[ \left|\mathbb{E}_{(x,y,z) \sim μ^{\otimes n}}[f(x)g(y)h(z)]\right| \geq \varepsilon. \] In this setting, our local inverse theorem asserts that there is $δ:=\textsf{exp}(-\varepsilon^{-O_μ(1)})$ such that with probability at least $δ$, a random restriction of $f$ down to $δn$ coordinates $δ$-correlates to a product function. To get a global inverse theorem, we prove a restriction inverse theorem for general product functions, stating that if a random restriction of $f$ down to $δn$ coordinates is $δ$-correlated with a product function with probability at least $δ$, then $f$ is $2^{-\textsf{poly}(\log(1/δ))}$-correlated with a function of the form $L\cdot P$, where $L$ is a function of degree $\textsf{poly}(1/δ)$, $\|L\|_2\leq 1$, and $P$ is a product function. We show applications to property testing and to additive combinatorics. In particular, we show the following result via a density increment argument. Let $Σ$ be a finite set and $S \subseteq Σ\times Σ\times Σ$ such that: (1) $(x, x, x) \in S$ for all $x \in S$, and (2) the supports of $S_{xy}$, $S_{xz}$, and $S_{yz}$ are all connected. Then, any set $A \subseteq Σ^n$ with $|Σ|^{-n}|A| \geq Ω((\log \log \log n)^{-c})$ contains $x, y, z \in A$, not all equal, such that $(x_i,y_i,z_i) \in S$ for all $i$. This gives the first reasonable bounds for the restricted 3-AP problem over finite fields.

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arXiv2024

On Approximability of Satisfiable $k$-CSPs: VII

Let $Σ_1,\ldots,Σ_k$ be finite alphabets, and let $μ$ be a distribution over $Σ_1 \times \dots \times Σ_k$ in which the probability of each atom is at least $α$. We prove that if $μ$ does not admit Abelian embeddings, and $f_i: Σ_i \to \mathbb{C}$ are $1$-bounded functions (for $i=1,\ldots,k$) such that \[ \left|\mathbb{E}_{(x_1,\dots,x_k) \sim μ^{\otimes n}}\Big[f_1(x_1) \dots f_k(x_k)\Big]\right| \geq \varepsilon, \] then there exists $L\colon Σ_1^n\to\mathbb{C}$ of degree at most $d$ and $\|L\|_2\leq 1$ such that $|\langle f_1, L\rangle|\geq δ$, where $d$ and $δ>0$ depend only on $k, α$ and $\varepsilon$. This answers the analytic question posed by Bhangale, Khot, and Minzer (STOC 2022). We also prove several extensions of this result that are useful in subsequent applications.

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arXiv2024

Reasonable Bounds for Combinatorial Lines of Length Three

We prove that any subset $A \subseteq [3]^n$ with $3^{-n}|A| \ge (\log\log\log\log n)^{-c}$ contains a combinatorial line of length $3$, i.e., $x, y, z \in A$, not all equal, with $x_i=y_i=z_i$ or $(x_i,y_i,z_i)=(0,1,2)$ for all $i = 1, 2, \dots, n$. This improves on the previous best bound of $3^{-n}|A| \ge Ω((\log^* n)^{-1/2})$ of [D.H.J. Polymath, Ann. of Math. 2012].

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arXiv2024

Quasi-Linear Size PCPs with Small Soundness from HDX

We construct 2-query, quasi-linear size probabilistically checkable proofs (PCPs) with arbitrarily small constant soundness, improving upon Dinur's 2-query quasi-linear size PCPs with soundness $1-Ω(1)$. As an immediate corollary, we get that under the exponential time hypothesis, for all $ε>0$ no approximation algorithm for $3$-SAT can obtain an approximation ratio of $7/8+ε$ in time $2^{n/\log^C n}$, where $C$ is a constant depending on $ε$. Our result builds on a recent line of independent works by Bafna, Lifshitz and Minzer, and Dikstein, Dinur and Lubotzky, that showed the existence of linear size direct product testers with small soundness. The main new ingredient in our proof is a technique that embeds a given 2-CSP into a 2-CSP on a prescribed graph, provided that the latter is a graph underlying a sufficiently good high-dimensional expander (HDX). We achieve this by establishing a novel connection between PCPs and fault-tolerant distributed computing, more precisely, to the almost-everywhere reliable transmission problem introduced by Dwork, Peleg, Pippenger and Upfal (1986). We instantiate this connection by showing that graphs underlying HDXs admit routing protocols that are tolerant to adversarial edge corruptions, also improving upon the state of the art constructions of sparse edge-fault-tolerant networks in the process. Our PCP construction requires variants of the aforementioned direct product testers with poly-logarithmic degree. The existence and constructability of these variants is shown in an appendix by Zhiwei Yun.

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arXiv2024

Parallel Repetition for $3$-Player XOR Games

In a $3$-$\mathsf{XOR}$ game $\mathcal{G}$, the verifier samples a challenge $(x,y,z)\sim μ$ where $μ$ is a probability distribution over $Σ\timesΓ\timesΦ$, and a map $t\colon Σ\timesΓ\timesΦ\to\mathcal{A}$ for a finite Abelian group $\mathcal{A}$ defining a constraint. The verifier sends the questions $x$, $y$ and $z$ to the players Alice, Bob and Charlie respectively, receives answers $f(x)$, $g(y)$ and $h(z)$ that are elements in $\mathcal{A}$ and accepts if $f(x)+g(y)+h(z) = t(x,y,z)$. The value, $\mathsf{val}(\mathcal{G})$, of the game is defined to be the maximum probability the verifier accepts over all players' strategies. We show that if $\mathcal{G}$ is a $3$-$\mathsf{XOR}$ game with value strictly less than $1$, whose underlying distribution over questions $μ$ does not admit Abelian embeddings into $(\mathbb{Z},+)$, then the value of the $n$-fold repetition of $\mathcal{G}$ is exponentially decaying. That is, there exists $c=c(\mathcal{G})>0$ such that $\mathsf{val}(\mathcal{G}^{\otimes n})\leq 2^{-cn}$. This extends a previous result of [Braverman-Khot-Minzer, FOCS 2023] showing exponential decay for the GHZ game. Our proof combines tools from additive combinatorics and tools from discrete Fourier analysis.

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arXiv2024

On Approximability of Satisfiable k-CSPs: IV

We prove a stability result for general $3$-wise correlations over distributions satisfying mild connectivity properties. More concretely, we show that if $Σ,Γ$ and $Φ$ are alphabets of constant size, and $μ$ is a pairwise connected distribution over $Σ\timesΓ\timesΦ$ with no $(\mathbb{Z},+)$ embeddings in which the probability of each atom is $Ω(1)$, then the following holds. Any triplets of $1$-bounded functions $f\colon Σ^n\to\mathbb{C}$, $g\colon Γ^n\to\mathbb{C}$, $h\colon Φ^n\to\mathbb{C}$ satisfying \[ \left|\mathbb{E}_{(x,y,z)\sim μ^{\otimes n}}\big[f(x)g(y)h(z)\big]\right|\geq \varepsilon \] must arise from an Abelian group associated with the distribution $μ$. More specifically, we show that there is an Abelian group $(H,+)$ of constant size such that for any such $f,g$ and $h$, the function $f$ (and similarly $g$ and $h$) is correlated with a function of the form $\tilde{f}(x) = χ(σ(x_1),\ldots,σ(x_n)) L (x)$, where $σ\colon Σ\to H$ is some map, $χ\in \hat{H}^{\otimes n}$ is a character, and $L\colon Σ^n\to\mathbb{C}$ is a low-degree function with bounded $2$-norm. En route we prove a few additional results that may be of independent interest, such as an improved direct product theorem, as well as a result we refer to as a ``restriction inverse theorem'' about the structure of functions that, under random restrictions, with noticeable probability have significant correlation with a product function. In companion papers, we show applications of our results to the fields of Probabilistically Checkable Proofs, as well as various areas in discrete mathematics such as extremal combinatorics and additive combinatorics.

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arXiv2024

A Dense Model Theorem for the Boolean Slice

The (low soundness) linearity testing problem for the middle slice of the Boolean cube is as follows. Let $\varepsilon>0$ and $f$ be a function on the middle slice on the Boolean cube, such that when choosing a uniformly random quadruple $(x,y,z ,x\oplus y\oplus z)$ of vectors of $2n$ bits with exactly $n$ ones, the probability that $f(x\oplus y \oplus z) = f(x) \oplus f(y) \oplus f(z)$ is at least $1/2+\varepsilon$. The linearity testing problem, posed by David, Dinur, Goldenberg, Kindler and Shinkar, asks whether there must be an actual linear function that agrees with $f$ on $1/2+\varepsilon'$ fraction of the inputs, where $\varepsilon' = \varepsilon'(\varepsilon)>0$. We solve this problem, showing that $f$ must indeed be correlated with a linear function. To do so, we prove a dense model theorem for the middle slice of the Boolean hypercube for Gowers uniformity norms. Specifically, we show that for every $k\in\mathbb{N}$, the normalized indicator function of the middle slice of the Boolean hypercube $\{0,1\}^{2n}$ is close in Gowers norm to the normalized indicator function of the union of all slices with weight $t = n\pmod{2^{k-1}}$. Using our techniques we also give a more general `low degree test' and a biased rank theorem for the slice.

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arXiv2024

Constant Degree Direct Product Testers with Small Soundness

Let $X$ be a $d$-dimensional simplicial complex. A function $F\colon X(k)\to \{0,1\}^k$ is said to be a direct product function if there exists a function $f\colon X(1)\to \{0,1\}$ such that $F(σ) = (f(σ_1), \ldots, f(σ_k))$ for each $k$-face $σ$. In an effort to simplify components of the PCP theorem, Goldreich and Safra introduced the problem of direct product testing, which asks whether one can test if $F\colon X(k)\to \{0,1\}^k$ is correlated with a direct product function by querying $F$ on only $2$ inputs. Dinur and Kaufman conjectured that there exist bounded degree complexes with a direct product test in the small soundness regime. We resolve their conjecture by showing that for all $δ>0$, there exists a family of high-dimensional expanders with degree $O_δ(1)$ and a $2$-query direct product tester with soundness $δ$. We use the characterization given by a subset of the authors and independently by Dikstein and Dinur, who showed that some form of non-Abelian coboundary expansion (which they called "Unique-Games coboundary expansion") is a necessary and sufficient condition for a complex to admit such direct product testers. Our main technical contribution is a general technique for showing coboundary expansion of complexes with coefficients in a non-Abelian group. This allows us to prove that the high dimensional expanders constructed by Chapman and Lubotzky satisfies the necessary conditions, thus admitting a 2-query direct product tester with small soundness.

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arXiv2024

Product Mixing in Compact Lie Groups

If $G$ is a group, we say a subset $S$ of $G$ is product-free if the equation $xy=z$ has no solutions with $x,y,z \in S$. For $D \in \mathbb{N}$, a group $G$ is said to be $D$-quasirandom if the minimal dimension of a nontrivial complex irreducible representation of $G$ is at least $D$. Gowers showed that in a $D$-quasirandom finite group $G$, the maximal size of a product-free set is at most $|G|/D^{1/3}$. This disproved a longstanding conjecture of Babai and Sós from 1985. For the special unitary group, $G=SU(n)$, Gowers observed that his argument yields an upper bound of $n^{-1/3}$ on the measure of a measurable product-free subset. In this paper, we improve Gowers' upper bound to $\exp(-cn^{1/3})$, where $c>0$ is an absolute constant. In fact, we establish something stronger, namely, product-mixing for measurable subsets of $SU(n)$ with measure at least $\exp(-cn^{1/3})$; for this product-mixing result, the $n^{1/3}$ in the exponent is sharp. Our approach involves introducing novel hypercontractive inequalities, which imply that the non-Abelian Fourier spectrum of the indicator function of a small set concentrates on high-dimensional irreducible representations. Our hypercontractive inequalities are obtained via methods from representation theory, harmonic analysis, random matrix theory and differential geometry. We generalize our hypercontractive inequalities from $SU(n)$ to an arbitrary $D$-quasirandom compact connected Lie group for $D$ at least an absolute constant, thereby extending our results on product-free sets to such groups. We also demonstrate various other applications of our inequalities to geometry (viz., non-Abelian Brunn-Minkowski type inequalities), mixing times, and the theory of growth in compact Lie groups.

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arXiv2024

Characterizing Direct Product Testing via Coboundary Expansion

A $d$-dimensional simplicial complex $X$ is said to support a direct product tester if any locally consistent function defined on its $k$-faces (where $k\ll d$) 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 $F\colon X(k)\to\{0,1\}^k$ it samples $(A,A')\sim μ$ and checks that $F[A]|_{A\cap A'} = F[A']|_{A\cap A'}$. 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 \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.

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arXiv2023

Parallel Repetition of k-Player Projection Games

We study parallel repetition of k-player games where the constraints satisfy the projection property. We prove exponential decay in the value of a parallel repetition of projection games with value less than 1.

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arXiv2023

Adversarial Low Degree Testing

In the $t$-online-erasure model in property testing, an adversary is allowed to erase $t$ values of a queried function for each query the tester makes. This model was recently formulated by Kalemaj, Raskhodnikova andVarma, who showed that the properties of linearity of functions as well as quadraticity can be tested in$O_t(1)$ many queries: $O(\log (t))$ for linearity and $2^{2^{O(t)}}$ for quadraticity. They asked whether the more general property of low-degreeness can be tested in the online erasure model, whether better testers exist for quadraticity, and if similar results hold when ``erasures'' are replaced with ``corruptions''. We show that, in the $t$-online-erasure model, for a prime power $q$, given query access to a function $f: \mathbb{F}_q^n \xrightarrow[]{} \mathbb{F}_q$, one can distinguish in $\mathrm{poly}(\log^{d+q}(t)/δ)$ queries between the case that $f$ is degree at most $d$, and the case that $f$ is $δ$-far from any degree $d$ function (with respect to the fractional hamming distance). This answers the aforementioned questions and brings the query complexity to nearly match the query complexity of low-degree testing in the classical property testing model. Our results are based on the observation that the property of low-degreeness admits a large and versatile family of query efficient testers. Our testers operates by querying a uniformly random, sufficiently large set of points in a large enough affine subspace, and finding a tester for low-degreeness that only utilizes queries from that set of points. We believe that this tester may find other applications to algorithms in the online-erasure model or other related models, and may be of independent interest.

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arXiv2023

On $t$-Intersecting Families of Permutations

We prove that there exists a constant $c_0$ such that for any $t \in \mathbb{N}$ and any $n\geq c_0 t$, if $A \subset S_n$ is a $t$-intersecting family of permutations then$|A|\leq (n-t)!$. Furthermore, if $|A|\ge 0.75(n-t)!$ then there exist $i_1,\ldots,i_t$ and $j_1,\ldots,j_t$ such that $σ(i_1)=j_1,\ldots,σ(i_t)=j_t$ holds for any $σ\in A$. This shows that the conjectures of Deza and Frankl (1977) and of Cameron (1988) on $t$-intersecting families of permutations hold for all $t \leq c_0 n$. Our proof method, based on hypercontractivity for global functions, does not use the specific structure of permutations, and applies in general to $t$-intersecting sub-families of `pseudorandom' families in $\{1,2,\ldots,n\}^n$, like $S_n$.

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arXiv2023

Influences in Mixing Measures

The theory of influences in product measures has profound applications in theoretical computer science, combinatorics, and discrete probability. This deep theory is intimately connected to functional inequalities and to the Fourier analysis of discrete groups. Originally, influences of functions were motivated by the study of social choice theory, wherein a Boolean function represents a voting scheme, its inputs represent the votes, and its output represents the outcome of the elections. Thus, product measures represent a scenario in which the votes of the parties are randomly and independently distributed, which is often far from the truth in real-life scenarios. We begin to develop the theory of influences for more general measures under mixing or correlation decay conditions. More specifically, we prove analogues of the KKL and Talagrand influence theorems for Markov Random Fields on bounded degree graphs with correlation decay. We show how some of the original applications of the theory of in terms of voting and coalitions extend to general measures with correlation decay. Our results thus shed light both on voting with correlated voters and on the behavior of general functions of Markov Random Fields (also called ``spin-systems") with correlation decay.

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arXiv2023

On Perfectly Friendly Bisections of Random Graphs

We prove that there exists a constant $γ_{\mathrm{crit}}\approx .17566$ such that if $G\sim \mathbb{G}(n,1/2)$ then for any $\varepsilon > 0$ with high probability $G$ has a equipartition such that each vertex has $(γ_{\mathrm{crit}}-\varepsilon)\sqrt{n}$ more neighbors in its own part than in the other part and with high probability no such partition exists for a separation of $(γ_{\mathrm{crit}}+\varepsilon)\sqrt{n}$. The proof involves a number of tools ranging from isoperimetric results on vertex-transitive sets of graphs coming from Boolean functions, switchings, degree enumeration formulas, and the second moment method. Our results substantially strengthen recent work of Ferber, Kwan, Narayanan, and the last two authors on a conjecture of Füredi from 1988 and in particular prove the existence of fully-friendly bisections in $\mathbb{G}(n,1/2)$

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arXiv2023

Solving Unique Games over Globally Hypercontractive Graphs

We 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. Our algorithm shows how to round "low-entropy" solutions to sum-of-squares (SoS) semidefinite programs, broadly extending the algorithmic framework of [BBKSS'21]. We give a new rounding scheme for SoS, which eliminates global correlations in a given pseudodistribution so that it retains various good properties even after conditioning. Getting structural control over a pseudodistribution after conditioning is a fundamental challenge in many SoS based algorithms. Due to these challenges, [BBKSS] were not able to establish strong algorithms for globally hypercontractive graphs, and could only do so for certifiable small-set expanders. Our results improve upon the results of [BBKSS] in various aspects: we are able to deal with instances with arbitrarily small (but constant) completeness, and most importantly, their algorithm gets a soundness guarantee that degrades with other parameters of the graph (which in all PCP constructions grow with the alphabet size), whereas our doesn't. 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.

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arXiv2023

Optimal Testing of Generalized Reed-Muller Codes in Fewer Queries

A local tester for an error correcting code $C\subseteq Σ^{n}$ is a tester that makes $Q$ oracle queries to a given word $w\in Σ^n$ and decides to accept or reject the word $w$. 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 $1$ if $w\in C$. By optimal soundness, we mean that if the tester accepts with probability at least $1-ε$ (where $ε$ is small), then it must be the case that $w$ is $O(ε/Q)$-close to some codeword $c\in C$ in Hamming distance. We show that Generalized Reed-Muller codes admit optimal testers with $Q = (3q)^{\lceil{ \frac{d+1}{q-1}\rceil}+O(1)}$ queries. Here, for a prime power $q = p^{k}$, the Generalized Reed-Muller code, RM[n,q,d], consists of the evaluations of all $n$-variate degree $d$ polynomials over $\mathbb{F}_q$. 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 $q^{\lceil{\frac{d+1}{q-q/p} \rceil}}$ queries. Our tester achieves query complexity which is polynomially better than by a power of $p/(p-1)$, 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).

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arXiv2022

Parallel Repetition for the GHZ Game: Exponential Decay

We show that the value of the $n$-fold repeated GHZ game is at most $2^{-Ω(n)}$, improving upon the polynomial bound established by Holmgren and Raz. Our result is established via a reduction to approximate subgroup type questions from additive combinatorics.

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