We propose a combinatorial hypothesis regarding a subspace vs. subspace agreement test, and prove that if correct it leads to a proof of the 2-to-1 Games Conjecture, albeit with imperfect completeness. This paper presents the second installment in a line of work by various subsets of the authors (with additional contributions by Barak, Kothari, and Steurer (ITCS'19)), which led to a proof of the 2-to-2 Games Conjecture.

We give an alternative, simple method to prove isoperimetric inequalities over the hypercube. In particular, we show: 1. An elementary proof of classical isoperimetric inequalities of Talagrand, as well as a stronger isoperimetric result conjectured by Talagrand and recently proved by Eldan and Gross. 2. A strengthening of the Friedgut junta theorem, asserting that if the p-moment of the sensitivity of a function is constant for some 1/2 + epsilon <= p <= 1, then the function is close to a junta. In this language, Friedgut's theorem is the special case that p = 1.

It is a classical result that the inner product function cannot be computed by an AC0 circuit [17, 1, 22]. It is conjectured that this holds even if we allow arbitrary preprocessing of each of the two inputs separately. We prove this conjecture when the preprocessing of one of the inputs is limited to output n + n/(logω(1)n) bits. Our methods extend to many other functions, including pseudorandom functions, and imply a (weak but nontrivial) limitation on the power of encoding inputs in low-complexity cryptography. Finally, under cryptographic assumptions, we relate the question of proving variants of the main conjecture with the question of learning AC0 under simple input distributions.

We prove a polynomial Bogolyubov type lemma for the special linear group over finite fields. Specifically, we show that there exists an absolute constant C>0, such that if A is a density α subset of the special linear group, then the set AA^-1AA^-1 contains a subgroup H of density α^C. Moreover, this subgroup is isomorphic to a special linear group of a smaller rank. We also show that if A is an approximate subgroups then it can be covered by the union of few cosets of H. Our proof makes use of the Gurevich–Howe notion of tensor rank, and of a strengthened Bonami type Lemma for global functions on the bilinear scheme. We also present applications to spectral bounds for global convolution operators, global product free sets, and covering numbers corresponding to global sets.

The hypercontractive inequality is a fundamental result in analysis, with many applications throughout discrete mathematics, theoretical computer science, combinatorics and more. So far, variants of this inequality have been proved mainly for product spaces, which raises the question of whether analogous results hold over non-product domains. We consider the symmetric group, $S_n$, one of the most basic non-product domains, and establish hypercontractive inequalities on it. Our inequalities are most effective for the class of \emph{global functions} on $S_n$, which are functions whose $2$-norm remains small when restricting $O(1)$ coordinates of the input, and assert that low-degree, global functions have small $q$-norms, for $q>2$. As applications, we show: 1. An analog of the level-$d$ inequality on the hypercube, asserting that the mass of a global function on low-degrees is very small. We also show how to use this inequality to bound the size of global, product-free sets in the alternating group $A_n$. 2. Isoperimetric inequalities on the transposition Cayley graph of $S_n$ for global functions, that are analogous to the KKL theorem and to the small-set expansion property in the Boolean hypercube. 3. Hypercontractive inequalities on the multi-slice, and stability versions of the Kruskal--Katona Theorem in some regimes of parameters.

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 ∈ S.In 1985, Babai and Sós [] asked, for a finite group G, how large a subset S⊆ G can be if it is product-free. The main tool (hitherto) for studying this problem has been the notion of a quasirandom group. For D ∈ 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|/D1/3. This disproved a longstanding conjecture of Babai and Sós from 1985. For the special unitary group, G=(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(−cn1/3), where c>0 is an absolute constant. In fact, we establish something stronger, namely, product-mixing for measurable subsets of (n) with measure at least exp(−cn1/3); for this product-mixing result, the n1/3 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 (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. A subsequent work due to Arunachalam, Girish and Lifshitz uses our inequalities to establish new separation results between classical and quantum communication complexity.

We prove an analogue of Bonami's (hypercontractive) lemma for complex-valued functions on L (𝑉 ,𝑊 ), where 𝑉 and 𝑊 are vector spaces over a finite field. This inequality is useful for functions on L (𝑉 ,𝑊 ) whose 'generalised influences' are small, in an appropriate sense. It leads to a significant shortening of the proof of a recent seminal result by Khot, Minzer and Safra that pseudorandom sets in Grassmann graphs have near-perfect expansion, which (in combination with the work of Dinur, Khot, Kindler, Minzer and Safra) implies the 2-2 Games conjecture (the variant, that is, with imperfect completeness)

Forbidden intersection problems for families of linear maps, Discrete Analysis 2023:19, 32 pp. A central problem in extremal combinatorics is to determine the maximal size of a set system given constraints on the sizes of the sets in the system and on the sizes of their intersections. For example, a special case of the famous Erdős-Ko-Rado theorem states that the largest family of subsets of $\{1,2,\dots,n\}$ of size $k$ such that any two members of the family have a non-empty intersection is, provided that $k\leq n/2$, $\binom{n-1}{k-1}$, and that if $kn/2$, then the question is trivial since any two sets of size $k$ intersect, and if $k=n/2$ one can take as an alternative best possible construction all sets of size greater than $n/2$ together with exactly one of $A$ and $A^c$ for each set $A$ of size $n/2$.) A family is called $t$-_intersecting_ if any two sets in the family intersect in a set of size at least $t$, and $s$-_intersection free_ if no two sets in the family have an intersection of size exactly $s$. The general form of the Erdős-Ko-Rado theorem is that for sufficiently large $n$, a maximal-sized $t$-intersecting family of sets of size $k$ must consist of all sets of size $k$ that contain some given set of size $t$. A remarkable theorem of Ahlswede and Khachatrian, which solved a long-standing open problem, answers the question of what happens when we drop the condition that $n$ is sufficiently large. The obvious way to force two sets to have an intersection of size at least $t$ is to insist that the contains some given set of size $t$, but a more general way is to insist that they each intersect a given set of size $r$ in at least $(t+r)/2$ elements, and sometimes this leads to larger families (as indeed it did in the trivial case mentioned above, which corresponds to taking $t=1$ and $r=n$ with $n<2k$). Ahlswede and Khachatrian showed that every $t$-intersecting family of maximal size is given by one of these more general examples. The study of $s$-intersection-free systems is harder than that of $t$-intersecting families, because the condition is not monotone, which rules out many approaches. Some impressive results have nevertheless been obtained, with the help of a wide variety of methods, including dimension arguments, Fourier analysis, and purely combinatorial techniques. Of course, a $t$-intersecting family is necessarily $(t-1)$-intersection free, so the maximum size of a $(t-1)$-intersection-free family is necessarily at least the maximum size of a $t$-intersecting family. A heuristic meta-conjecture is that in many cases these maxima are in fact equal: that is, the best way to ensure that a set system does not have any intersections of size $t-1$ is to ensure that all intersections are of size at least $t$. In this paper the authors consider an analogue of the problem for linear maps between vector spaces. If $V$ and $W$ are finite-dimensional vector spaces over the field $\mathbb F_q$ for some prime power $q$, and if $\alpha$ and $\beta$ are linear maps from $V$ to $W$, then the analogue of the intersection that they consider is the dimension of the kernel of $\alpha-\beta$ -- that is, the dimension of the subspace of $V$ on which $\alpha$ and $\beta$ agree. Thus, a family $\mathcal F$ of linear maps from $V$ to $W$ is $t$-_intersecting_ if $\dim\ker(\alpha-\beta)\geq t$ for every $\alpha,\beta\in\mathcal F$, and it is $s$-_intersection free_ if it is not possible to find $\alpha,\beta\in\mathcal F$ with $\dim\ker(\alpha-\beta)=s$. Suppose one wishes to find a large $t$-intersecting collection of linear maps from $V$ to $W$. As with families of sets, there is an obvious example: simply fix a $t$-dimensional subspace $U$ of $V$ and take all linear maps $\alpha$ such that $U\subset\ker\alpha$. Slightly more generally, one can take a translate of this example: that is, one can fix some linear map $\alpha_0$ and take all linear maps $\alpha$ such that $U\subset\ker(\alpha-\alpha_0)$, or in other words, a maximal family of linear maps that all agree on $U$. Unlike with the sets case, there is a second natural example, only slightly less obvious than the first. Since the rank of a linear map is equal to the rank of its adjoint (or in matrix terms, row-rank equals column-rank), we can pick a $t$-dimensional subspace $Y^*$ of $W^*$ and take a maximal family of linear maps $\alpha$ such that the adjoint maps $\alpha^*$ all agree on $Y^*$. The precise problem considered by the authors concerns invertible maps, in which case we may as well take $V$ and $W$ to be equal. They show that if $n$ is sufficiently large, and if $\mathcal F$ is a $(t-1)$-intersection-free family of invertible linear maps from $V$ to $V$ of maximal size, then either there must exist a subspace $U\subset V$ of dimension $t$ such that all the maps in $\mathcal F$ agree on $U$, or there must be a subspace $U^*\subset V^*$ of dimension $t$ such that all the adjoints of the maps in $\mathcal F$ agree on $U^*$. This considerably strengthens a recent result of Ernst and Schmidt, who proved the same result for $t$-intersecting families, but without characterizing the equality cases and with a more complicated proof. The proof follows a similar structure to that of a result of two of the authors concerning intersecting families of permutations (where the "intersection" of two permutations is taken to be the set on which they agree). However, in this context an additional ingredient is required, namely a difficult new hypercontractivity inequality for functions defined on $L(V,W)$, the space of linear maps from $V$ to $W$. In its strongest form, the inequality can be found in another paper by the same authors, but here a weaker version with a simpler proof, given in the paper, suffices. This is used to show that a $(t-1)$-intersection-free family is approximately contained in a $t$-intersecting "junta", which is roughly speaking a family $\mathcal F$ of linear maps such that whether or not $\alpha$ belongs to $\mathcal F$ depends only on a few values of $\alpha$ and $\alpha^*$.

We study an analogue of the Erd\H{o}s-S\'os forbidden intersection problem, for families of linear maps. If $V$ and $W$ are vector spaces over the same field, we say a family $\mathcal{F}$ of linear maps from $V$ to $W$ is \emph{$(t-1)$-intersection-free} if for any two linear maps $\sigma_1,\sigma_2 \in \mathcal{F}$, $\dim(\{v \in V:\ \sigma_1(v)=\sigma_2(v)\}) \neq t-1$. We prove that if $n$ is sufficiently large depending on $t$, $q$ is any prime power, $V$ is an $n$-dimensional vector space over $\mathbb{F}_q$, and $\mathcal{F} \subset \textrm{GL}(V)$ is $(t-1)$-intersection-free, then $|\mathcal{F}| \leq \prod_{i=1}^{n-t}(q^n - q^{i+t-1})$. Equality holds only if there exists a $t$-dimensional subspace of $V$ on which all elements of $\mathcal{F}$ agree, or a $t$-dimensional subspace of $V^*$ on which all elements of $\{\sigma^*:\ \sigma \in \mathcal{F}\}$ agree. Our main tool is a `junta approximation' result for families of linear maps with a forbidden intersection: namely, that if $V$ and $W$ are finite-dimensional vector spaces over the same finite field, then any $(t-1)$-intersection-free family of linear maps from $V$ to $W$ is essentially contained in a $t$-intersecting \emph{junta} (meaning, a family $\mathcal{J}$ of linear maps from $V$ to $W$ such that the membership of $\sigma$ in $\mathcal{J}$ is determined by $\sigma(v_1),\ldots,\sigma(v_M),\sigma^*(a_1),\ldots,\sigma^*(a_N)$, where $v_1,\ldots,v_M \in V$, $a_1,\ldots,a_N \in W^*$ and $M+N$ is bounded). The proof of this in turn relies on a variant of the `junta method' (originally introduced by Dinur and Friedgut, and powefully extended by Keller and the last author), together with spectral techniques and a hypercontractive inequality.

Lionel Levine's hat challenge has t players, each with a (very large, or infinite) stack of hats on their head, each hat independently colored at random black or white. The players are allowed to coordinate before the random colors are chosen, but not after. Each player sees all hats except for those on her own head. They then proceed to simultaneously try and each pick a black hat from their respective stacks. They are proclaimed successful only if they are all correct. Levine's conjecture is that the success probability tends to zero when the number of players grows. We prove that this success probability is strictly decreasing in the number of players, and present some connections to problems in graph theory: relating the size of the largest independent set in a graph and in a random induced subgraph of it, and bounding the size of a set of vertices intersecting every maximum-size independent set in a graph.

We show improved monotonicity testers for the Boolean hypercube under the $p$-biased measure, as well as over the hypergrid $[m]^n$. Our results are: 1. For any $p\in (0,1)$, for the $p$-biased hypercube we show a non-adaptive tester that makes $\tilde{O}(\sqrt{n}/\varepsilon^2)$ queries, accepts monotone functions with probability $1$ and rejects functions that are $\varepsilon$-far from monotone with probability at least $2/3$. 2. For all $m\in\mathbb{N}$, we show an $\tilde{O}(\sqrt{n}m^3/\varepsilon^2)$ query monotonicity tester over $[m]^n$. We also establish corresponding directed isoperimetric inequalities in these domains. Previously, the best known tester due to Black, Chakrabarty and Seshadhri had $\Omega(n^{5/6})$ query complexity. Our results are optimal up to poly-logarithmic factors and the dependency on $m$. Our proof uses a notion of monotone embeddings of measures into the Boolean hypercube that can be used to reduce the problem of monotonicity testing over an arbitrary product domains to the Boolean cube. The embedding maps a function over a product domain of dimension $n$ into a function over a Boolean cube of a larger dimension $n'$, while preserving its distance from being monotone; an embedding is considered efficient if $n'$ is not much larger than $n$, and we show how to construct efficient embeddings in the above mentioned settings.

We give alternate proofs for three related results in analysis of Boolean functions, namely the KKL Theorem, Friedgut’s Junta Theorem, and Talagrand’s strengthening of the KKL Theorem. We follow a new approach: looking at the first Fourier level of the function after a suitable random restriction and applying the Log-Sobolev inequality appropriately. In particular, we avoid using the hypercontractive inequality that is common to the original proofs. Our proofs might serve as an alternate, uniform exposition to these theorems and the techniques might benefit further research.

We study the structure of non-expanding sets in the Grassmann graph. We put forth a hypothesis stating that every small set whose expansion is smaller than 1– δ must be correlated with one of a specified list of sets which are isomorphic to smaller Grassmann graphs. We develop a framework of Fourier analysis for analyzing functions over the Grassmann graph, and prove that our hypothesis holds for all sets whose expansion is below 3/4. Our work is motivated by [DKK + 18], wherein the authors show that a linearity agreement hypothesis implies an NP-hardness gap of 1/2– ε vs. ε for Unique Games and other inapproximability results. Barak, Kothari and Steurer show that the hypothesis in this work implies the linearity agreement hypothesis [DKK + 18]. Following initial publication of this work, our hypothesis was proved in [KMS18].

Lionel Levine’s hat challenge has t players, each with a (very large, or infinite) stack of hats on their head, each hat independently colored at random black or white. The players are allowed to coordinate before the random colors are chosen, but not after. Each player sees all hats except for those on her own head. They then proceed to simultaneously try and each pick a black hat from their respective stacks. They are proclaimed successful only if they are all correct. Levine’s conjecture was the success probability tends to zero when the number of players grows. We prove that this success probability is strictly decreasing in the number of players, and present some connections to questions in graph theory.

We explore quantum-inspired interactive proof systems where the prover is limited. Namely, we improve on a result by \cite{AG17} showing a quantum-inspired interactive protocol ($\IP$) for $PreciseBQP$ where the prover is only assumed to be a $\PreciseBQP$ machine, and show that the result can be strengthened to show an $\IP$ for $\NP^{\PP}$ with a prover which is only assumed to be an $\NP^{\PP}$ machine - which was not known before. We also show how the protocol can be used to directly verify $\QMA$ computations, thus connecting the sum-check protocol by \cite{AAV13} with the result of \cite{AG17,LFKN90}. Our results shed light on a quantum-inspired proof for $\IP=\PSPACE$, as $\PreciseQMA$ captures the full $\PSPACE$ power.

The total influence of a function is a central notion in analysis of Boolean functions, and characterizing functions that have small total influence is one of the most fundamental questions associated with it. The KKL theorem and the Friedgut junta theorem give a strong characterization of such functions whenever the bound on the total influence is $$o(\log n)$$. However, both results become useless when the total influence of the function is $$\omega (\log n)$$. The only case in which this logarithmic barrier has been broken for an interesting class of functions was proved by Bourgain and Kalai, who focused on functions that are symmetric under large enough subgroups of $$S_n$$. In this paper, we build and improve on the techniques of the Bourgain–Kalai paper and establish new concentration results on the Fourier spectrum of Boolean functions with small total influence. Our results include: Our concentration result for the Fourier spectrum of functions with small total influence also has new implications in learning theory. More specifically, we conclude that the class of functions whose total influence is at most K is agnostically learnable in time $$2^{O(K\log K)}$$ using membership queries. Thus, the class of functions with total influence $$O(\log n/\log \log n)$$ is agnostically learnable in $$\mathsf{poly}(n)$$ time.

We present a polynomial time reduction from gap-3LIN to label cover with 2-to-1 constraints. In the “yes” case the fraction of satisfied constraints is at least 1 −ε, and in the “no” case we show that this fraction is at most ε, assuming a certain (new) combinatorial hypothesis on the Grassmann graph. In other words, we describe a combinatorial hypothesis that implies the 2-to-1 conjecture with imperfect completeness. The companion submitted paper [Dinur, Khot, Kindler, Minzer and Safra, STOC 2018] makes some progress towards proving this hypothesis. Our work builds on earlier work by a subset of the authors [Khot, Minzer and Safra, STOC 2017] where a slightly different hypothesis was used to obtain hardness of approximating vertex cover to within factor of √2−ε. The most important implication of this work is (assuming the hypothesis) an NP-hardness gap of 1/2−ε vs. ε for unique games . In addition, we derive optimal NP-hardness for approximating the max-cut-gain problem, NP-hardness of coloring an almost 4-colorable graph with any constant number of colors, and the same √2−ε NP-hardness for approximate vertex cover that was already obtained based on a slightly different hypothesis. Recent progress towards proving our hypothesis [Barak, Kothari and Steurer, ECCC TR18-077], [Dinur, Khot, Kindler, Minzer and Safra, STOC 2018] directly implies some new unconditional NP-hardness results. These include new points of NP-hardness for unique games and for 2-to-1 and 2-to-2 games. More recently, the full version of our hypothesis was proven [Khot, Minzer and Safra, ECCC TR18-006].

We study the problem of matrix isomorphism of matrix Lie algebras (MatIsoLie). Lie algebras arise centrally in areas as diverse as differential equations, particle physics, group theory, and the Mulmuley -- Sohoni Geometric Complexity Theory program. A matrix Lie algebra is a set L of matrices that is closed under linear combinations and the operation [A, B] = AB - BA. Two matrix Lie algebras L, L' are matrix isomorphic if there is an invertible matrix M such that conjugating every matrix in L by M yields the set L'. We show that certain cases of MatIsoLie -- for the wide and widely studied classes of semi simple and abelian Lie algebras -- are equivalent to graph isomorphism and linear code equivalence, respectively. On the other hand, we give polynomial-time algorithms for other cases of MatIsoLie, which allow us to mostly derandomize a recent result of Kayal on affine equivalence of polynomials.

The k -fold direct sum encoding of a string α ∈ --0,1} n is a function f α that takes as input sets S ⊆ [ n ] of size k and outputs f α (S) = ∑ i ∈ S α i (mod 2. In this paper we prove a Direct Sum Testing theorem. We describe a three query test that accepts with probability one any function of the form f α for some α, and rejects with probability Ω(ε) functions f that are ε being a direct sum encoding. This theorem has a couple of additional guises: Linearity testing: By identifying the subsets of [ n ] with vectors in --0,1} n e natural way, our result can be thought of as a linearity testing theorem for functions whose domain is restricted to the k 'th layer of the hypercube (i.e. the set of n -bit strings with Hamming weight k ). Tensor power testing: By moving to --1,1 notation, the direct sum encoding is equivalent (up to a difference that is negligible when k « √n) to a tensor power. Thus our theorem implies a three query test for deciding if a given tensor α ∈ --- 1,1} n k is a tensor power of a single dimensional vector α ∈ ----1,1} n , i.e. whether there is some α such that f = α k . We also provide a four query test for checking if a given ±1 matrix has rank 1. Our test naturally extends the linearity test of Blum, Luby, and Rubinfeld (STOC '90). Our analysis proceeds by first handling the k = n /2 case, and then reducing this case to the general k < n /2 case, using a recent direct product testing theorem of Dinur and Steurer (CCC '2014). The k < n /2 case is proven via a new proof for linearity testing on the hypercube, which we extend to the restricted domain of the n /2-th layer of the hypercube.