We study the power of randomness in the Number-on-Forehead (NOF) model in communication complexity. We construct an explicit 3-player function $f:[N]^3 \to \{0,1\}$, such that: (i) there exist a randomized NOF protocol computing it that sends a constant number of bits; but (ii) any deterministic or nondeterministic NOF protocol computing ... more >>>

Frequency estimation in data streams is one of the classical problems in streaming algorithms. Following much research, there are now almost matching upper and lower bounds for the trade-off needed between the number of samples and the space complexity of the algorithm, when the data streams are adversarial. However, in ... more >>>

A folklore conjecture in quantum computing is that the acceptance probability of a quantum query algorithm can be approximated by a classical decision tree, with only a polynomial increase in the number of queries. Motivated by this conjecture, Aaronson and Ambainis (Theory of Computing, 2014) conjectured that this should hold ... more >>>

Hypercontractivity is one of the most powerful tools in Boolean function analysis. Originally studied over the discrete hypercube, recent years have seen increasing interest in extensions to settings like the $p$-biased cube, slice, or Grassmannian, where variants of hypercontractivity have found a number of breakthrough applications including the resolution of ... more >>>

Higher order random walks (HD-walks) on high dimensional expanders have played a crucial role in a number of recent breakthroughs in theoretical computer science, perhaps most famously in the recent resolution of the Mihail-Vazirani conjecture (Anari et al. STOC 2019), which focuses on HD-walks on one-sided local-spectral expanders. In this ... more >>>

Let $f: \{0,1\}^n \to \{0, 1\}$ be a boolean function, and let $f_\land (x, y) = f(x \land y)$ denote the AND-function of $f$, where $x \land y$ denotes bit-wise AND. We study the deterministic communication complexity of $f_\land$ and show that, up to a $\log n$ factor, it is ... more >>>

Lifting theorems are a generic way to lift lower bounds in query complexity to lower bounds in communication complexity, with applications in diverse areas, such as combinatorial optimization, proof complexity, game theory. Lifting theorems rely on a gadget, where smaller gadgets give stronger lower bounds. However, existing proof techniques are ... more >>>

A decision list is an ordered list of rules. Each rule is specified by a term, which is a conjunction of literals, and a value. Given an input, the output of a decision list is the value corresponding to the first rule whole term is satisfied by the input. Decision ... more >>>

There are two natural complexity measures associated with DNFs: their size, which is the number of clauses; and their width, which is the maximal number of variables in a clause. It is a folklore result that DNFs of small size can be approximated by DNFs of small width (logarithmic in ... more >>>

We study the relation between streaming algorithms and linear sketching algorithms, in the context of binary updates. We show that for inputs in $n$ dimensions,
the existence of efficient streaming algorithms which can process $\Omega(n^2)$ updates implies efficient linear sketching algorithms with comparable cost.
This improves upon the previous work ... more >>>

The Erdos-Rado sunflower theorem (Journal of Lond. Math. Soc. 1960) is a fundamental result in combinatorics, and the corresponding sunflower conjecture is a central open problem. Motivated by applications in complexity theory, Rossman (FOCS 2010) extended the result to quasi-sunflowers, where similar conjectures emerge about the optimal parameters for which ... more >>>

Let $H$ be an arbitrary family of hyper-planes in $d$-dimensions. We show that the point-location problem for $H$
can be solved by a linear decision tree that only uses a special type of queries called \emph{generalized comparison queries}. These queries correspond to hyperplanes that can be written as a linear ... more >>>

The GM-MDS conjecture of Dau et al. (ISIT 2014) speculates that the MDS condition, which guarantees the existence of MDS matrices with a prescribed set of zeros over large fields, is in fact sufficient for existence of such matrices over small fields. We prove this conjecture.

We propose a new framework for constructing pseudorandom generators for $n$-variate Boolean functions. It is based on two new notions. First, we introduce fractional pseudorandom generators, which are pseudorandom distributions taking values in $[-1,1]^n$. Next, we use a fractional pseudorandom generator as steps of a random walk in $[-1,1]^n$ that ... more >>>

We study an extension of active learning in which the learning algorithm may ask the annotator to compare the distances of two examples from the boundary of their label-class. For example, in a recommendation system application (say for restaurants), the annotator may be asked whether she liked or disliked a ... more >>>

We construct near optimal linear decision trees for a variety of decision problems in combinatorics and discrete geometry.
For example, for any constant $k$, we construct linear decision trees that solve the $k$-SUM problem on $n$ elements using $O(n \log^2 n)$ linear queries.
Moreover, the queries we use are comparison ... more >>>

A new probabilistic technique for establishing the existence of certain regular combinatorial structures has been introduced by Kuperberg, Lovett, and Peled (STOC 2012). Using this technique, it can be shown that under certain conditions, a randomly chosen structure has the required properties of a $t-(n,k,?)$ combinatorial design with tiny, yet ... more >>>

Assume that the edges of the complete bipartite graph $K_{n,n}$ are labeled with elements of $\mathbb{F}_2^d$, such that the sum over
any simple cycle is nonzero. What is the smallest possible value of $d$? This problem was raised by Gopalan et al. [SODA 2017] as it characterizes the alphabet size ... more >>>

The sensitivity conjecture is one of the central open problems in boolean complexity. A recent work of Gopalan et al. [CCC 2016] conjectured a robust analog of the sensitivity conjecture, which relates the decay of the Fourier mass of a boolean function to moments of its sensitivity. We prove this ... more >>>

Zero knowledge proof systems have been widely studied in cryptography. In the statistical setting, two classes of proof systems studied are Statistical Zero Knowledge (SZK) and Non-Interactive Statistical Zero Knowledge (NISZK), where the difference is that in NISZK only very limited communication is allowed between the verifier and the prover. ... more >>>

Let $f:\{0,1\}^n \to \{0,1\}$ be a boolean function. Its associated XOR function is the two-party function $f_\oplus(x,y) = f(x \oplus y)$.
We show that, up to polynomial factors, the deterministic communication complexity of $f_{\oplus}$ is equal to the parity decision tree complexity of $f$.
This relies on a novel technique ... more >>>

We study the structure of the Fourier coefficients of low degree multivariate polynomials over finite fields. We consider three properties: (i) the number of nonzero Fourier coefficients; (ii) the sum of the absolute value of the Fourier coefficients; and (iii) the size of the linear subspace spanned by the nonzero ... more >>>

The noisy population recovery problem is a statistical inference problem, which is a special case of the problem of learning mixtures of product distributions. Given an unknown distribution on $n$-bit strings with support of size $k$, and given access only to noisy samples from it, where each bit is flipped ... more >>>

Motivated by the Beck-Fiala conjecture, we study discrepancy bounds for random sparse set systems. Concretely, these are set systems $(X,\Sigma)$, where each element $x \in X$ lies in $t$ randomly selected sets of $\Sigma$, where $t$ is an integer parameter. We provide new bounds in two regimes of parameters. We ... more >>>

The study of seeded randomness extractors is a major line of research in theoretical computer science. The goal is to construct deterministic algorithms which can take a ``weak" random source $X$ with min-entropy $k$ and a uniformly random seed $Y$ of length $d$, and outputs a string of length close ... more >>>

Suppose that you have $n$ truly random bits $x=(x_1,\ldots,x_n)$ and you wish to use them to generate $m\gg n$ pseudorandom bits $y=(y_1,\ldots, y_m)$ using a local mapping, i.e., each $y_i$ should depend on at most $d=O(1)$ bits of $x$. In the polynomial regime of $m=n^s$, $s>1$, the only known solution, ... more >>>

Let $f$ be a polynomial of degree $d$ in $n$ variables over a finite field $\mathbb{F}$. The polynomial is said to be unbiased if the distribution of $f(x)$ for a uniform input $x \in \mathbb{F}^n$ is close to the uniform distribution over $\mathbb{F}$, and is called biased otherwise. The polynomial ... more >>>

The problem of constructing explicit functions which cannot be approximated by low degree polynomials has been extensively studied in computational complexity, motivated by applications in circuit lower bounds, pseudo-randomness, constructions of Ramsey graphs and locally decodable codes. Still, most of the known lower bounds become trivial for polynomials of ... more >>>

We develop a new method to prove communication lower bounds for composed functions of the form $f\circ g^n$ where $f$ is any boolean function on $n$ inputs and $g$ is a sufficiently ``hard'' two-party gadget. Our main structure theorem states that each rectangle in the communication matrix of $f \circ ... more >>>

Network coding studies the capacity of networks to carry information, when internal nodes are allowed to actively encode information. It is known that for multi-cast networks, the network coding capacity can be achieved by linear codes. It is also known not to be true for general networks. The best separation ... more >>>

The noisy population recovery problem is a basic statistical inference problem. Given an unknown distribution in $\{0,1\}^n$ with support of size $k$,
and given access only to noisy samples from it, where each bit is flipped independently with probability $1/2-\eps$,
estimate the original probability up to an additive error of ... more >>>

The list decoding problem for a code asks for the maximal radius up to which any ball of that radius contains only a constant number of codewords. The list decoding radius is not well understood even for well studied codes, like Reed-Solomon or Reed-Muller codes.

Fix a finite field $\mathbb{F}$. ... more >>>

We show that there exists a family of groups $G_n$ and nontrivial irreducible representations $\rho_n$ such that, for any constant $t$, the average of $\rho_n$ over $t$ uniformly random elements $g_1, \ldots, g_t \in G_n$ has operator norm $1$ with probability approaching 1 as $n \rightarrow \infty$. More quantitatively, we ... more >>>

The log-rank conjecture is one of the fundamental open problems in communication complexity. It speculates that the deterministic communication complexity of any two-party function is equal to the log of the rank of its associated matrix, up to polynomial factors. Despite much research, we still know very little about this ... more >>>

The densities of small linear structures (such as arithmetic progressions) in subsets of Abelian groups can be expressed as certain analytic averages involving linear forms. Higher-order Fourier analysis examines such averages by approximating the indicator function of a subset by a function of bounded number of polynomials. Then, to approximate ... more >>>

We give an exact algorithm for the 0-1 Integer Linear Programming problem with a linear number of constraints that improves over exhaustive search by an exponential factor. Specifically, our algorithm runs in time $2^{(1-\text{poly}(1/c))n}$ where $n$ is the
number of variables and $cn$ is the number of constraints. The key ... more >>>

A $t$-$(n,k,\lambda)$ design over $\mathbb{F}_q$ is a collection of $k$-dimensional subspaces of $\mathbb{F}_q^n$, ($k$-subspaces, for short), called blocks, such that each $t$-dimensional subspace of $\mathbb{F}_q^n$ is contained in exactly $\lambda$ blocks. Such $t$-designs over $\mathbb{F}_q$ are the $q$-analogs of conventional combinatorial designs. Nontrivial $t$-$(n,k,\lambda)$ designs over $\mathbb{F}_q$ are currently known ... more >>>

Let $\cal{P}$ be an affine invariant property of functions $\mathbb{F}_p^n \to [R]$ for fixed $p$ and $R$. We show that if $\cal{P}$ is locally testable with a constant number of queries, then one can estimate the distance of a function $f$ from $\cal{P}$ with a constant number of queries. This ... more >>>

We prove that any total boolean function of rank $r$ can be computed by a deterministic communication protocol of complexity $O(\sqrt{r} \cdot \log(r))$. Equivalently, any graph whose adjacency matrix has rank $r$ has chromatic number at most $2^{O(\sqrt{r} \cdot \log(r))}$. This gives a nearly quadratic improvement in the dependence on ... more >>>

Non-malleable codes provide a useful and meaningful security guarantee in situations where traditional error-correction (and even error-detection) is impossible; for example, when the attacker can completely overwrite the encoded message. Informally, a code is non-malleable if the message contained in a modified codeword is either the original message, or a ... more >>>

We prove that several measures in communication complexity are equivalent, up to polynomial factors in the logarithm of the rank of the associated matrix: deterministic communication complexity, randomized communication complexity, information cost and zero-communication cost. This shows that in order to prove the log-rank conjecture, it suffices to show that ... more >>>

Let $\mathbb{F} = \mathbb{F}_p$ for any fixed prime $p \geq 2$. An affine-invariant property is a property of functions on $\mathbb{F}^n$ that is closed under taking affine transformations of the domain. We prove that all affine-invariant property having local characterizations are testable. In fact, we show a proximity-oblivious test for ... more >>>

Let $G$ be a finite abelian group of torsion $r$ and let $A$ be a subset of $G$.
The Freiman-Ruzsa theorem asserts that if $|A+A| \le K|A|$
then $A$ is contained in a coset of a subgroup of $G$ of size at most $K^2 r^{K^4} |A|$. It was ... more >>>

We prove a Chernoff-like large deviation bound on the sum of non-independent random variables that have the following dependence structure. The variables $Y_1,\ldots,Y_r$ are arbitrary Boolean functions of independent random variables $X_1,\ldots,X_m$, modulo a restriction that every $X_i$ influences at most $k$ of the variables $Y_1,\ldots,Y_r$.

There has been considerable interest lately in the complexity of distributions. Recently, Lovett and Viola (CCC 2011) showed that the statistical distance between a uniform distribution over a good code, and any distribution which can be efficiently sampled by a small bounded-depth AC0 circuit, is inverse-polynomially close to one. That ... more >>>

We prove new lower bounds on the encoding length of Matching Vector (MV) codes. These recently discovered families of Locally Decodable Codes (LDCs) originate in the works of Yekhanin [Yek] and Efremenko [Efr] and are the only known families of LDCs with a constant number of queries and sub-exponential encoding ... more >>>

The polynomial Freiman-Ruzsa conjecture is one of the important conjectures in additive combinatorics. It asserts than one can switch between combinatorial and algebraic notions of approximate subgroups with only a polynomial loss in the underlying parameters. This conjecture has also already found several applications in theoretical computer science. Recently, Tom ... more >>>

Invariance with respect to linear or affine transformations of the domain is arguably the most common symmetry exhibited by natural algebraic properties. In this work, we show that any low complexity affine-invariant property of multivariate functions over finite fields is testable with a constant number of queries. This immediately reproves, ... more >>>

For a {0,1}-valued matrix $M$ let CC($M$) denote the deterministic communication complexity of the boolean function associated with $M$. The log-rank conjecture of Lovasz and Saks [FOCS 1988] states that CC($M$) is at most $\log^c({\mbox{rank}}(M))$ for some absolute constant $c$ where rank($M$) denotes the rank of $M$ over the field ... more >>>

We show the existence of rigid combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, $t$-designs, and $t$-wise permutations. In all cases, the sizes of the objects are optimal up to polynomial overhead. ... more >>>

In this work we describe an explicit, simple, construction of large subsets of F^n, where F is a finite field, that have small intersection with every k-dimensional affine subspace. Interest in the explicit construction of such sets, termed subspace-evasive sets, started in the work of Pudlak and Rodl (2004) ... more >>>

A folklore result in arithmetic complexity shows that the number of multiplications required to compute some $n$-variate polynomial of degree $d$ is $\sqrt{{n+d \choose n}}$. We complement this by an almost matching upper bound, showing that any $n$-variate polynomial of degree $d$ over any field can be computed with only ... more >>>

A family of permutations in $S_n$ is $k$-wise independent if a uniform permutation chosen from the family maps any distinct $k$ elements to any distinct $k$ elements equally likely. Efficient constructions of $k$-wise independent permutations are known for $k=2$ and $k=3$, but are unknown for $k \ge 4$. In fact, ... more >>>

We consider a system of linear constraints over any finite Abelian group $G$ of the following form: $\ell_i(x_1,\ldots,x_n) \equiv \ell_{i,1}x_1+\cdots+\ell_{i,n}x_n \in A_i$ for $i=1,\ldots,t$ and each $A_i \subset G$, $\ell_{i,j}$ is an element of $G$ and $x_i$'s are Boolean variables. Our main result shows that the subset of the Boolean ... more >>>

Recently there has been much interest in Gowers uniformity norms from the perspective of theoretical computer science. This is mainly due to the fact that these norms provide a method for testing whether the maximum correlation of a function $f:\mathbb{F}_p^n \rightarrow \mathbb{F}_p$ with polynomials of degree at most $d \le ... more >>>

Recently there has been much interest in polynomial threshold functions in the context of learning theory, structural results and pseudorandomness. A crucial ingredient in these works is the understanding of the distribution of low-degree multivariate polynomials evaluated over normally distributed inputs. In particular, the two important properties are exponential tail ... more >>>

In this article we are interested in the density of small linear structures (e.g. arithmetic progressions) in subsets $A$ of the group $\mathbb{F}_p^n$. It is possible to express these densities as certain analytic averages involving $1_A$, the indicator function of $A$. In the higher-order Fourier analytic approach, the function $1_A$ ... more >>>

We study a variant of the classical circuit-lower-bound problems: proving lower bounds for sampling distributions given random bits. We prove a lower bound of $1-1/n^{\Omega(1)}$ on the statistical distance between (i) the output distribution of any small constant-depth (a.k.a.~$\mathrm{AC}^0$) circuit $f : \{0,1\}^{\mathrm{poly}(n)} \to \{0,1\}^n$, and (ii) the uniform distribution ... more >>>

An approximate membership data structure is a randomized data
structure for representing a set which supports membership
queries. It allows for a small false positive error rate but has
no false negative errors. Such data structures were first
introduced by Bloom in the 1970's, and have since had numerous
applications, ... more >>>

In this work we consider linear codes which are locally testable
in a sublinear number of queries. We give the first general family
of locally testable codes of exponential size. Previous results of
this form were known only for codes of quasi-polynomial size (e.g.
Reed-Muller codes). We accomplish this by ... more >>>

In this paper we give the first construction of a pseudorandom generator, with seed length $O(\log n)$, for $\mathrm{CC}_0[p]$, the class of constant-depth circuits with unbounded fan-in $\mathrm{MOD}_p$ gates, for some prime $p$. More accurately, the seed length of our generator is $O(\log n)$ for any constant error $\epsilon>0$. In ... more >>>

We study two conjectures in additive combinatorics. The first is the polynomial Freiman-Ruzsa conjecture, which relates to the structure of sets with small doubling. The second is the inverse Gowers conjecture for $U^ $, which relates to functions which locally look like quadratics. In both cases a weak form, with ... more >>>

We study the computational power of polynomial threshold functions, that is, threshold functions of real polynomials over the boolean cube. We provide two new results bounding the computational power of this model.
Our first result shows that low-degree polynomial threshold functions cannot approximate any function with many influential variables. We ... more >>>

Recently, Viola (CCC'08) showed that the sum of $d$ small-biased distributions fools degree-$d$ polynomial tests; that is, every polynomial expression of degree at most $d$ in the bits of the sum has distribution very close to that induced by this expression evaluated on uniformly selected random bits. We show that ... more >>>

Every Boolean function on $n$ variables can be expressed as a unique multivariate polynomial modulo $p$ for every prime $p$. In this work, we study how the degree of a function in one characteristic affects its complexity in other characteristics. We establish the following general principle: functions with low degree ... more >>>

We study the density of the weights of Generalized Reed--Muller codes. Let $RM_p(r,m)$ denote the code of multivariate polynomials over $\F_p$ in $m$ variables of total degree at most $r$. We consider the case of fixed degree $r$, when we let the number of variables $m$ tend to infinity. We ... more >>>

In this work we study the list-decoding size of Reed-Muller codes. Given a received word and a distance parameter, we are interested in bounding the size of the list of Reed-Muller codewords that are within that distance from the received word. Previous bounds of Gopalan, Klivans and Zuckerman~\cite{GKZ08} on the ... more >>>

We study the problem of how well a typical multivariate polynomial can be approximated by lower degree polynomials over $\F$.
We prove that, with very high probability, a random degree $d$ polynomial has only an exponentially small correlation with all polynomials of degree $d-1$, for all degrees $d$ up to ... more >>>

A degree-d polynomial p in n variables over a field F is equidistributed if it takes on each of its |F| values close to equally often, and biased otherwise. We say that p has low rank if it can be expressed as a function of a small number of lower ... more >>>

Let $p$ be a fixed prime number, and $N$ be a large integer.
The 'Inverse Conjecture for the Gowers norm' states that if the "$d$-th Gowers norm" of a function $f:\F_p^N \to \F_p$ is non-negligible, that is larger than a constant independent of $N$, then $f$ can be non-trivially ... more >>>

Linearity tests are randomized algorithms which have oracle access to the truth table of some function $f$,
which are supposed to distinguish between linear functions and functions which are far from linear. Linearity tests were first introduced by Blum, Luby and Rubenfeld in \cite{BLR93}, and were later used in the ... more >>>

We give an explicit construction of pseudorandom
generators against low degree polynomials over finite fields. We
show that the sum of $2^d$ small-biased generators with error
$\epsilon^{2^{O(d)}}$ is a pseudorandom generator against degree $d$
polynomials with error $\epsilon$. This gives a generator with seed
length $2^{O(d)} \log{(n/\epsilon)}$. Our construction follows ... more >>>

It is well known that $\R^N$ has subspaces of dimension
proportional to $N$ on which the $\ell_1$ norm is equivalent to the
$\ell_2$ norm; however, no explicit constructions are known.
Extending earlier work by Artstein--Avidan and Milman, we prove that
such a subspace can be generated using $O(N)$ random bits.