The log-rank conjecture is a longstanding open problem with multiple equivalent formulations in complexity theory and mathematics. In its linear-algebraic form, it asserts that the rank and partitioning number of a Boolean matrix are quasi-polynomially related. We propose a relaxed but still equivalent version of the conjecture based on a new matrix parameter, signed rectangle rank: the minimum number of all-1 rectangles needed to express the Boolean matrix as a $\pm 1$-sum. Signed rectangle rank lies between rank and partition number, and our main result shows that it is in fact equivalent to rank up to a logarithmic factor. Additionally, we extend the main result to tensors. This reframes the log-rank conjecture as: can every signed decomposition of a Boolean matrix be made positive with only quasi-polynomial blowup? As an application, we prove an equivalence between the log-rank conjecture and a conjecture of Lovett and Singer-Sudan on cross-intersecting set systems.

We prove that for any 3-player game G, whose query distribution has the same support as the GHZ game (i.e., all x,y,z ∈ {0,1} satisfying x+y+z = 0 (mod 2)), the value of the n-fold parallel repetition of G decays exponentially fast: val(G^{⊗ n}) ≤ exp(-n^c) for all sufficiently large n, where c > 0 is an absolute constant. We also prove a concentration bound for the parallel repetition of the GHZ game: For any constant ε > 0, the probability that the players win at least a (3/4+ε) fraction of the n coordinates is at most exp(-n^c), where c = c(ε) > 0 is a constant. In both settings, our work exponentially improves upon the previous best known bounds which were only polynomially small, i.e., of the order n^{-Ω(1)}. Our key technical tool is the notion of algebraic spreadness adapted from the breakthrough work of Kelley and Meka (FOCS '23) on sets free of 3-term progressions.

Exact and point-wise approximating representations of Boolean functions by real polynomials have been of great interest in the theory of computing. We focus on the study of sparsity of such representations. Our results include the following: First, we show that for every total Boolean function, its exact and approximate sparsity in the De Morgan basis are polynomially related to each other in the log scale, ignoring poly-log(n) factors. This answers an open question posed by Knop, Lovett, McGuire and Yuan (STOC 2021). It builds on and is analogous to the seminal result of Nisan and Szegedy (Computational Complexity 1994) who proved the same for degree and approximate degree. Second, we consider more powerful representations using generalized monomials, where each monomial is an indicator of a sub-cube. There are 3n such monomials, where n is the number of variables. We prove that even for these representations, the sparsity and approximate sparsity of total Boolean functions remain polynomially related to each other in the log scale, ignoring poly-log(n) factors. Third, we show that for every total Boolean function f, the log of its De Morgan sparsity characterizes up to polynomial loss and ignoring poly-log(n) factors, the quantum and classical 2-party bounded-error communication complexity of f ∘ EQ4, where EQ4 is Equality of two 2-bit strings, one held by Alice and the other by Bob. As a consequence, we show that bounded-error quantum protocols cannot exhibit super-polynomial cost advantage over their classical counterparts, for computing such functions.

We consider (almost) k-wise independent hash functions, whose evaluations on any k inputs are (almost) uniformly random, for very large values of k. Such hash functions need to have a large key that grows linearly with k. However, it may be possible to evaluate them in sub-linear time by only reading a small subset of t ≪ k locations during each evaluation; we call such hash functions t-local. Such hash functions have applications to nearly optimal bounded-use information-theoretic cryptography. Local hash functions were previously studied in several works starting with Siegel (FOCS’89, SICOMP’04). For a hash function with n-bit input and output size, we get the following new results:

The Fundamental Theorem of PAC Learning asserts that learnability of a concept class $H$ is equivalent to the *uniform convergence* of empirical error in $H$ to its mean, or equivalently, to the problem of *density estimation*, learnability of the underlying marginal distribution with respect to events in $H$. This seminal equivalence relies strongly on PAC learning’s ‘distribution-free’ assumption, that the adversary may choose any marginal distribution over data. Unfortunately, the distribution-free model is known to be overly adversarial in practice, failing to predict the success of modern machine learning algorithms, but without the Fundamental Theorem our theoretical understanding of learning under distributional constraints remains highly limited. In this work, we revisit the connection between PAC learning, uniform convergence, and density estimation beyond the distribution-free setting when the adversary is restricted to choosing a marginal distribution from a known family $\mathscr{P}$. We prove that while the traditional Fundamental Theorem fails, a finer-grained connection between the three fundamental notions continues to hold: 1. PAC-Learning is strictly sandwiched between two relaxed models of density estimation, differing only in whether the learner knows the set of well-estimated events in $H$. 2. Under reasonable assumptions on $H$ and $\mathscr{P}$, density estimation is equivalent to *uniform estimation*, a weakening of uniform convergence allowing non-empirical estimators. Together, our results give a clearer picture of how the Fundamental Theorem extends beyond the distribution-free setting and shed new light on the classically challenging problem of learning under arbitrary distributional assumptions.

Let G be a finite abelian group and A be a subset of $G \times G$ which is corner-free, meaning that there are no $x, y \in G$ and $d \in G \backslash\{0\}$ such that $(x, y),(x+d, y),(x, y+d) \in A$. We prove that \begin{equation*}|A| \leq|G|^{2} \cdot \exp \left(-(\log |G|)^{\Omega{1}}\right)\end{equation*}As a consequence, we obtain polynomial (in the input length) lower bounds on the non-deterministic communication complexity of Exactly-N in the 3-player Number-on-Forehead model. We also obtain the first “reasonable” lower bounds on the coloring version of the 3 -dimensional corners problem, as well as on the non-deterministic communication complexity of Exactly-N in the 4-player Number-on-Forehead model. This is an extended abstract. The full version of the paper can be found at https://arxiv.org/abs/2504.07006.

The log-rank conjecture, a longstanding problem in communication complexity, has persistently eluded resolution for decades. Consequently, some recent efforts have focused on potential approaches for establishing the conjecture in the special case of XOR functions, where the communication matrix is lifted from a boolean function, and the rank of the matrix equals the Fourier sparsity of the function, which is the number of its nonzero Fourier coefficients. In this note, we refute two conjectures. The first has origins in Montanaro and Osborne (arXiv'09) and is considered in Tsang et al. (FOCS'13), and the second one is due to Mande and Sanyal (FSTTCS'20). These conjectures were proposed in order to improve the best-known bound of Lovett (STOC'14) regarding the log-rank conjecture in the special case of XOR functions. Both conjectures speculate that the set of nonzero Fourier coefficients of the boolean function has some strong additive structure. We refute these conjectures by constructing two specific boolean functions tailored to each.

We revisit the fundamental Boolean Matrix Multiplication (BMM) problem. With the invention of algebraic fast matrix multiplication over 50 years ago, it also became known that BMM can be solved in truly subcubic O(nω) time, where ω<3; much work has gone into bringing ω closer to 2. Since then, a parallel line of work has sought comparably fast combinatorial algorithms but with limited success. The na'ive O(n3)-time algorithm was initially improved by a log2n factor [Arlazarov et al.; RAS’70], then by log2.25n [Bansal and Williams; FOCS’09], then by log3n [Chan; SODA’15], and finally by log4n [Yu; ICALP’15].

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 → {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 it requires sending about (logN)1/3 many bits. This exponentially improves upon the previously best-known such separation. At the core of our proof is an extension of a recent result on sets of integers without 3-term arithmetic progressions into a non-arithmetic setting.

A fundamental question in reinforcement learning theory is: suppose the optimal value functions are linear in given features, can we learn them efficiently? This problem’s counterpart in supervised learning, linear regression, can be solved both statistically and computationally efficiently. Therefore, it was quite surprising when a recent work \cite{kane2022computational} showed a computational-statistical gap for linear reinforcement learning: even though there are polynomial sample-complexity algorithms, unless NP = RP, there are no polynomial time algorithms for this setting.In this work, we build on their result to show a computational lower bound, which is exponential in feature dimension and horizon, for linear reinforcement learning under the Randomized Exponential Time Hypothesis. To prove this we build a round-based game where in each round the learner is searching for an unknown vector in a unit hypercube. The rewards in this game are chosen such that if the learner achieves large reward, then the learner’s actions can be used to simulate solving a variant of 3-SAT, where (a) each variable shows up in a bounded number of clauses (b) if an instance has no solutions then it also has no solutions that satisfy more than (1-$\epsilon$)-fraction of clauses. We use standard reductions to show this 3-SAT variant is approximately as hard as 3-SAT. Finally, we also show a lower bound optimized for horizon dependence that almost matches the best known upper bound of $\exp(\sqrt{H})$.

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 the case where the data stream is given in a random order, or is stochastic, only weaker lower bounds exist. In this work we close this gap, up to logarithmic factors. In order to do so we consider the needle problem, which is a natural hard problem for frequency estimation studied in (Andoni et al. 2008, Crouch et al. 2016). Here, the goal is to distinguish between two distributions over data streams with t samples. The first is uniform over a large enough domain. The second is a planted model; a secret “needle“ is uniformly chosen, and then each element in the stream equals the needle with probability p, and otherwise is uniformly chosen from the domain. It is simple to design streaming algorithms that distinguish the distributions using space $s \approx 1 /\left(p^{2} t\right)$. It was unclear if this is tight, as the existing lower bounds are weaker. We close this gap and show that the trade-off is near optimal, up to a logarithmic factor. Our proof builds and extends classical connections between streaming algorithms and communication complexity, concretely multi-party unique set-disjointness. We introduce two new ingredients that allow us to prove sharp bounds. The first is a lower bound for an asymmetric version of multi-party unique set-disjointness, where players receive input sets of different sizes, and where the communication of each player is normalized relative to their input length. The second is a combinatorial technique that allows to sample needles in the planted model by first sampling intervals, and then sampling a uniform needle in each interval.

Learning and equilibrium computation in games are fundamental problems across computer science and economics, with applications ranging from politics to machine learning. Much of the work in this area revolves around a simple algorithm termed randomized weighted majority (RWM), also known as “Hedge” or “Multiplicative Weights Update,” which is well known to achieve statistically optimal rates in adversarial settings (Littlestone and Warmuth '94, Freund and Schapire '99). Unfortunately, RWM comes with an inherent computational barrier: it requires maintaining and sampling from a distribution over all possible actions. In typical settings of interest the action space is exponentially large, seemingly rendering RWM useless in practice. In this work, we refute this notion for a broad variety of structured games, showing it is possible to efficiently (approximately) sample the action space in RWM in polylogarithmic time. This gives the first efficient no-regret algorithms for problems such as the (discrete) Colonel Blotto game, matroid congestion, matroid security, and basic dueling games. As an immediate corollary, we give a polylogarithmic time meta-algorithm to compute approximate Nash Equilibria for these games that is exponentially faster than prior methods in several important settings. Further, our algorithm is the first to efficiently compute equilibria for more involved variants of these games with general sums, more than two players, and, for Colonel Blotto, multiple resource types. Our results also greatly generalize earlier work on efficient RWM-based techniques for exponential strategy sets from (Cesa-Bianchi and Lugosi '09).

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 is said to have low rank if it can be expressed as a composition of a few lower degree polynomials. Green and Tao [Contrib. Discrete Math 2009] and Kaufman and Lovett [FOCS 2008] showed that bias implies low rank for fixed degree polynomials over fixed prime fields. This lies at the heart of many tools in higher order Fourier analysis. In this work, we extend this result to all prime fields (of size possibly growing with$n$). We also provide a generalization to nonprime fields in the large characteristic case. However, we state all our applications in the prime field setting for the sake of simplicity of presentation. Using the above generalization to large fields as a starting point, we are also able to settle the list decoding radius of fixed degree Reed-Muller codes over growing fields. The case of fixed size fields was solved by Bhowmick and Lovett [STOC 2015], which resolved a conjecture of Gopalan-Klivans-Zuckerman [STOC 2008]. Here, we show that the list decoding radius is equal the minimum distance of the code for all fixed degrees, even when the field size is possibly growing with$n$. Additionally, we effectively resolve the weight distribution problem for Reed-Muller codes of fixed degree over all fields, first raised in 1977 in the classic textbook by MacWilliams and Sloane [Research Problem 15.1 in Theory of Error Correcting Codes].

Fast mixing of random walks on hypergraphs (simplicial complexes) has recently led to myriad breakthroughs throughout theoretical computer science. Many important applications, however, (e.g. to LTCs, 2-2 games) rely on a more general class of underlying structures called posets, and crucially take advantage of non-simplicial structure. These works make it clear that the global expansion properties of posets depend strongly on their underlying architecture (e.g. simplicial, cubical, linear algebraic), but the overall phenomenon remains poorly understood. In this work, we quantify the advantage of different poset architectures in both a spectral and combinatorial sense, highlighting how regularity controls the spectral decay and edge-expansion of corresponding random walks. We show that the spectra of walks on expanding posets (Dikstein, Dinur, Filmus, Harsha APPROX-RANDOM 2018) concentrate in strips around a small number of approximate eigenvalues controlled by the regularity of the underlying poset. This gives a simple condition to identify poset architectures (e.g. the Grassmann) that exhibit strong (even exponential) decay of eigenvalues, versus architectures like hypergraphs whose eigenvalues decay linearly - a crucial distinction in applications to hardness of approximation and agreement testing such as the recent proof of the 2-2 Games Conjecture (Khot, Minzer, Safra FOCS 2018). We show these results lead to a tight characterization of edge-expansion on expanding posets in the 𝓁₂-regime (generalizing recent work of Bafna, Hopkins, Kaufman, and Lovett (SODA 2022)), and pay special attention to the case of the Grassmann where we show our results are tight for a natural set of sparsifications of the Grassmann graphs. We note for clarity that our results do not recover the characterization of expansion used in the proof of the 2-2 Games Conjecture which relies on 𝓁_∞ rather than 𝓁₂-structure.

The equivalence of realizable and agnostic learnability is a fundamental phenomenon in learning theory. With variants ranging from classical settings like PAC learning and regression to recent trends such as adversarially robust and private learning, it’s surprising we still lack a unified theory; traditional proofs of the equivalence tend to be disparate, and rely on strong model-specific assumptions like uniform convergence and sample compression. In this work, we give the first model-independent framework explaining the equivalence of realizable and agnostic learnability: a three-line blackbox reduction that simplifies, unifies, and extends our understanding across a wide variety of settings. This includes models with no known characterization of learnability such as learning with arbitrary distributional assumptions or general loss, as well as a host of other popular settings such as robust learning, partial learning, fair learning, and the statistical query model. More generally, we argue that the equivalence of realizable and agnostic learning is actually a special case of a broader phenomenon we call property generalization: any desirable property of a learning algorithm (e.g. noise tolerance, privacy, stability) that can be satisfied over finite hypothesis classes extends (possibly in some variation) to any learnable hypothesis class.

Reinforcement learning with function approximation has recently achieved tremendous results in applications with large state spaces. This empirical success has motivated a growing body of theoretical work proposing necessary and sufficient conditions under which efficient reinforcement learning is possible. From this line of work, a remarkably simple minimal sufficient condition has emerged for sample efficient reinforcement learning: MDPs with optimal value function V* and Q* linear in some known low-dimensional features. In this setting, recent works have designed sample efficient algorithms which require a number of samples polynomial in the feature dimension and independent of the size of state space. They however leave finding computationally efficient algorithms as future work and this is considered a major open problem in the community. In this work, we make progress on this open problem by presenting the first computational lower bound for RL with linear function approximation: unless NP=RP, no randomized polynomial time algorithm exists for deterministic transition MDPs with a constant number of actions and linear optimal value functions. To prove this, we show a reduction from Unique-Sat, where we convert a CNF formula into an MDP with deterministic transitions, constant number of actions and low dimensional linear optimal value functions. This result also exhibits the first computational-statistical gap in reinforcement learning with linear function approximation, as the underlying statistical problem is information-theoretically solvable with a polynomial number of queries, but no computationally efficient algorithm exists unless NP=RP. Finally, we also prove a quasi-polynomial time lower bound under the Randomized Exponential Time Hypothesis.

Higher order random walks (HD-walks) on high dimensional expanders (HDX) have seen an incredible amount of study and application since their introduction by Kaufman and Mass (ITCS 2016), yet their broader combinatorial and spectral properties remain poorly understood. We develop a combinatorial characterization of the spectral structure of HD-walks on two-sided local-spectral expanders (Dinur and Kaufman FOCS 2017), which offer a broad generalization of the well-studied Johnson and Grassmann graphs. Our characterization, which shows that the spectra of HD-walks lie tightly concentrated in a few combinatorially structured strips, leads to novel structural theorems such as a tight ℓ2-characterization of edge-expansion, as well as to a new understanding of local-to-global graph algorithms on HDX. Towards the latter, we introduce a novel spectral complexity measure called Stripped Threshold Rank, and show how it can replace the (much larger) threshold rank as a parameter controlling the performance of algorithms on structured objects. Combined with a sum-of-squares proof for the former ℓ2-characterization, we give a concrete application of this framework to algorithms for unique games on HD-walks, where in many cases we improve the state of the art (Barak, Raghavendra, and Steurer FOCS 2011, and Arora, Barak, and Steurer JACM 2015) from nearly-exponential to polynomial time (e.g. for sparsifications of Johnson graphs or of slices of the q-ary hypercube). Our characterization of expansion also holds an interesting connection to hardness of approximation, where an ℓ∞-variant for the Grassmann graphs was recently used to resolve the 2-2 Games Conjecture (Khot, Minzer, and Safra FOCS 2018). We give a reduction from a related ℓ∞-variant to our ℓ2-characterization, but it loses factors in the regime of interest for hardness where the gap between ℓ2 and ℓ∞ structure is large. Nevertheless, our results open the door for further work on the use of HDX in hardness of approximation and their general relation to unique games.

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 Khot’s 2-2 Games Conjecture (Khot, Minzer, Safra FOCS 2018). In this work, we develop a new theory of hypercontractivity on high dimensional expanders (HDX), an important class of expanding complexes that has recently seen similarly impressive applications in both coding theory and approximate sampling. Our results lead to a new understanding of the structure of Boolean functions on HDX, including a tight analog of the KKL Theorem and a new characterization of non-expanding sets.

We prove new results on the polarizing random walk framework introduced in recent works of Chattopadhyay et al. [Chattopadhyay et al., 2019; Eshan Chattopadhyay et al., 2019] that exploit L₁ Fourier tail bounds for classes of Boolean functions to construct pseudorandom generators (PRGs). We show that given a bound on the k-th level of the Fourier spectrum, one can construct a PRG with a seed length whose quality scales with k. This interpolates previous works, which either require Fourier bounds on all levels [Chattopadhyay et al., 2019], or have polynomial dependence on the error parameter in the seed length [Eshan Chattopadhyay et al., 2019], and thus answers an open question in [Eshan Chattopadhyay et al., 2019]. As an example, we show that for polynomial error, Fourier bounds on the first O(log n) levels is sufficient to recover the seed length in [Chattopadhyay et al., 2019], which requires bounds on the entire tail. We obtain our results by an alternate analysis of fractional PRGs using Taylor’s theorem and bounding the degree-k Lagrange remainder term using multilinearity and random restrictions. Interestingly, our analysis relies only on the level-k unsigned Fourier sum, which is potentially a much smaller quantity than the L₁ notion in previous works. By generalizing a connection established in [Chattopadhyay et al., 2020], we give a new reduction from constructing PRGs to proving correlation bounds. Finally, using these improvements we show how to obtain a PRG for 𝔽₂ polynomials with seed length close to the state-of-the-art construction due to Viola [Emanuele Viola, 2009].

The explosive growth of easily-accessible unlabeled data has lead to growing interest in \emph{active learning}, a paradigm in which data-hungry learning algorithms adaptively select informative examples in order to lower prohibitively expensive labeling costs. Unfortunately, in standard worst-case models of learning, the active setting often provides no improvement over non-adaptive algorithms. To combat this, a series of recent works have considered a model in which the learner may ask \emph{enriched} queries beyond labels. While such models have seen success in drastically lowering label costs, they tend to come at the expense of requiring large amounts of memory. In this work, we study what families of classifiers can be learned in \emph{bounded memory}. To this end, we introduce a novel streaming-variant of enriched-query active learning along with a natural combinatorial parameter called \emph{lossless sample compression} that is sufficient for learning not only with bounded memory, but in a query-optimal and computationally efficient manner as well. Finally, we give three fundamental examples of classifier families with small, easy to compute lossless compression schemes when given access to basic enriched queries: axis-aligned rectangles, decision trees, and halfspaces in two dimensions.

This work introduces Bilinear Classes, a new structural framework, which permit generalization in reinforcement learning in a wide variety of settings through the use of function approximation. The framework incorporates nearly all existing models in which a polynomial sample complexity is achievable, and, notably, also includes new models, such as the Linear Q*/V* model in which both the optimal Q-function and the optimal V-function are linear in some known feature space. Our main result provides an RL algorithm which has polynomial sample complexity for Bilinear Classes; notably, this sample complexity is stated in terms of a reduction to the generalization error of an underlying supervised learning sub-problem. These bounds nearly match the best known sample complexity bounds for existing models. Furthermore, this framework also extends to the infinite dimensional (RKHS) setting: for the the Linear Q*/V* model, linear MDPs, and linear mixture MDPs, we provide sample complexities that have no explicit dependence on the explicit feature dimension (which could be infinite), but instead depends only on information theoretic quantities.

Let f: {0, 1}n → {0, 1} be a boolean function, and let f∧(x, y) = f(x ∧ y) denote the AND-function of f, where x ∧ y denotes bit-wise AND. We study the deterministic communication complexity of f∧ and show that, up to a logn factor, it is bounded by a polynomial in the logarithm of the real rank of the communication matrix of f∧. This comes within a logn factor of establishing the log-rank conjecture for AND-functions with no assumptions on f. Our result stands in contrast with previous results on special cases of the log-rank conjecture, which needed significant restrictions on f such as monotonicity or low F2-degree. Our techniques can also be used to prove (within a logn factor) a lifting theorem for AND-functions, stating that the deterministic communication complexity of f∧ is polynomially related to the AND-decision tree complexity of f.

A $k \times n$ matrix over a field is called an MDS (maximum distance separable) matrix if it satisfies the following property: Any $k$ columns of it are linearly independent. Equivalently, its rows span an MDS code. A question arising in coding theory is what zero patterns MDS matrices can have. There is a natural combinatorial condition, called the rectangle condition, which is necessary over any field, and sufficient over exponentially large fields, concretely of size ${n-1 \choose k-1}$. The GM-MDS conjecture of Dau, Song, and Yuen [ On the existence of MDS codes over small fields with constrained generator matrices, in 2014 IEEE International Symposium on Information Theory (ISIT), pp. 1787--1791] speculated that whenever the rectangle condition holds, there exist algebraic constructions over much smaller fields of size $n+k-1$, and gave an algebraic conjecture that implies this. In this work, we prove this algebraic conjecture. In an independent and parallel work, Yildiz and Hassibi [ Optimum linear codes with support constraints over small fields, in 2018 IEEE Information Theory Workshop (ITW), pp. 1--5] found an alternative proof for the algebraic conjecture.

Sign-rank and discrepancy are two central notions in communication complexity. The seminal work of Babai, Frankl, and Simon from 1986 initiated an active line of research that investigates the gap between these two notions. In this article, we establish the strongest possible separation by constructing a boolean matrix whose sign-rank is only 3, and yet its discrepancy is 2^{-Ω(n)}. We note that every matrix of sign-rank 2 has discrepancy n^{-O(1)}. Our result in particular implies that there are boolean functions with O(1) unbounded error randomized communication complexity while having Ω(n) weakly unbounded error randomized communication complexity.

With the explosion of massive, widely available unlabeled data in the past years, finding label and time efficient, robust learning algorithms has become ever more important in theory and in practice. We study the paradigm of active learning, in which algorithms with access to large pools of data may adaptively choose what samples to label in the hope of exponentially increasing efficiency. By introducing comparisons, an additional type of query comparing two points, we provide the first time and query efficient algorithms for learning non-homogeneous linear separators robust to bounded (Massart) noise. We further provide algorithms for a generalization of the popular Tsybakov low noise condition, and show how comparisons provide a strong reliability guarantee that is often impractical or impossible with only labels - returning a classifier that makes no errors with high probability.

Given a finite set X ⊂ Rdand a binary linear classifier c: Rd→ {0,1}, how many queries of the form c(x) are required to learn the label of every point in X? Known as point location, this problem has inspired over 35 years of research in the pursuit of an optimal algorithm. Building on the prior work of Kane, Lovett, and Moran (ICALP 2018), we provide the first nearly optimal solution, a randomized linear decision tree of depth Õ(dlog(|X|)), improving on the previous best of Õ(d2log(|X|)) from Ezra and Sharir (Discrete and Computational Geometry, 2019). As a corollary, we also provide the first nearly optimal algorithm for actively learning halfspaces in the membership query model. En route to these results, building on the work of Carlen, Lieb, and Loss (J. Geometric Analysis 2004), as well as Dvir, Saraf, and Wigderson (STOC 2014), we prove a novel characterization of Barthe's Theorem (Inventiones Mathematicae, 1998) of independent interest. In particular, we show that X may be transformed into approximate isotropic position if and only if there exists no k-dimensional subspace with more than a k/d-fraction of X, and provide a similar characterization for exact isotropic position. The below is an extended abstract. The full work can be found at https://arxiv.org/abs/2004.11380.

Combinatorial dimensions play an important role in the theory of machine learning. For example, VC dimension characterizes PAC learning, SQ dimension characterizes weak learning with statistical queries, and Littlestone dimension characterizes online learning. In this paper we aim to develop combinatorial dimensions that characterize bounded memory learning. We propose a candidate solution for the case of realizable strong learning under a known distribution, based on the SQ dimension of neighboring distributions. We prove both upper and lower bounds for our candidate solution, that match in some regime of parameters. This is the first characterization of strong learning under space constraints in any regime. In this parameter regime there is an equivalence between bounded memory and SQ learning. We conjecture that our characterization holds in a much wider regime of parameters.

In the world of big data, large but costly to label datasets dominate many fields. Active learning, a semi-supervised alternative to the standard PAC-learning model, was introduced to explore whether adaptive labeling could learn concepts with exponentially fewer labeled samples. While previous results show that active learning performs no better than its supervised alternative for important concept classes such as linear separators, we show that by adding weak distributional assumptions and allowing comparison queries, active learning requires exponentially fewer samples. Further, we show that these results hold as well for a stronger model of learning called Reliable and Probably Useful (RPU) learning. In this model, our learner is not allowed to make mistakes, but may instead answer ``I don't know.'' While previous negative results showed this model to have intractably large sample complexity for label queries, we show that comparison queries make RPU-learning at worst logarithmically more expensive in both the passive and active regimes.

A sunflower with r petals is a collection of r sets so that the intersection of each pair is equal to the intersection of all. Erdős and Rado proved the sunflower lemma: for any fixed r, any family of sets of size w, with at least about w w sets, must contain a sunflower. The famous sunflower conjecture is that the bound on the number of sets can be improved to c w for some constant c. In this paper, we improve the bound to about (logw) w . In fact, we prove the result for a robust notion of sunflowers, for which the bound we obtain is tight up to lower order terms.

A major challenge in complexity theory is to explicitly construct functions that have small correlation with low-degree polynomials over F2. We introduce a new technique to prove such correlation bounds with F2 polynomials. Using this technique, we bound the correlation of an XOR of Majorities with constant degree polynomials. In fact, we prove a more general XOR lemma that extends to arbitrary resilient functions. We conjecture that the technique generalizes to higher degree polynomials as well.

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 whose term is satisfied by the input. Decision lists generalize both CNFs and DNFs and have been studied both in complexity theory and in learning theory. The size of a decision list is the number of rules, and its width is the maximal number of variables in a term. We prove that decision lists of small width can always be approximated by decision lists of small size, where we obtain sharp bounds for such approximation. This also resolves a conjecture of Gopalan, Meka, and Reingold (Computational Complexity, 2013) on DNF sparsification. An ingredient in our proof is a new random restriction lemma, which allows to analyze how DNFs (and more generally, decision lists) simplify if a small fraction of the variables are fixed. This is in contrast to the more commonly used switching lemma, which requires most of the variables to be fixed.

The canonical problem that gives an exponential separation between deterministic and randomized communication complexity in the classical two-party communication model is "Equality". In this work we show that even allowing access to an "Equality" oracle, deterministic protocols remain exponentially weaker than randomized ones. More precisely, we exhibit a total function on n bits with randomized one-sided communication complexity O(log n), but such that every deterministic protocol with access to "Equality" oracle needs Omega(n) cost to compute it. Additionally we exhibit a natural and strict infinite hierarchy within BPP, starting with the class P^{EQ} at its bottom.

A major open problem in quantum communication complexity is whether quantum protocols can be exponentially more efficient than classical protocols for computing total Boolean functions; the prevailing conjecture is that they cannot be so. In a seminal work, Razborov (2002) resolved this question for And-functions of the form F(x,y) = f(x₁ ∧ y₁, …, x_n ∧ y_n), when the outer function f is symmetric, by proving that their bounded-error quantum and classical communication complexities are polynomially related. Since then, extending this result to all And-functions has remained open and has been posed by several authors. In this work, we settle this problem in a strong way. We show that for every Boolean function f, the bounded-error quantum and classical deterministic communication complexities of the function f∘And₂ are polynomially related, up to polylogarithmic factors in n. We prove this by showing that both are characterized - up to polynomial loss - by the logarithm of the De Morgan sparsity of f. Our results build on the recent work of Chattopadhyay, Dahiya, and Lovett [Arkadev Chattopadhyay et al., 2026] on structural characterizations of non-sparse Boolean functions, which we extend to resolve the conjecture for general And-functions.

The sunflower conjecture is one of the most well-known open problems in combinatorics. It has several applications in theoretical computer science, one of which is DNF compression, due to Gopalan, Meka and Reingold (Computational Complexity, 2013). In this paper, we show that improved bounds for DNF compression imply improved bounds for the sunflower conjecture, which is the reverse direction of the DNF compression result. The main approach is based on regularity of set systems and a structure-vs-pseudorandomness approach to the sunflower conjecture.

We propose an algebraic approach to proving circuit lower bounds for ACC0 by defining and studying the notion of torus polynomials. We show how currently known polynomial-based approximation results for AC0 and ACC0 can be reformulated in this framework, implying that ACC0 can be approximated by low-degree torus polynomials. Furthermore, as a step towards proving ACC0 lower bounds for the majority function via our approach, we show that MAJORITY cannot be approximated by low-degree symmetric torus polynomials. We also pose several open problems related to our framework.

A recent work of Chattopadhyay et al. (CCC 2018) introduced a new framework for the design of pseudorandom generators for Boolean functions. It works under the assumption that the Fourier tails of the Boolean functions are uniformly bounded for all levels by an exponential function. In this work, we design an alternative pseudorandom generator that only requires bounds on the second level of the Fourier tails. It is based on a derandomization of the work of Raz and Tal (ECCC 2018) who used the above framework to obtain an oracle separation between BQP and PH. As an application, we give a concrete conjecture for bounds on the second level of the Fourier tails for low degree polynomials over the finite field F_2. If true, it would imply an efficient pseudorandom generator for AC^0[oplus], a well-known open problem in complexity theory. As a stepping stone towards resolving this conjecture, we prove such bounds for the first level of the Fourier tails.

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 the size). The other direction is much less clear.

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 queries, which compare the sums of two k -subsets; when viewed as linear queries, comparison queries are 2 k -sparse and have only { −1,0,1} coefficients. We give similar constructions for sorting sumsets A+B and for solving the SUBSET-SUM problem, both with optimal number of queries, up to poly-logarithmic terms. Our constructions are based on the notion of “inference dimension,” recently introduced by the authors in the context of active classification with comparison queries. This can be viewed as another contribution to the fruitful link between machine learning and discrete geometry, which goes back to the discovery of the VC dimension.

Maximally recoverable codes are codes designed for distributed storage which combine quick recovery from single node failure and optimal recovery from catastrophic failure. Gopalan et al. [ Maximally recoverable codes for grid-like topologies, in Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2017, pp. 2092--2108] studied the alphabet size needed for such codes in grid topologies and gave a combinatorial characterization for it. Consider a labeling of the edges of the complete bipartite graph $K_{n,n}$, with labels coming from $\mathbb{F}_2^d$, that satisfies the following condition: for any simple cycle, the sum of the labels over its edges is nonzero. The minimal $d$ where this is possible controls the alphabet size needed for maximally recoverable codes in $n \times n$ grid topologies. Prior to the current work, it was known that $d$ is between $(\log n)^2$ and $n \log n$. We improve both bounds and show that $d$ is linear in $n$. The upper bound is a recursive construction which beats the random construction. The lower bound follows by first relating the problem to the independence number of the Birkhoff polytope graph, and then providing tight bounds for it using the representation theory of the symmetric group.

The Erdös-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 it holds. In this work, we exhibit a surprising connection between the existence of sunflowers and quasi-sunflowers in large enough set systems, and the problem of constructing (or existing) certain randomness extractors. This allows us to re-derive the known results in a systematic manner, and to reduce the relevant conjectures to the problem of obtaining improved constructions of the randomness extractors.

We show that proving mildly super-linear lower bounds on non-commutative arithmetic circuits implies exponential lower bounds on non-commutative circuits. That is, non-commutative circuit complexity is a threshold phenomenon: an apparently weak lower bound actually suffices to show the strongest lower bounds we could desire. This is part of a recent line of inquiry into why arithmetic circuit complexity, despite being a heavily restricted version of Boolean complexity, still cannot prove super-linear lower bounds on general devices. One can view our work as positive news (it suffices to prove weak lower bounds to get strong ones) or negative news (it is as hard to prove weak lower bounds as it is to prove strong ones). We leave it to the reader to determine their own level of optimism.

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 converges to {-1,1}^n. We prove that this random walk converges fast (in time logarithmic in n) due to polarization. As an application, we construct pseudorandom generators for Boolean functions with bounded Fourier tails. We use this to obtain a pseudorandom generator for functions with sensitivity s, whose seed length is polynomial in s. Other examples include functions computed by branching programs of various sorts or by bounded depth circuits.

An MDS matrix is a matrix whose minors all have full rank. A question arising in coding theory is, what zero patterns can MDS matrices have. There is a natural combinatorial necessary condition (called the MDS condition) which is necessary over any field, and sufficient over very large fields by a probabilistic argument. Dau et al. (ISIT 2014) conjectured that the MDS condition is sufficient over small fields as well, and gave an algebraic conjecture which would imply this. In this work, we prove this conjecture.

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 generalized comparison queries. These queries correspond to hyperplanes that can be written as a linear combination of two hyperplanes from H; in particular, if all hyperplanes in H are k-sparse then generalized comparisons are 2k-sparse. The depth of the obtained linear decision tree is polynomial in d and logarithmic in |H|, which is comparable to previous results in the literature that use general linear queries. This extends the study of comparison trees from a previous work by the authors [Kane {et al.}, FOCS 2017]. The main benefit is that using generalized comparison queries allows to overcome limitations that apply for the more restricted type of comparison queries. Our analysis combines a seminal result of Forster regarding sets in isotropic position [Forster, JCSS 2002], the margin-based inference dimension analysis for comparison queries from [Kane {et al.}, FOCS 2017], and compactness arguments.

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 the robust sensitivity conjecture in this work with near optimal parameters.

An important result in discrepancy due to Banaszczyk states that for any set of n vectors in ℝm of ℓ2 norm at most 1 and any convex body K in ℝm of Gaussian measure at least half, there exists a ± 1 combination of these vectors which lies in 5K. This result implies the best known bounds for several problems in discrepancy. Banaszczyk’s proof of this result is non-constructive and an open problem has been to give an efficient algorithm to find such a ± 1 combination of the vectors.

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 log2 n) linear queries. Moreover, the queries we use are comparison queries, which compare the sums of two k-subsets; when viewed as linear queries, comparison queries are 2k-sparse and have only {−1,0,1} coefficients. We give similar constructions for sorting sumsets A+B and for solving the SUBSET-SUM problem, both with optimal number of queries, up to poly-logarithmic terms.

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 completely unrelated value. Although such codes do not exist if the family of “tampering functions” ${\mathcal F}$ is completely unrestricted, they are known to exist for many broad tampering families ${\mathcal F}$. One such natural family is the family of tampering functions in the so-called split-state model. Here the message $m$ is encoded into two shares $L$ and $R$, and the attacker is allowed to arbitrarily tamper with $L$ and $R$ individually. The split-state tampering arises in many realistic applications, such as the design of non-malleable secret sharing schemes, motivating the question of designing efficient non-malleable codes in this model. Prior to this work, non-malleable codes in the split-state model received considerable attention in the literature but either (1) were constructed in the random oracle model, or (2) relied on advanced cryptographic assumptions (such as noninteractive zero-knowledge proofs and leakage-resilient encryption), or (3) could only encode 1-bit messages. As our main result, we build the first efficient, multi-bit, information-theoretically-secure non-malleable code in the split-state model. The heart of our construction uses the following new property of the inner-product function $\langle{L,R\rangle}$ over the vector space ${{F}_p}^n$ (for a prime $p$ and large enough dimension $n$): if $L$ and $R$ are uniformly random over ${\mathbb{F}_p}^n$, and $f,g:{\mathbb{F}_p}^n\rightarrow {\mathbb{F}_p}^n$ are two arbitrary functions on $L$ and $R$, then the joint distribution $(\langle{L,R\rangle},\langle{f(L),g(R)\rangle})$ is “close” to the convex combination of “affine distributions” $\{(U,aU+b)\mid a,b\in \mathbb{F}_p\}$, where $U$ is uniformly random in ${\mathbb{F}_p}$. In turn, the proof of this surprising property of the inner product function critically relies on some results from additive combinatorics, including the so-called quasi-polynomial Freiman--Ruzsa theorem, which was recently established by Sanders [Anal. PDE, 5 (2012), pp. 627--655] as a step toward resolving the polynomial Freiman--Ruzsa conjecture [B. Green, in Surveys in Combinatorics, London Mathematical Society, London, 2005, pp. 1--29].

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, originating from [Goldreich, Electronic Colloquium on Computational Complexity (ECCC), 2000], is based on random local functions: Compute $y_i$ by applying some fixed (public) $d$-ary predicate $P$ to a random (public) tuple of distinct input indices $(x_{i_1},\ldots,x_{i_d})$. Our goal in this paper is to understand, for any value of $s$, how the pseudorandomness of the resulting sequence depends on the choice of the underlying predicate. We derive the following results: (1) We show that pseudorandomness against $\mathbb{F}_2$-linear adversaries (i.e., the distribution $y$ has small bias) is achieved if the predicate is (a) $k=\Omega(s)$-resilient, i.e., uncorrelated with any $k$-subset of its inputs, and (b) has algebraic degree of $\Omega(s)$ even after fixing $\Omega(s)$ of its inputs. We also show that these requirements are necessary, and so they form a tight characterization (up to constants) of security against linear attacks. Our positive result shows that a $d$-local small-biased generator can have output length of $n^{\Omega(d)}$, answering an open question of Mossel, Shpilka, and Trevisan [ Proceedings of FOCS, 2003]. Our negative result shows that a candidate for a pseudorandom generator proposed by Applebaum [ Comput. Complexity, 25 (2016), pp. 667--722] and by O'Donnell and Witmer [ Proceedings of CCC 2014] is insecure. We use similar techniques to refute a conjecture of Feldman, Perkins, and Vempala [ Proceedings of STOC 2015] regarding the hardness of planted constraint satisfaction problems. (2) Motivated by the cryptanalysis literature, we consider security against algebraic attacks. We provide the first theoretical treatment of such attacks by formalizing a general notion of algebraic inversion and distinguishing attacks based on the polynomial calculus proof system. We show that algebraic attacks succeed if and only if the predicate $P$ has rational degree $e=\Theta(s)$, where the rational degree of a predicate $P$ is the smallest integer $e$ for which there exist degree $e$ polynomials $Q,R$, not both zero, such that $PQ=R$. As a corollary, we obtain the first example of a predicate $P$ for which the generated sequence $y$ passes all linear tests but fails to pass some polynomial-time computable test, answering an open question posed by Applebaum [ Comput. Complexity, 25 (2016), pp. 667--722].

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 of entropy reduction for protocols, combined with existing techniques in Fourier analysis and additive combinatorics.

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 F. The Reed-Muller code RMF(n, d) is defined by n-variate degree-d polynomials over F. In this paper, we study the list decoding radius of Reed-Muller codes over a constant prime field F = Fp, constant degree d, and large n. We show that the list decoding radius is equal to the minimal distance of the code. That is, if we denote by S(d) the normalized minimal distance of RMF(n, d), then the number of codewords in any ball of radius δ(d) - ε is bounded by c = c(p, d, e) independent of n. This resolves a conjecture of Gopalan et al., who among other results proved it in the special case of F = F2; and extends the work of Gopalan who proved the conjecture in the case of d = 2. We also analyse the number of codewords in balls of radius exceeding the minimal distance of the code. For e ≤ d, we show that the number of codewords of RMF(n, d) in a ball of radius δ(e)-ε is bounded by exp(c · nd-e), where c = c(p, d, ε) is independent of n. The dependence on n is tight. This extends the work of Kaufman et al. who proved similar bounds over F2. The proof relies on several new ingredients: an extension of the Frieze- Kannan weak regularity to general function spaces, higher order Fourier analysis, and an extension of the Schwartz-Zippel lemma to the compositions of polynomials.

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 independently with some unknown noise probability, estimate from a few samples the underlying parameters of the model. Previous work [De et al., FOCS 2016] designed polynomial time algorithms which work under the assumption that the noise parameters are known exactly. In this work, we remove this assumption, and show how to recover the underlying parameters, even when the noise is unknown, in quasi-polynomial time.

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 specific restaurant (a label query); or which one of two restaurants did she like more (a comparison query). We focus on the class of half spaces, and show that under natural assumptions, such as large margin or bounded bit-description of the input examples, it is possible to reveal all the labels of a sample of size n using approximately O(log n) queries. This implies an exponential improvement over classical active learning, where only label queries are allowed. We complement these results by showing that if any of these assumptions is removed then, in the worst case, Ω(n) queries are required. Our results follow from a new general framework of active learning with additional queries. We identify a combinatorial dimension, called the inference dimension, that captures the query complexity when each additional query is determined by O(1) examples (such as comparison queries, each of which is determined by the two compared examples). Our results for half spaces follow by bounding the inference dimension in the cases discussed above.

Maximally recoverable codes are codes designed for distributed storage which combine quick recovery from single node failure and optimal recovery from catastrophic failure. Gopalan et al [SODA 2017] studied the alphabet size needed for such codes in grid topologies and gave a combinatorial characterization for it. Consider a labeling of the edges of the complete bipartite graph Kn,nwith labels coming from F2d, that satisfies the following condition: for any simple cycle, the sum of the labels over its edges is nonzero. The minimal d where this is possible controls the alphabet size needed for maximally recoverable codes in n × n grid topologies. Prior to the current work, it was known that d is between log(n)2and n log n. We improve both bounds and show that d is linear in n. The upper bound is a recursive construction which beats the random construction. The lower bound follows by first relating the problem to the independence number of the Birkhoff polytope graph, and then providing tight bounds for it using the representation theory of the symmetric group.

An important theorem of Banaszczyk (Random Structures & Algorithms 1998) states that for any sequence of vectors of l_2 norm at most 1/5 and any convex body K of Gaussian measure 1/2 in R^n, there exists a signed combination of these vectors which lands inside K. A major open problem is to devise a constructive version of Banaszczyk's vector balancing theorem, i.e. to find an efficient algorithm which constructs the signed combination. We make progress towards this goal along several fronts. As our first contribution, we show an equivalence between Banaszczyk's theorem and the existence of O(1)-subgaussian distributions over signed combinations. For the case of symmetric convex bodies, our equivalence implies the existence of a universal signing algorithm (i.e. independent of the body), which simply samples from the subgaussian sign distribution and checks to see if the associated combination lands inside the body. For asymmetric convex bodies, we provide a novel recentering procedure, which allows us to reduce to the case where the body is symmetric. As our second main contribution, we show that the above framework can be efficiently implemented when the vectors have length O(1/sqrt{log n}), recovering Banaszczyk's results under this stronger assumption. More precisely, we use random walk techniques to produce the required O(1)-subgaussian signing distributions when the vectors have length O(1/sqrt{log n}), and use a stochastic gradient ascent method to implement the recentering procedure for asymmetric bodies.

Let f be a boolean function on n variables. Its associated XOR function is the two-party function F(x, y) = f(x xor y). We show that, up to polynomial factors, the deterministic communication complexity of F is equal to the parity decision tree complexity of f. This relies on a novel technique of entropy reduction for protocols, combined with existing techniques in Fourier analysis and additive combinatorics.

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 to k that is close to uniform and independent of Y. Dodis and Wichs [DW09] introduced a generalization of randomness extractors called non-malleable extractors (nmExt) where nmExt(X, Y) is close to uniform and independent of Y and nmExt(X, f(Y)) for any function f with no fixed points. We relax the notion of a non-malleable extractor and introduce what we call an affine-malleable extractor (AmExt : Fnx Fd→ F) where AmExt(X, Y ) is close to uniform and independent of Y and has some limited dependence of AmExt(X, f(Y )) - that conditioned on Y , (AmExt(X, Y ), AmExt(X, f(Y ))) is ε-close to (U, A · U + B) where U is uniformly distributed in F and A, B E F are random variables independent of U. We show that the inner-product function (·, ·) : Fn×Fn→ F is an affine-malleable extractor for min-entropy k = n/2 + Ω(log(1/ε)). Moreover, under a plausible conjecture in additive combinatorics (called the Spectrum Doubling Conjecture), we show that this holds for k = Ω(log n log(1/ε)). As a modest justification of the conjecture, we show that a weaker version of the conjecture is implied by the widely believed Polynomial Freiman-Ruzsa conjecture. We also study the classical problem of privacy amplification, where two parties Alice and Bob share a weak secret X of min-entropy k, and wish to agree on secret key R of length m over a public communication channel completely controlled by a computationally unbounded attacker Eve. The main application of non-malleable extractors and their many variants has been in constructing secure privacy amplification protocols. We show that affine-malleable extractors along with affine-evasive sets can also be used to construct efficient privacy amplification protocols. This gives a much simpler protocol for min-entropy k = n/2 + Ω(log(1/ε)), and additionally, under the Spectrum Doubling Conjecture, achieves near optimal parameters and achieves additional security properties like source privacy that have been the focus of some recent results in privacy amplification.

We prove that any total boolean function of rank r can be computed by a deterministic communication protocol of complexity O (√ ċ log( r )). Equivalently, any graph whose adjacency matrix has rank r has chromatic number at most 2 O (√ r ċlog( r )) . This gives a nearly quadratic improvement in the dependence on the rank over previous results.

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 $f \circ g^n$ can be simulated by a nonnegative combination of juntas. This is a new formalization for the intuition that each low-communication randomized protocol can only “query” a few inputs of $f$ as encoded by the gadget $g$. Consequently, we characterize the communication complexity of $f\circ g^n$ in all known one-sided (i.e., not closed under complement) zero-communication models by a corresponding query complexity measure of $f$. These models in turn capture important lower bound techniques such as corruption, smooth rectangle bound, relaxed partition bound, and extended discrepancy. As applications, we resolve several open problems from prior work. We show that $\mathsf{SBP}^{\sf cc}$ (a class characterized by corruption) is not closed under intersection. An immediate corollary is that $\mathsf{MA}^{\sf cc} \neq \mathsf{SBP}^{\sf cc}$. These results answer questions of Klauck [Proceedings of the 18th Conference on Computational Complexity (CCC), IEEE Computer Society, Los Alamitos, CA, 2003, pp. 118--134] and Böhler, Glasser, and Meister [J. Comput. System Sci., 72 (2006), pp. 1043--1076]. We also show that the approximate nonnegative rank of partial boolean matrices does not admit efficient error reduction. This answers a question of Kol et al. [Proceedings of the 41st International Colloquium on Automata, Languages, and Programming (ICALP), Springer, Berlin, 2014, pp. 701--712] for partial matrices. In subsequent work, our structure theorem has been applied to resolve the communication complexity of the clique versus independent set problem.

We prove that for any constant k and any epsilon < 1, there exist bimatrix win-lose games for which every epsilon-WSNE requires supports of cardinality greater than k. To do this, we provide a graph-theoretic characterization of win-lose games that possess epsilon-WSNE with constant cardinality supports. We then apply a result in additive number theory of Haight to construct win-lose games that do not satisfy the requirements of the characterization. These constructions disprove graph theoretic conjectures of Daskalakis, Mehta and Papadimitriou and Myers.

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 super-logarithmic degree. Here, we suggest a new barrier explaining this phenomenon. We show that many of the existing lower bound proof techniques extend to nonclassical polynomials, an extension of classical polynomials which arose in higher order Fourier analysis. Moreover, these techniques are tight for nonclassical polynomials of logarithmic degree.

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 F. The Reed-Muller code RMF(n,d) is defined by n-variate degree-d polynomials over F. In this work, we study the list decoding radius of Reed-Muller codes over a constant prime field F=Fp, constant degree d and large n. We show that the list decoding radius is equal to the minimal distance of the code.

We develop a new method to prove communication lower bounds for composed functions of the form f o gn 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 o gn can be simulated by a nonnegative combination of juntas. This is the strongest yet formalization for the intuition that each low-communication randomized protocol can only "query" few inputs of f as encoded by the gadget g. Consequently, we characterize the communication complexity of f o gn in all known one-sided zero-communication models by a corresponding query complexity measure of f. These models in turn capture important lower bound techniques such as corruption, smooth rectangle bound, relaxed partition bound, and extended discrepancy. As applications, we resolve several open problems from prior work: We show that SBPcc (a class characterized by corruption) is not closed under intersection. An immediate corollary is that MAcc ≠ SBPcc. These results answer questions of Klauck (CCC 2003) and Bohler et al. (JCSS 2006). We also show that approximate nonnegative rank of partial boolean matrices does not admit efficient error reduction. This answers a question of Kol et al. (ICALP) for partial matrices.

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-μ)/2, estimate the original probability up to an additive error of ε. We give an algorithm which solves this problem in time polynomial in (klog log k, n, 1/ε). This improves on the previous algorithm of Wigderson and Yehudayoff [FOCS 2012] which solves the problem in time polynomial in (klog k, n, 1/ε). Our main technical contribution, which facilitates the algorithm, is a new reverse Bonami-Beckner inequality for the L1 norm of sparse functions.

Minimizing the discrepancy of a set system is a fundamental problem in combinatorics. One of the cornerstones in this area is the celebrated six standard deviations result of Spencer [Trans. Amer. Math. Soc., 289 (1985), pp. 679--706]: In any system of $n$ sets in a universe of size $n$, there always exists a coloring which achieves discrepancy $6\sqrt{n}$. The original proof of Spencer was existential in nature and did not give an efficient algorithm to find such a coloring. Recently, a breakthrough work of Bansal [Proceedings of FOCS, 2010, pp. 3--10] gave an efficient algorithm which finds such a coloring. His algorithm was based on an SDP relaxation of the discrepancy problem and a clever rounding procedure. In this work we give a new randomized algorithm to find a coloring as in Spencer's result based on a restricted random walk we call Edge-Walk. Our algorithm and its analysis use only basic linear algebra and is truly constructive in that it does not appeal to the existential arguments, giving a new proof of Spencer's theorem and the partial coloring lemma.

Non-malleable codes provide a useful and meaningful security guarantee in situations where traditional errorcorrection (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 completely unrelated value. Although such codes do not exist if the family of "tampering functions" F is completely unrestricted, they are known to exist for many broad tampering families F. One such natural family is the family of tampering functions in the so called split-state model. Here the message m is encoded into two shares L and R, and the attacker is allowed to arbitrarily tamper with L and R individually. The split-state tampering arises in many realistic applications, such as the design of non-malleable secret sharing schemes, motivating the question of designing efficient non-malleable codes in this model.

We prove that any total boolean function of rank r can be computed by a deterministic communication protocol of complexity O(√r · log(r)). Similarly, any graph whose adjacency matrix has rank r has chromatic number at most 2O(√r · log(r)). This gives a nearly quadratic improvement in the dependence on the rank over previous results.

Identifying complexity measures that bound the communication complexity of a {0,1}-valued matrix M is one the most fundamental problems in communication complexity. Mehlhorn and Schmidt [1982] were the first to suggest matrix-rank as one such measure. Among other things, they showed log rank F(M) CC(M) rankF2(M), where CC ( M ) denotes the (deterministic) communication complexity of the function associated with M , and the rank on the left-hand side is over any field F and on the right-hand side it is over the two-element field F 2. For certain matrices M , communication complexity equals the right-hand side, and this completely settles the question of “communication complexity vs. F 2-rank”. Here we reopen this question by pointing out that, when M has an additional natural combinatorial property---high discrepancy with respect to distributions which are uniform over submatrices---then communication complexity can be sublinear in F 2-rank. Assuming the Polynomial Freiman-Ruzsa (PFR) conjecture in additive combinatorics, we show that CC(M) O(rank F2(M)/log rank F2(M)) for any matrix M which satisfies this combinatorial property. We also observe that if M has low rank over the reals, then it has low rank over F 2 and it additionally satisfies this combinatorial property. As a corollary, our results also give the first (conditional) sublinear bound on communication complexity in terms of rank over the reals, a result improved later by Lovett [2014]. Our proof is based on the study of the “approximate duality conjecture” which was suggested by Ben-Sasson and Zewi [2011] and studied there in connection to the PFR conjecture. First, we improve the bounds on approximate duality assuming the PFR conjecture. Then, we use the approximate duality conjecture (with improved bounds) to get our upper bound on the communication complexity of low-rank matrices.

A matching vector (MV) family modulo $m$ is a pair of ordered lists $U=(u_1,\ldots,u_t)$ and $V=(v_1,\ldots,v_t)$ where $u_i$, $v_j \in \mathbb{Z}_m^n$ with the following inner product pattern: for any $i$, $\langle u_i,v_i\rangle=0$, and for any $i \ne j$, $\langle u_i,v_j\rangle \ne 0$. An MV family is called $q$-restricted if inner products $\langle u_i,v_j\rangle$ take at most $q$ different values. Our interest in MV families stems from their recent application in the construction of subexponential locally decodable codes (LDCs). There, $q$-restricted MV families are used to construct LDCs with $q$ queries, and there is special interest in the regime where $q$ is constant. When $m$ is a prime it is known that such constructions yield codes with exponential block length. However, for composite $m$ the behavior is dramatically different. A recent work by Efremenko [SIAM J. Comput., 40 (2011), pp. 1154--1178] (based on an approach initiated by Yekhanin [J. ACM, 55 (2008), pp. 1--16]) gives the first subexponential LDC with constant queries. It is based on a construction of an MV family of superpolynomial size by Grolmusz [Combinatorica, 20 (2000), pp. 71--86] modulo composite $m$. In this work, we prove two lower bounds on the block length of LDCs which are based on black box construction using MV families. When $q$ is constant (or sufficiently small), we prove that such LDCs must have a quadratic block length. When the modulus $m$ is constant (as it is in the construction of Efremenko) we prove a superpolynomial lower bound on the block-length of the LDCs, assuming a well-known conjecture in additive combinatorics, the polynomial Freiman--Ruzsa conjecture over $\mathbb{Z}_m$.

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 $f:\mathbb{F}_p^n\to\mathbb{F}_p$ with polynomials of degree at most $d\leq p$ is nonnegligible, while making only a constant number of queries to the function. This is an instance of correlation testing. In this framework, a fixed test is applied to a function, and the acceptance probability of the test is dependent on the correlation of the function from the property. This is an analogue of proximity oblivious testing, a notion coined by Goldreich and Ron, in the high error regime. In this work, we study general properties which are affine invariant and which are correlation testable using a constant number of queries. We show that any such property (as long as the field size is not too small) can in fact be tested by Gowers uniformity tests, and hence having correlation with the property is equivalent to having correlation with degree $d$ polynomials for some fixed $d$. We stress that our result holds also for nonlinear properties which are affine invariant. This completely classifies affine invariant properties which are correlation testable. The proof is based on higher-order Fourier analysis. Another ingredient is a nontrivial extension of a graph theoretical theorem of Erdös, Lovász, and Spencer to the context of additive number theory.

Let P be an affine invariant property of multivariate functions over a constant size finite field. We show that if P is locally testable with a constant number of queries, then one can estimate the distance of a function f from P with a constant number of queries. This was previously unknown even for simple properties such as cubic polynomials over the binary field. Our test is simple: take a restriction of f to a constant dimensional affine subspace, and measure its distance from P. We show that by choosing the dimension large enough, this approximates with high probability the global distance of f from P. The analysis combines the approach of Fischer and Newman [SIAM J. Comp 2007] who established a similar result for graph properties, with recently developed tools in higher order Fourier analysis, in particular those developed in Bhattacharyya et al. [STOC 2013].

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, for instance, that the Reed-Muller code over Fp of degree d < p is testable, with an argument that uses no detailed algebraic information about polynomials, except that having low degree is preserved by composition with affine maps. The complexity of an affine-invariant property refers to the maximum complexity, as defined by Green and Tao (Ann. Math. 2008), of the sets of linear forms used to characterize . A more precise statement of our main result is that for any fixed prime p ≥ 2 and fixed integer R ≥ 2, any affine-invariant property of functions f : Fnp → [R] is testable, if the complexity of the property is less than p. Our proof involves developing analogs of graph-theoretic techniques in an algebraic setting, using tools from higher-order Fourier analysis.

Set F = Fp for any fixed prime p ≥ 2. An affine-invariant property is a property of functions over Fn that is closed under taking affine transformations of the domain. We prove that all affine-invariant properties having local characterizations are testable. In fact, we show a proximity-oblivious test for any such property cP, meaning that given an input function f, we make a constant number of queries to f, always accept if f satisfies cP, and otherwise reject with probability larger than a positive number that depends only on the distance between f and cP. More generally, we show that any affine-invariant property that is closed under taking restrictions to subspaces and has bounded complexity is testable.

A Matching Vector (MV) family modulo m is a pair of ordered lists U=(u1,...,ut) and V=(v1,...,vt) where ui,vj ∈ Zmn with the following inner product pattern: for any i, {ui,vi}=0, and for any i ≠ j, {ui,vj} ≠ 0. A MV family is called q-restricted if inner products {ui,vj} take at most q different values.

An approximate membership data structure is a randomized data structure 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 1970s and have since had numerous applications, mainly in distributed systems, database systems, and networks. The algorithm of Bloom (known as a Bloom filter) is quite effective: it can store an approximation of a set $S$ of size $n$ by using only $\approx 1.44 n \log_2(1/\varepsilon)$ bits while having false positive error $\varepsilon$. This is within a constant factor of the information-theoretic lower bound of $n \log_2(1/\varepsilon)$ for storing such sets. Closing this gap is an important open problem, as Bloom filters are widely used in situations where storage is at a premium. Bloom filters have another property: they are dynamic. That is, they support the iterative insertions of up to $n$ elements. In fact, if one removes this requirement, there exist static data structures that receive the entire set at once and can almost achieve the information-theoretic lower bound; they require only $(1+o(1)) n \log_2(1/\varepsilon)$ bits. Our main result is a new lower bound for the space requirements of any dynamic approximate membership data structure. We show that for any constant $\varepsilon>0$, any such data structure that achieves false positive error rate of $\varepsilon$ must use at least $C(\varepsilon) \cdot n \log_2(1/\varepsilon)$ memory bits, where $C(\varepsilon)>1$ depends only on $\varepsilon$. This shows that the information-theoretic lower bound cannot be achieved by dynamic data structures for any constant error rate.

We consider the problem of constructing pseudorandom generators for read-once circuits. We give an explicit construction of a pseudorandom generator for the class of read-once constant depth circuits with unbounded fan-in AND, OR, NOT and generalized modulo m gates, where m is an arbitrary fixed constant. The seed length of our generator is poly-logarithmic in the number of variables and the error.

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 is, such distributions are very far from each other. We strengthen their result, and show that the distance is in fact exponentially close to one. This allows us to strengthen the parameters in their application for data structure lower bounds for succinct data structures for codes. From a technical point of view, we develop new large deviation bounds for functions computed by small depth decision trees, which we then apply to obtain bounds for AC0 circuits via the switching lemma. We show that if such functions are Lipschitz on average in a certain sense, then they are in fact Lipschitz almost everywhere. This type of result falls into the extensive line of research which studies large deviation bounds for the sum of random variables, where while not independent, exhibit large deviation bounds similar to these obtained by independent random variables.

For a {0, 1}-valued matrix M let CC(M) denote the deterministic communication complexity of the boolean function associated with M. It is well-known since the work of Mehlhorn and Schmidt [STOC 1982] that CC(M) is bounded from above by rank(M) and from below by log rank(M) where rank(M) denotes the rank of M over the field of real numbers. Determining where in this range lies the true worst-case value of CC(M) is a fundamental open problem in communication complexity. The state of the art is log1.631rank(M) ≤ CC(M) ≤ 0.415 rank(M), the lower bound is by Kushilevitz [unpublished, 1995] and the upper bound is due to Kotlov [Journal of Graph Theory, 1996]. Lovasz and Saks [FOCS 1988] conjecture that CC(M) is closer to the lower bound, i.e., CC(M)≤ logcrank(M)) for some absolute constant c - this is the famous "log-rank conjecture'' - but so far there has been no evidence to support it, even giving a slightly non-trivial (o(rank(M))) upper bound on the communication complexity. Our main result is that, assuming the Polynomial Freiman-Ruzsa (PFR) conjecture in additive combinatorics, there exists a universal constant c such that CC(M) ≤ c ·rank(M)/log rank(M). Although our bound is stated using the rank of M over the reals, our proof goes by studying the problem over the finite field of size 2, and there we bring to bear a number of new tools from additive combinatorics which we hope will facilitate further progress on this perplexing question. In more detail, our proof is based on the study of the "approximate duality conjecture'' which was suggested by Ben-Sasson and Zewi [STOC 2011] and studied there in connection to the PFR conjecture. First we improve the bounds on approximate duality assuming the PFR conjecture. Then we use the approximate duality conjecture (with improved bounds) to get our upper bound on the communication complexity of low-rank martices.

Minimizing the discrepancy of a set system is a fundamental problem in combinatorics. One of the cornerstones in this area is the celebrated six standard deviations result of Spencer (AMS 1985): In any system of n sets in a universe of size n, there always exists a coloring which achieves discrepancy 6√n. The original proof of Spencer was existential in nature, and did not give an efficient algorithm to find such a coloring. Recently, a breakthrough work of Bansal (FOCS 2010) gave an efficient algorithm which finds such a coloring. His algorithm was based on an SDP relaxation of the discrepancy problem and a clever rounding procedure. In this work we give a new randomized algorithm to find a coloring as in Spencer's result based on a restricted random walk we call Edge-Walk. Our algorithm and its analysis use only basic linear algebra and is “truly” constructive in that it does not appeal to the existential arguments, giving a new proof of Spencer's theorem and the partial coloring lemma.

We construct explicit subspace-evasive sets. These are subsets of Fn of size |F|(1-ε)n whose intersection with any k-dimensional subspace is bounded by a constant c(k,ε). This problem was raised by Guruswami (CCC 2011) as it leads to optimal rate list-decodable codes of constant list size. The main technical ingredient is the construction of k low-degree polynomials whose common set of zeros has small intersection with any k-dimensional subspace.

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. The proof of existence is probabilistic. We show that a randomly chosen such object has the required properties with positive yet tiny probability. The main technical ingredient is a special local central limit theorem for suitable lattice random walks with finitely many steps.

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 on the statistical distance between (i) the output distribution of any small constant-depth (a.k.a. AC0) circuit, and (ii) the uniform distribution over any code that is ``good'', i.e. has constant relative distance and rate. This seems to be the first lower bound of this kind. We give two simple applications of this result: (1) any data structure for storing codewords of a good code requires an additive logarithmic redundancy, if each bit of the codeword can be retrieved by a small AC0 circuit; (2) for some choice of the underlying combinatorial designs, the output distribution of Nisan's pseudorandom generator against AC0 circuits of depth d cannot be sampled by small AC0 circuits of depth less than d.

The weight distribution and list-decoding size of Reed-Muller codes are studied in this work. Given a weight parameter, we are interested in bounding the number of Reed-Muller codewords with weight up to the given parameter; and 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. Obtaining tight bounds for the weight distribution of Reed-Muller codes has been a long standing open problem in coding theory, dating back to 1976. In this work, we make a new connection between computer science techniques used to study low-degree polynomials and these coding theory questions. This allows us to resolve the weight distribution and list-decoding size of Reed-Muller codes for all distances. Previous results could only handle bounded distances: Azumi, Kasami, and Tokura gave bounds on the weight distribution which hold up to 2.5 times the minimal distance of the code; and Gopalan, Klivans, and Zuckerman gave bounds on the list-decoding size which hold up to the Johnson bound.

We consider a system of linear constraints over any finite Abelian group G of the following form: ℓi(x1, ..., xn) ≡ ℓi,1x1+ ⋯ + ℓi,nxn∈ Aifor i=1, ..., N and each Ai⊂ G, ℓi,jis an element of G and xi's are Boolean variables. Our main result shows that the subset of the Boolean cube that satisfies these constraints has exponentially small correlation with the MODqboolean function, when the order of G and q are co-prime numbers. Our work extends the recent result of Chattopadhyay and Wigderson (FOCS'09) who obtain such a correlation bound for linear systems over cyclic groups whose order is a product of two distinct primes or has at most one prime factor. Our result also immediately yields the first exponential bounds on the size of boolean depth-four circuits of the form MAJ ο AND ο ANY ο(1)ο MODmfor computing the MODqfunction, when m, q are co-prime. No superpolynomial lower bounds were known for such circuits for computing any explicit function. This completely solves an open problem posed by Beigel and Maciel (Complexity'97).

The Weil bound for character sums is a deep result in Algebraic Geometry with many applications both in mathematics and in the theoretical computer science. The Weil bound states that for any polynomial f(x) over a finite field F and any additive character χ : F → ℂ, either χ(f(x)) is a constant function or it is distributed close to uniform. The Weil bound is quite effective as long as deg (f) ≪ √|F|, but it breaks down when the degree of f exceeds √|F|. As the Weil bound plays a central role in many areas, finding extensions for polynomials of larger degree is an important problem with many possible applications. In this work we develop such an extension over finite fields Fpn of small characteristic: we prove that if f(x) = g(x) + h(x) where deg(g) ≪ √|F| and h(x) is a sparse polynomial of arbitrary degree but bounded weight degree, then the same conclusion of the classical Weil bound still holds: either χ(f(x)) is constant or its distribution is close to uniform. In particular, this shows that the subcode of Reed-Muller codes of degree ω(1) generated by traces of sparse polynomials is a code with near optimal distance, while Reed-Muller of such a degree has no distance (i.e. o(1) distance) ; this is one of the few examples where one can prove that sparse polynomials behave differently from non-sparse polynomials of the same degree. As an application we prove new general results for affine invariant codes. We prove that any affine-invariant subspace of quasi-polynomial size is (1) indeed a code (i.e. has good distance) and (2) is locally testable. Previous results for general affine invariant codes were known only for codes of polynomial size, and of length 2nwhere n needed to be a prime. Thus, our techniques are the first to extend to general families of such codes of super- polynomial size, where we also remove the requirement from n to be a prime. The proof is based on two main ingredients: the extension of the Weil bound for character sums, and a new Fourier-analytic approach for estimating the weight distribution of general codes with large dual distance, which may be of independent interest.

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:Fpn -> Fp with polynomials of degree at most d ≤ p is non-negligible, while making only a constant number of queries to the function. This is an instance of correlation testing. In this framework, a fixed test is applied to a function, and the acceptance probability of the test is dependent on the correlation of the function from the property. This is an analog of proximity oblivious testing, a notion coined by Goldreich and Ron, in the high error regime. We study in this work general properties which are affine invariant and which are correlation testable using a constant number of queries. We show that any such property (as long as the field size is not too small) can in fact be tested by the Gowers uniformity test, and hence having correlation with the property is equivalent to having correlation with degree d polynomials for some fixed d. We stress that our result holds also for non-linear properties which are affine invariant. This completely classifies affine invariant properties which are correlation testable.

In this paper we give the first construction of a pseudorandom generator, with seed length O(log n), for CC0[p], the class of constant-depth circuits with unbounded fan-in MODpgates, for some prime p. More accurately, the seed length of our generator is O(log n) for any constant error ϵ > 0. In fact, we obtain our generator by fooling distributions generated by low degree polynomials, over Fp, when evaluated on the Boolean cube. This result significantly extends previous constructions that either required a long seed or that could only fool the distribution generated by linear functions over Fp, when evaluated on the Boolean cube. Enroute of constructing our PRG, we prove two structural results for low degree polynomials over finite fields that can be of independent interest. 1) Let f be an n-variate degree d polynomial over Fp. Then, for every ϵ > 0 there exists a subset S ⊂ [n], whose size depends only on d and ϵ, such that Σα∈Fpn:α≠0,αS=0|f̂(α)|2≤ ϵ. Namely, there is a constant size subset S such that the total weight of the nonzero Fourier coefficients that do not involve any variable from S is small. 2) Let f be an n-variate degree d polynomial over Fp. If the distribution of f when applied to uniform zero-one bits is ϵ-far (in statistical distance) from its distribution when applied to biased bits, then for every δ > 0, f can be approximated over zero-one bits, up to error δ, by a function of a small number (depending only on ϵ, δ and d) of lower degree polynomials.

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, mainly in distributed systems, database systems, and networks. The algorithm of Bloom is quite effective: it can store a set S of size n by using only ≈1.44nlog2(1/ε) bits while having false positive error ε. This is within a constant factor of the entropy lower bound of nlog2(1/ε) for storing such sets. Closing this gap is an important open problem, as Bloom filters are widely used is situations were storage is at a premium. Bloom filters have another property: they are dynamic. That is, they support the iterative insertions of up to n elements. In fact, if one removes this requirement, there exist static data structures which receive the entire set at once and can almost achieve the entropy lower bound; they require only nlog2(1/ε)(1 + o(1)) bits. Our main result is a new lower bound for the memory requirements of any dynamic approximate membership data structure. We show that for any constant ε > 0, any such data structure which achieves false positive error rate of ε must use at least C(ε) · nlog2(1/ε) memory bits, where C(ε) > 1 depends only on ε. This shows that the entropy lower bound cannot be achieved by dynamic data structures for any constant error rate. In fact, our lower bound holds even in the setting where the insertion and query algorithms may use shared randomness, and where they are only required to perform well on average.

The possible relative weights of codewords of Generalized Reed-Muller codes are studied. LetRMq(r,m) denote the code of polynomials over the finite fieldFqinmvariables of total degree at mostr. The relative weight of a codewordf¿ RMq(r,m) is the fraction of nonzero entries inf. The possible relative weights are studied, when the fieldFqand the degreerare fixed, and the number of variablesmtends to infinity. It is proved that the set of possible weights is sparse-for any¿which is not rational of the form¿ = ¿/qk, there exists some¿ > 0such that no weights fall in the interval(¿-¿,¿+¿). This demonstrates a new property of the weight distribution of Generalized Reed-Muller codes.

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 modulo p must have high complexity in every other characteristic q. More precisely, we show the following results about Boolean functions f : {0,1}nrarr {0,1} which depend on all n variables, and distinct primes p, q: (1) If f has degree o(log n) modulo p, then it must have degree Omega(n1-o(1)) modulo q. Thus a Boolean function has degree o(log n) in only one characteristic. This result is essentially tight as there exist functions that have degree log n in every characteristic. (2) If f has degree d = o(log n) modulo p, it cannot be computed correctly on more than 1 - p-O(d)fraction of the hypercube by polynomials of degree n1/2-isinmodulo q. As a corollary of the above results it follows that if f has degree o(log n) modulo p, then it requires super-polynomial size A C0[q] circuits. This gives a lower bound for a broad and natural class of functions.

We study the question of designing cryptographic schemes which are secure even if an arbitrary function f(sk) of the secret key is leaked, as long as the secret key sk is still (exponentially) hard to compute from this auxiliary input. This setting of auxiliary input is more general than the more traditional setting, which assumes that some of information about the secret key sk may be leaked, but sk still has high min-entropy left. In particular, we deal with situations where f(sk) information-theoretically determines the entire secret key sk.

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 degree polynomials. Green and Tao [GT07] have shown that over large fields (i.e when d <|F|) a biased polynomial must have low rank. They have also conjectured that bias implies low rank over general fields, but their proof technique fails to show that. In this work we affirmatively answer their conjecture. Using this result we obtain a general worst case to average case reductions for polynomials. That is, we show that a polynomial that can be approximated by a few polynomials of bounded degree (i.e. a polynomial with non negligible correlation with a function of few bounded degree polynomials), can be computed by a few polynomials of bounded degree. We derive some relations between our results to the construction of pseudorandom generators. Our work provides another evidence to the structure vs. randomness dichotomy.

Linearity tests are randomized algorithms which have oracle access to the truth table of some function f, and are supposed to distinguish between linear functions and functions which are far from linear. Linearity tests were first introduced by (Blum, Luby and Rubenfeld, 1993), and were later used in the PCP theorem, among other applications. The quality of a linearity test is described by its correctness c - the probability it accepts linear functions, its soundness s - the probability it accepts functions far from linear, and its query complexity q - the number of queries it makes. Linearity tests were studied in order to decrease the soundness of linearity tests, while keeping the query complexity small (for one reason, to improve PCP constructions). Samorodnitsky and Trevisan (Samorodnitsky and Trevisan 2000) constructed the Complete Graph Test, and prove that no Hyper Graph Test can perform better than the Complete Graph Test. Later in (Samorodnitsky and Trevisan 2006) they prove, among other results, that no non-adaptive linearity test can perform better than the Complete Graph Test. Their proof uses the algebraic machinery of the Gowers Norm. A result by (Ben-Sasson, Harsha and Raskhodnikova 2005) allows to generalize this lower bound also to adaptive linearity tests. We also prove the same optimal lower bound for adaptive linearity test, but our proof technique is arguably simpler and more direct than the one used in (Samorodnitsky and Trevisan 2006). We also study, like (Samorodnitsky and Trevisan 2006), the behavior of linearity tests on quadratic functions. However, instead of analyzing the Gowers Norm of certain functions, we provide a more direct combinatorial proof, studying the behavior of linearity tests on random quadratic functions. This proof technique also lets us prove directly the lower bound also for adaptive linearity tests.

We give an explicit construction of pseudorandom generators against low degree polynomials over finite fields. We show that the sum of 2d small-biased generators with error ε2O(d) is a pseudorandom generator against degree d polynomials with error ε. This gives a generator with seed length 2O(d) log(n/ε). Our construction follows the recent breakthrough result of Bogadnov and Viola. Their work shows that the sum of d small-biased generators is a pseudo-random generator against degree d polynomials, assuming the Inverse Gowers Conjecture. However, this conjecture is only proven for d=2,3. The main advantage of our work is that it does not rely on any unproven conjectures.

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:FNp to Fp is non-negligible, that is larger than a constant independent of N, then f has a non-trivial correlation with a degree d-1 polynomial. The conjecture is known to hold for d=2,3 and for any prime p. In this paper we show the conjecture to be false for p=2 and for d=4, by presenting an explicit function whose 4-th Gowers norm is non-negligible, but whose correlation with any polynomial of degree 3 is exponentially small. Essentially the same result, with different bounds for correlation, was independently obtained by Green and Tao. Their analysis uses a modification of a Ramsey-type argument of Alon and Beigel to show inapproximability of certain functions by low-degree polynomials. We observe that a combination of our results with the argument of Alon and Beigel implies the inverse conjecture to be false for any prime p, for d = p2.