Shachar Lovett

Abstract: We give an explicit construction of a pseudorandom,generator against low- degree polynomials over finite fields. Pseudorandom generators against linear polynomials, known as small-bias generators, were first introduced by Naor and Naor (STOC 1990). We show that the sum of 2,(d)log(n/e) against degree-d polynomails. Our construction follows the break- through result of Bogdanov and Viola (FOCS 2007). Their work shows

... Shachar Lovett Department of Computer Science The Weizmann Institute of Science Rehovot 76100, Israel shachar.lovett@weizmann.ac.il ... we do not use in this paper to avoid confusion), was first studied (over F = F2) by Naor and Naor [11] and later by Alon, Goldreich, Hastad ...

ABSTRACT We prove that several measures in communication complexity are equivalent, up to polynomial factors in the logarithm of the rank of the associated matrix: deterministic communication complexity, randomized communication complexity, information cost and zero-communication cost. This shows that in order to prove the log-rank conjecture, it suffices to show that low-rank matrices have efficient protocols in any of the aforementioned measures. Furthermore, we show that the notion of zero-communication complexity is equivalent to an extension of the common discrepancy bound. Linial et al. [Combinatorica, 2007] showed that the discrepancy of a sign matrix is lower-bounded by an inverse polynomial in the logarithm of the associated matrix. We show that if these results can be generalized to the extended discrepancy, this will imply the log-rank conjecture.

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.

ABSTRACT Recently there has been much interest in Gowers uniformity norms from the perspective of theoretical computer science. This is mainly due to the fact that these norms provide a method for testing whether the maximum correlation of a function $f:\mathbb{F}_p^n \rightarrow \mathbb{F}_p$ with polynomials of degree at most $d \le p$ is non-negligible, while making only a constant number of queries to the function. This is an instance of {\em 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 {\em 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 non-linear 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 Erd\"os, Lov\'asz and Spencer to the context of additive number theory.

ABSTRACT A $t$-$(n,k,\lambda)$ design over $\F_q$ is a collection of $k$-dimensional subspaces of $\F_q^n$, called blocks, such that each $t$-dimensional subspace of $\F_q^n$ is contained in exactly $\lambda$ blocks. Such $t$-designs over $\F_q$ are the $q$-analogs of conventional combinatorial designs. Nontrivial $t$-$(n,k,\lambda)$ designs over $\F_q$ are currently known to exist only for $t \leq 3$. Herein, we prove that simple (meaning, without repeated blocks) nontrivial $t$-$(n,k,\lambda)$ designs over $\F_q$ exist for all $t$ and $q$, provided that $k > 12t$ and $n$ is sufficiently large. This may be regarded as a $q$-analog of the celebrated Teirlinck theorem for combinatorial designs.

Let p be a fixed prime number, and N be a large integer. The ’Inverse Conjecture for the Gowers norm’ states that if the ”d-th Gowers norm” of a function f : F,. 1,Introduction We consider multivariate,functions over finite fields. The main question of interest here would

... I thank my supervisor, Omer Reingold, for useful comments and for his constant support and interest in the work. I thank Alex Samorodnitsky and Prahladh Harsha for helpful discussions. References [ALM+98] S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. ...

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 in (BLR93), 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 constructed in (ST00) the Complete Graph Test, and prove that no Hyper Graph Test can perform better than the Complete Graph Test. Later in (ST06) they prove, among other results, that no non-adaptive li...

The possible relative weights of codewords of Generalized Reed-Muller codes are studied. Let RMq(r,m) denote the code of polynomials over the finite field Fq in m variables of total degree at most r. The relative weight of a codeword f ¿ RMq(r,m) is the fraction of nonzero entries in f. The possible relative weights are studied, when the field Fq

ABSTRACT Let $\cal{P}$ be an affine invariant property of functions $\mathbb{F}_p^n \to [R]$ for fixed $p$ and $R$. We show that if $\cal{P}$ is locally testable with a constant number of queries, then one can estimate the distance of a function $f$ from $\cal{P}$ with a constant number of queries. This was previously unknown even for simple properties such as cubic polynomials over $\mathbb{F}_2$. Our test is simple: take a restriction of $f$ to a constant dimensional affine subspace, and measure its distance from $\cal{P}$. We show that by choosing the dimension large enough, this approximates with high probability the global distance of $f$ from $\cP$. 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].

ABSTRACT 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.