Avishay Tal

On the structure of boolean functions with small spectral norm

In this paper we prove results regarding Boolean functions with small spectral norm (the spectral norm of ƒ is ||ƒ||1 = ∑α|ƒ(α)|). Specifically, we prove the following results for functions ƒ :{0, 1}n → [0, 1}with ||ƒ||1 = A. There is a subspace V of co-dimension at most A2 such that ƒ|v is constant. ƒ can be computed by a parity decision tree of size 2A2n2a. (a parity decision tree is a decision tree whose nodes are labeled with arbitrary linear functions.) ƒ can be computed by a De Morgan formula of size O(2A2 n2A+2) and by a De Morgan formula of depth O( A2 + log(n) • A). If in addition ƒ has at most s nonzero Fourier coefficients, then ƒ can be computed by a parity decision tree of depth A2log s. For every ε > 0 there is a parity decision tree of depth O(A2 + log(1/ε)) and size 2O(A2)} • min{ 1/ε2,O(log(1/ε))2A} that ε-approximates ƒ. Furthermore, this tree can be learned, with probability 1--δ, using poly(n, exp(A2), 1/ε,log(1/δ)) membership queries. All the results above (except ref{abs:DeMorgan}) also hold (with a slight change in parameters) for functions f : Znp → {0, 1}.

Tight bounds on The Fourier Spectrum of AC 0

We show that AC^0 circuits on n variables with depth d and size m have at most 2^{-Omega(k/log^{d-1} m)} of their Fourier mass at level k or above. Our proof builds on a previous result by Hastad (SICOMP, 2014) who proved this bound for the special case k=n. Our result improves the seminal result of Linial, Mansour and Nisan (JACM, 1993) and is tight up to the constants hidden in the Omega notation. As an application, we improve Bravermans celebrated result (JACM, 2010). Braverman showed that any r(m,d,epsilon)-wise independent distribution epsilon-fools AC^0 circuits of size m and depth d, for r(m,d,epsilon) = O(log(m/epsilon))^{2d^2+7d+3}. Our improved bounds on the Fourier tails of AC^0 circuits allows us to improve this estimate to r(m,d,epsilon) = O(log(m/epsilon))^{3d+3}. In contrast, an example by Mansour (appearing in Luby and Velickovics paper - Algorithmica, 1996) shows that there is a log^{d-1}(m)\log(1/epsilon)-wise independent distribution that does not epsilon-fool AC^0 circuits of size m and depth d. Hence, our result is tight up to the factor

Two Structural Results for Low Degree Polynomials and Applications

3

On the minimal fourier degree of symmetric Boolean functions

in the exponent.

On the degree of univariate polynomials over the integers

In this paper, two structural results concerning low degree polynomials over finite fields are given. The first states that over any finite field F, for any polynomial f on n variables with degree d > log(n)/10, there exists a subspace of F^n with dimension at least d n^(1/(d-1)) on which f is constant. This result is shown to be tight. Stated differently, a degree d polynomial cannot compute an affine disperser for dimension smaller than the stated dimension. Using a recursive argument, we obtain our second structural result, showing that any degree d polynomial f induces a partition of F^n to affine subspaces of dimension n^(1/(d-1)!), such that f is constant on each part. We extend both structural results to more than one polynomial. We further prove an analog of the first structural result to sparse polynomials (with no restriction on the degree) and to functions that are close to low degree polynomials. We also consider the algorithmic aspect of the two structural results. Our structural results have various applications, two of which are: * Dvir [CC 2012] introduced the notion of extractors for varieties, and gave explicit constructions of such extractors over large fields. We show that over any finite field any affine extractor is also an extractor for varieties with related parameters. Our reduction also holds for dispersers, and we conclude that Shaltiels affine disperser [FOCS 2011] is a disperser for varieties over the binary field. * Ben-Sasson and Kopparty [SIAM J. C 2012] proved that any degree 3 affine disperser over a prime field is also an affine extractor with related parameters. Using our structural results, and based on the work of Kaufman and Lovett [FOCS 2008] and Haramaty and Shpilka [STOC 2010], we generalize this result to any constant degree.

On the Minimal Fourier Degree of Symmetric Boolean Functions

In this paper we give a new upper bound on the minimal degree of a nonzero Fourier coefficient in any non-linear symmetric Boolean function. Specifically, we prove that for every non-linear and symmetric f: {0, 1}k → {0, 1} there exists a set;

Matrix rigidity of random toeplitz matrices

\not 0 \ne S \subset [k]

Extractor-based time-space lower bounds for learning

such that ¦S¦ = O(Γ(k)+√k, and

Formula lower bounds via the quantum method

\hat f(S) \ne 0

Improved Average-Case Lower Bounds for De Morgan Formula Size: Matching Worst-Case Lower Bound

where Γ(m)≤m0.525 is the largest gap between consecutive prime numbers in {1,..., m}. As an application we obtain a new analysis of the PAC learning algorithm for symmetric juntas, under the uniform distribution, of Mossel et al. [10].Our bound on the degree is a significant improvement over the previous result of Kolountzakis et al. [8] who proved that ¦S¦=O(k=log k).We also show a connection between lower-bounding the degree of non-constant functions that take values in {0,1,2} and the question that we study here.