Pseudorandomness via the discrete Fourier transform

We present a new approach to constructing unconditional pseudorandom generators against classes of functions that involve computing a linear function of the inputs. We give an explicit construction of a pseudorandom generator that fools the discrete Fourier transforms of linear functions with seed-length that is nearly logarithmic (up to polyloglog factors) in the input size and the desired error parameter. Our result gives a single pseudorandom generator that fools several important classes of tests computable in logspace that have been considered in the literature, including halfspaces (over general domains), modular tests and combinatorial shapes. For all these classes, our generator is the first that achieves near logarithmic seed-length in both the input length and the error parameter. Getting such a seed-length is a natural challenge in its own right, which needs to be overcome in order to derandomize RL - a central question in complexity theory. Our construction combines ideas from a large body of prior work, ranging from a classical construction of [NN93] to the recent gradually increasing independence paradigm of [KMN11, CRSW13, GMRTV12], while also introducing some novel analytic machinery which might find other applications

Bounded Independence Plus Noise Fools Products

Let D be a b-wise independent distribution over {0,1}^m. Let E be the "noise" distribution over {0,1}^m where the bits are independent and each bit is 1 with probability eta/2. We study which tests f: {0,1}^m -> [-1,1] are epsilon-fooled by D+E, i.e., |E[f(D+E)] - E[f(U)]| <= epsilon where U is the uniform distribution. We show that D+E epsilon-fools product tests f: ({0,1}^n)^k -> [-1,1] given by the product of k bounded functions on disjoint n-bit inputs with error epsilon = k(1-eta)^{Omega(b^2/m)}, where m = nk and b >= n. This bound is tight when b = Omega(m) and eta >= (log k)/m. For b >= m^{2/3} log m and any constant eta the distribution D+E also 0.1-fools log-space algorithms. We develop two applications of this type of results. First, we prove communication lower bounds for decoding noisy codewords of length m split among k parties. For Reed-Solomon codes of dimension m/k where k = O(1), communication Omega(eta m) - O(log m) is required to decode one message symbol from a codeword with eta m errors, and communication O(eta m log m) suffices. Second, we obtain pseudorandom generators. We can epsilon-fool product tests f: ({0,1}^n)^k -> [-1,1] under any permutation of the bits with seed lengths 2n + O~(k^2 log(1/epsilon)) and O(n) + O~(sqrt{nk log 1/epsilon}). Previous generators have seed lengths >= nk/2 or >= n sqrt{n k}. For the special case where the k bounded functions have range {0,1} the previous generators have seed length >= (n+log k)log(1/epsilon)

Fourier Bounds and Pseudorandom Generators for Product Tests

We study the Fourier spectrum of functions f : {0,1}^{mk} -> {-1,0,1} which can be written as a product of k Boolean functions f_i on disjoint m-bit inputs. We prove that for every positive integer d, sum_{S subseteq [mk]: |S|=d} |hat{f_S}| = O(min{m, sqrt{m log(2k)}})^d . Our upper bounds are tight up to a constant factor in the O(*). Our proof uses Schur-convexity, and builds on a new "level-d inequality" that bounds above sum_{|S|=d} hat{f_S}^2 for any [0,1]-valued function f in terms of its expectation, which may be of independent interest. As a result, we construct pseudorandom generators for such functions with seed length O~(m + log(k/epsilon)), which is optimal up to polynomial factors in log m, log log k and log log(1/epsilon). Our generator in particular works for the well-studied class of combinatorial rectangles, where in addition we allow the bits to be read in any order. Even for this special case, previous generators have an extra O~(log(1/epsilon)) factor in their seed lengths. We also extend our results to functions f_i whose range is [-1,1]

Fourier Growth of Regular Branching Programs

We analyze the Fourier growth, i.e. the L? Fourier weight at level k (denoted L_{1,k}), of read-once regular branching programs. We prove that every read-once regular branching program B of width w ? [1,?] with s accepting states on n-bit inputs must have its L_{1,k} bounded by min{Pr[B(U_n) = 1](w-1)^k, s ? O((n log n)/k)^{(k-1)/2}}. For any constant k, our result is tight up to constant factors for the AND function on w-1 bits, and is tight up to polylogarithmic factors for unbounded width programs. In particular, for k = 1 we have L_{1,1}(B) ? s, with no dependence on the width w of the program. Our result gives new bounds on the coin problem and new pseudorandom generators (PRGs). Furthermore, we obtain an explicit generator for unordered permutation branching programs of unbounded width with a constant factor stretch, where no PRG was previously known. Applying a composition theorem of B?asiok, Ivanov, Jin, Lee, Servedio and Viola (RANDOM 2021), we extend our results to "generalized group products," a generalization of modular sums and product tests

Multidimensional sampling for simulation and integration: measures, discrepancies, and quasi-random numbers

This is basically a review of the field of Quasi-Monte Carlo intended for computational physicists and other potential users of quasi-random numbers. As such, much of the material is not new, but is presented here in a style hopefully more accessible to physicists than the specialized mathematical literature. There are also some new results: On the practical side we give important empirical properties of large quasi-random point sets, especially the exact quadratic discrepancies; on the theoretical side, there is the exact distribution of quadratic discrepancy for random point sets.Comment: 51 pages. Full paper, including all figures also available at: ftp://ftp.nikhef.nl/pub/preprints/96-017.ps.gz Accepted for publication in Comp.Phys.Comm. Fixed some typos, corrected formula 108,figure 11 and table

Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness, and quantum computation. Many of the developements are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, quantum mechanics, representation theory, and the theory of error-correcting codes

Explicit Correlation Amplifiers for Finding Outlier Correlations in Deterministic Subquadratic Time

We derandomize G. Valiant\u27s [J.ACM 62(2015) Art.13] subquadratic-time algorithm for finding outlier correlations in binary data. Our derandomized algorithm gives deterministic subquadratic scaling essentially for the same parameter range as Valiant\u27s randomized algorithm, but the precise constants we save over quadratic scaling are more modest. Our main technical tool for derandomization is an explicit family of correlation amplifiers built via a family of zigzag-product expanders in Reingold, Vadhan, and Wigderson [Ann. of Math 155(2002), 157-187]. We say that a function f:{-1,1}^d ->{-1,1}^D is a correlation amplifier with threshold 0 = 1, and strength p an even positive integer if for all pairs of vectors x,y in {-1,1}^d it holds that (i) ||| | >= tau*d implies (/gamma^d})^p*D /d)^p*D