We prove several new results about bounded uniform and small-bias distributions. A main message is that, small-bias, even perturbed with noise, does not fool several classes of tests better than bounded uniformity. We prove this for threshold tests, small-space algorithms, and small-depth circuits. In particular, we obtain small-bias distributions that 1) achieve an optimal lower bound on their statistical distance to any bounded-uniform distribution. This closes a line of research initiated by Alon, Goldreich, and Mansour in 2003, and improves on a result by O'Donnell and Zhao. 2) have heavier tail mass than the uniform distribution. This answers a question posed by several researchers including Bun and Steinke. 3) rule out a popular paradigm for constructing pseudorandom generators, originating in a 1989 work by Ajtai and Wigderson. This again answers a question raised by several researchers. For branching programs, our result matches a bound by Forbes and Kelley. Our small-bias distributions above are symmetric. We show that the xor of any two symmetric small-bias distributions fools any bounded function. Hence our examples cannot be extended to the xor of two small-bias distributions, another popular paradigm whose power remains unknown. We also generalize and simplify the proof of a result of Bazzi.

An n-bit boolean function is resilient to coalitions of size q if any fixed set of q bits is unlikely to influence the function when the other n-q bits are chosen uniformly. We give explicit constructions of depth-3 circuits that are resilient to coalitions of size cn/log^2n with bias n^-c. Previous explicit constructions with the same resilience had constant bias. Our construction is simpler and we generalize it to biased product distributions. Our proof builds on previous work; the main differences are the use of a tail bound for expander walks in combination with a refined analysis based on Janson's inequality.

An n-bit boolean function is resilient to coalitions of size q if no fixed set of q bits is likely to influence the value of the function when the other n - q bits are chosen uniformly at random, even though the function is nearly balanced. We construct explicit functions resilient to coalitions of size q = n/(log n)(O(log log n)) = n(1-o)(1) computable by linear-size circuits and linear-time algorithms. We also obtain a tight size-depth tradeoff for computing such resilient functions. Constructions such as ours were not available even non-explicitly. It was known that functions resilient to coalitions of size q = n(0.63)... can be computed by linear-size circuits [BL85], and functions resilient to coalitions of size q = Theta(n/log(2) n) can be computed by quadratic-size circuits [AL93]. One component of our proofs is a new composition theorem for resilient functions.

We study the fundamental challenge of exhibiting explicit functions that have small correlation with low-degree polynomials over $\mathbb{F}_{2}$. Our main contributions include: 1. In STOC 2020, CHHLZ introduced a new technique to prove correlation bounds. Using their technique they established new correlation bounds for low-degree polynomials. They conjectured that their technique generalizes to higher degree polynomials as well. We give a counterexample to their conjecture, in fact ruling out weaker parameters and showing what they prove is essentially the best possible. 2. We propose a new approach for proving correlation bounds with the central "mod functions", consisting of two steps: (I) the polynomials that maximize correlation are symmetric and (II) symmetric polynomials have small correlation. Contrary to related results in the literature, we conjecture that (I) is true. We argue this approach is not affected by existing "barrier results". 3. We prove our conjecture for quadratic polynomials. Specifically, we determine the maximum possible correlation between quadratic polynomials modulo 2 and the functions $(x_{1},\dots,x_{n})\to z^{\sum x_{i}}$ for any $z$ on the complex unit circle; and show that it is achieved by symmetric polynomials. To obtain our results we develop a new proof technique: we express correlation in terms of directional derivatives and analyze it by slowly restricting the direction. 4. We make partial progress on the conjecture for cubic polynomials, in particular proving tight correlation bounds for cubic polynomials whose degree-3 part is symmetric.

We analyze the Fourier growth, i.e. the $L_1$ Fourier weight at level $k$ (denoted $L_{1,k}$), of various well-studied classes of"structured"$\mathbb{F}_2$-polynomials. This study is motivated by applications in pseudorandomness, in particular recent results and conjectures due to [CHHL19,CHLT19,CGLSS20] which show that upper bounds on Fourier growth (even at level $k=2$) give unconditional pseudorandom generators. Our main structural results on Fourier growth are as follows: - We show that any symmetric degree-$d$ $\mathbb{F}_2$-polynomial $p$ has $L_{1,k}(p) \le \Pr[p=1] \cdot O(d)^k$, and this is tight for any constant $k$. This quadratically strengthens an earlier bound that was implicit in [RSV13]. - We show that any read-$\Delta$ degree-$d$ $\mathbb{F}_2$-polynomial $p$ has $L_{1,k}(p) \le \Pr[p=1] \cdot (k \Delta d)^{O(k)}$. - We establish a composition theorem which gives $L_{1,k}$ bounds on disjoint compositions of functions that are closed under restrictions and admit $L_{1,k}$ bounds. Finally, we apply the above structural results to obtain new unconditional pseudorandom generators and new correlation bounds for various classes of $\mathbb{F}_2$-polynomials.

We study a simple and general template for constructing affine extractors by composing a linear transformation with resilient functions. Using this we show that good affine extractors can be computed by non-explicit circuits of various types, including AC0-Xor circuits: AC0 circuits with a layer of parity gates at the input. We also show that one-sided extractors can be computed by small DNF-Xor circuits, and separate these circuits from other well-studied classes. As a further motivation for studying DNF-Xor circuits we show that if they can approximate inner product then small AC0-Xor circuits can compute it exactly – a long-standing open problem.

Speeding up probabilistic inference remains a key challenge in probabilistic databases (PDBs) and the related area of statistical relational learning (SRL). Since computing probabilities for query answers is #P-hard, even for fairly simple conjunctive queries, both the PDB and SRL communities have proposed a number of approximation techniques over the years. The two prevalent techniques are either (i) MCMC-style sampling or (ii) branch-and-bound (B&B) algorithms that iteratively improve model-based bounds using a combination of variable substitution and elimination. We propose a new anytime B&B approximation scheme that encompasses all prior model-based approximation schemes proposed in the PDB and SRL literature. Our approach relies on the novel idea of "scaled dissociation" which can improve both the upper and lower bounds of existing modelbased algorithms. We apply our approach to the well-studied problem of evaluating self-join-free conjunctive queries over tuple-independent PDBs, and show a consistent reduction in approximation error in our experiments on TPC-H, Yago3, and a synthetic benchmark setting.

In this work is presented one new approach for processing of sequences of medical CT images, called Hierarchical Adaptive Karhunen-Loeve Transform (HAKLT). The aim is to achieve high decorrelation for each group of 9 consecutive CT images, obtained from the original larger sequence. In result, the main part of the energy of all images in one group is concentrated in a relatively small number of eigen images. This result could be obtained using the well-known Karhunen-Loeve Transform (KLT) with transformation matrix of size 9x9. However, for the implementation of the 2-levels HAKLT in each level are used 3 transform matrices of size 3x3, in result of which the computational complexity of the new algorithm is reduced in average 2 times, when compared to that of KLT with 9x9 matrix. One more advantage is that the algorithm permits parallel processing for each group of 3 images in every hierarchical level. In this work are also included the results of the algorithm modeling for sequences of real CT images, which confirm its ability to carry out efficient decorrelation. The HAKLT algorithm could be farther used as a basis for the creation of algorithms for efficient compression of sequences of CT images and for features space minimization in the regions of interest, which contain various classes of searched objects.

This paper present the results of computer simulations of pattern recognition using LPCC filters, designed for various types of distortions of input images (rotation, change of scale and rotation + change of scale).