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In this article we present a method to implement orthogonal polynomials and many other special functions in Computer Algebra systems enabling the user to work with those functions appropriately, and in particular to verify different types…
Fourier series multiscale method, a concise and efficient analytical approach for multiscale computation, will be developed out of this series of papers. The second paper is concerned with simultaneous approximation to functions and their…
We define tests of boolean functions which distinguish between linear (or quadratic) polynomials, and functions which are very far, in an appropriate sense, from these polynomials. The tests have optimal or nearly optimal trade-offs between…
This paper investigates the approximation properties of deep neural networks with piecewise-polynomial activation functions. We derive the required depth, width, and sparsity of a deep neural network to approximate any H\"{o}lder smooth…
We study the close connection between rational functions that approximate a given Boolean function, and quantum algorithms that compute the same function using postselection. We show that the minimal degree of the former equals (up to a…
We study the complexity of learning and approximation of self-bounding functions over the uniform distribution on the Boolean hypercube ${0,1}^n$. Informally, a function $f:{0,1}^n \rightarrow \mathbb{R}$ is self-bounding if for every $x…
The existence of string functions, which are not polynomial time computable, but whose graph is checkable in polynomial time, is a basic assumption in cryptography. We prove that in the framework of algebraic complexity, there are no such…
Let $g: \{-1,1\}^k \to \{-1,1\}$ be any Boolean function and $q_1,\dots,q_k$ be any degree-2 polynomials over $\{-1,1\}^n.$ We give a \emph{deterministic} algorithm which, given as input explicit descriptions of $g,q_1,\dots,q_k$ and an…
Chattopadhyay, Mande and Sherif (ECCC 2018) recently exhibited a total Boolean function, the sink function, that has polynomial approximate rank and polynomial randomized communication complexity. This gives an exponential separation…
A low-degree test is a collection of simple, local rules for checking the proximity of an arbitrary function to a low-degree polynomial. Each rule depends on the function's values at a small number of places. If a function satisfies many…
The vectorial Boolean functions are employed in cryptography to build block coding algorithms. An important criterion on these functions is their resistance to the differential cryptanalysis. Nyberg defined the notion of almost perfect…
It is already shown that a Boolean function for a NP-complete problem can be computed by a polynomial-sized circuit if its variables have enough number of automorphisms. Looking at this previous study from the different perspective gives us…
The origin of this study is based on not only explicit formulas of finite sums involving higher powers of binomial coefficients, but also explicit evaluations of generating functions for this sums. It should be emphasized that this study…
In this work, we consider the maximization of submodular functions constrained by independence systems. Because of the wide applicability of submodular functions, this problem has been extensively studied in the literature, on specialized…
We study approximation of Boolean functions by low-degree polynomials over the ring $\mathbb{Z}/2^k\mathbb{Z}$. More precisely, given a Boolean function $F:\{0,1\}^n \rightarrow \{0,1\}$, define its $k$-lift to be $F_k:\{0,1\}^n \rightarrow…
We prove lower bounds on the error incurred when approximating any oscillating function using piecewise polynomial spaces. The estimates are explicit in the polynomial degree and have optimal dependence on the meshwidth and frequency when…
Functions of one or more variables are usually approximated with a basis: a complete, linearly-independent system of functions that spans a suitable function space. The topic of this paper is the numerical approximation of functions using…
We study monotone neural networks with threshold gates where all the weights (other than the biases) are non-negative. We focus on the expressive power and efficiency of representation of such networks. Our first result establishes that…
Low-rank approximations of data matrices are an important dimensionality reduction tool in machine learning and regression analysis. We consider the case of categorical variables, where it can be formulated as the problem of finding…
The paper deals with the problem of approximating the functions of several variables by branched continued fractions, in particular, multidimensional A- and J-fractions with independent variables. A generalization of Gragg's algorithm is…