Related papers: Polynomial Threshold Functions: Structure, Approxi…
A method for approximating continuous functions $\mathbb{Z}_{p}^{n}\rightarrow\mathbb{Z}_{p}$ by a linear superposition of continuous functions $\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{p}$ is presented and a polynomial regression model is…
In this article we study F-pure thresholds (and, more generally, F-thresholds) of homogeneous polynomials in two variables over a field of characteristic p>0. Passing to a field extension, we factor such a polynomial into a product of…
We show that any $O_d(\epsilon^{-4d 7^d})$-independent family of Gaussians $\epsilon$-fools any degree-$d$ polynomial threshold function.
We consider multilinear Littlewood polynomials, polynomials in $n$ variables in which a specified set of monomials $U$ have $\pm 1$ coefficients, and all other coefficients are $0$. We provide upper and lower bounds (which are close for $U$…
Motivated by conforming finite element methods for elliptic problems of second order, we analyze the approximation of the gradient of a target function by continuous piecewise polynomial functions over a simplicial mesh. The main result is…
A key challenge in scientific machine learning is solving partial differential equations (PDEs) on complex domains, where the curved geometry complicates the approximation of functions and their derivatives required by differential…
Recent years have witnessed the introduction and development of extremely fast rational function algorithms. Many ideas in this realm arose from polynomial-based linear-algebraic algorithms. However, polynomial approximation is occasionally…
We introduce an index for measuring the influence of the k-th smallest variable on a pseudo-Boolean function. This index is defined from a weighted least squares approximation of the function by linear combinations of order statistic…
We contribute to a better understanding of the class of functions that can be represented by a neural network with ReLU activations and a given architecture. Using techniques from mixed-integer optimization, polyhedral theory, and tropical…
Let $\mathsf{TH}_k$ denote the $k$-out-of-$n$ threshold function: given $n$ input Boolean variables, the output is $1$ if and only if at least $k$ of the inputs are $1$. We consider the problem of computing the $\mathsf{TH}_k$ function…
Stochastic Boolean Function Evaluation is the problem of determining the value of a given Boolean function f on an unknown input x, when each bit of x_i of x can only be determined by paying an associated cost c_i. The assumption is that x…
We conjecture that bounded generalised polynomial functions cannot be generated by finite automata, except for the trivial case when they are periodic away from a finite set. Using methods from ergodic theory, we are able to partially…
Understanding the capacity of quantum circuits through the lens of approximation theory is essential for evaluating the complexity of quantum circuits required to solve various problems in scientific computation. We design quantum circuits…
We show (almost) separation between certain important classes of Boolean functions. The technique that we use is to show that the total influence of functions in one class is less than the total influence of functions in the other class. In…
We investigate the rational approximation of fractional powers of unbounded positive operators attainable with a specific integral representation of the operator function. We provide accurate error bounds by exploiting classical results in…
In many high-dimensional problems,polynomial-time algorithms fall short of achieving the statistical limits attainable without computational constraints. A powerful approach to probe the limits of polynomial-time algorithms is to study the…
In this paper, we investigate the problem of finding tight linear lower bounding functions for multivariate polynomials over boxes. These functions are obtained by the expansion of polynomials into Bernstein form and using the linear least…
Let us assume that $f$ is a continuous function defined on the unit ball of $\mathbb R^d$, of the form $f(x) = g (A x)$, where $A$ is a $k \times d$ matrix and $g$ is a function of $k$ variables for $k \ll d$. We are given a budget $m \in…
We consider the problem of exact identification for read-once functions over arbitrary Boolean bases. We introduce a new type of queries (subcube identity ones), discuss its connection to previously known ones, and study the complexity of…
We prove various theorems on approximation using polynomials with integer coefficients in the Bernstein basis of any given order. In the extreme, we draw the coefficients from $\{ \pm 1\}$ only. A basic case of our results states that for…