Related papers: Approximating Functions on Boxes
The approximate degree of a Boolean function f is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. Approximate degree is known to be a lower bound on quantum query complexity. We resolve or nearly…
We study sets $V$ in the tridisc that are relatively polynomially convex and have the polynomial extension property. If $V$ is one-dimensional, and is either algebraic, or has polynomially convex projections, we show that it is a retract.…
We study spaces $\mathcal{CV}^{k}(\Omega,E)$ of $k$-times continuously partially differentiable functions on an open set $\Omega\subset\mathbb{R}^{d}$ with values in a locally convex Hausdorff space $E$. The space…
Tame functions are a class of nonsmooth, nonconvex functions, which feature in a wide range of applications: functions encountered in the training of deep neural networks with all common activations, value functions of mixed-integer…
We classify the Boolean degree $1$ functions of $k$-spaces in a vector space of dimension $n$ (also known as Cameron-Liebler classes) over the field with $q$ elements for $n \geq n_0(k, q)$. This also implies that two-intersecting sets with…
Distance function to a compact set plays a central role in several areas of computational geometry. Methods that rely on it are robust to the perturbations of the data by the Hausdorff noise, but fail in the presence of outliers. The…
Pasting and Reversing operations have been used successfully over the set of integer numbers, simple permutations, rings and recently over a generalized vector product. In this paper, these operations are defined from a natural way to be…
In this work, we introduce new families of nonconforming approximation methods for reconstructing functions on general polygonal meshes. These methods are defined using degrees of freedom based on weighted moments of orthogonal polynomials…
Vector Fitting (VF) is a popular method of constructing rational approximants that provides a least squares fit to frequency response measurements. In an earlier work, we provided an analysis of VF for scalar-valued rational functions and…
A convex envelope for the problem of finding the best approximation to a given matrix with a prescribed rank is constructed. This convex envelope allows the usage of traditional optimization techniques when additional constraints are added…
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 give explicit polynomial-sized (in $n$ and $k$) semidefinite representations of the hyperbolicity cones associated with the elementary symmetric polynomials of degree $k$ in $n$ variables. These convex cones form a family of…
The level of a function f on an n-dimensional space encloses a region. The volume of a region between two such levels depends on both levels. Fixing one of them the volume becomes a function of the remaining level. We show that if the…
Given any square matrix or a bounded operator $A$ in a Hilbert space such that $p(A)$ is normal (or similar to normal), we construct a Banach algebra, depending on the polynomial $p$, for which a simple functional calculus holds. When the…
Near-vector spaces extend linear algebra tools to non-linear algebraic structures, enabling the study of non-linear problems. However, explicit constructions remain rare. This paper introduces a broad computable family of near-vector…
We use tools of additive combinatorics for the study of subvarieties defined by {\it high rank} families of polynomials in high dimensional $\mathbb{F} _q$-vector spaces. In the first, analytic part of the paper we prove a number properties…
A fast approximation to the Boys functions (related to the lower incomplete gamma function of half-integer parameter) by a single closed-form analytical expression for all argument values have been developed and tested. Besides the…
This paper concerns the approximation of smooth, high-dimensional functions from limited samples using polynomials. This task lies at the heart of many applications in computational science and engineering - notably, some of those arising…
A new representation is proposed for functions in a Sobolev space with dominating mixed smoothness on an $N$-dimensional hyperrectangle. In particular, it is shown that these functions can be expressed in terms of their highest-order mixed…
We present algorithms and heuristics to compute the characteristic polynomial of a matrix given its minimal polynomial. The matrix is represented as a black-box, i.e., by a function to compute its matrix-vector product. The methods apply to…