Polynomial Approximation of Symmetric Functions

TL;DR

This paper investigates how symmetry properties of multivariate functions can be leveraged to enhance polynomial approximation efficiency, providing bounds and rates that depend on input dimensions and set sizes.

Contribution

It introduces methods to exploit symmetry for improved polynomial approximation and derives approximation bounds for functions on multi-sets with variable sizes.

Findings

Symmetry exploitation reduces approximation cost for symmetric functions.

Derived bounds depend on input dimension and number of arguments.

Established approximation rates for multi-set functions with variable size N.

Abstract

We study the polynomial approximation of symmetric multivariate functions and of multi-set functions. Specifically, we consider $f(x_1, \dots, x_N)$, where $x_i \in \mathbb{R}^d$, and $f$ is invariant under permutations of its $N$ arguments. We demonstrate how these symmetries can be exploited to improve the cost versus error ratio in a polynomial approximation of the function $f$, and in particular study the dependence of that ratio on $d, N$ and the polynomial degree. These results are then used to construct approximations and prove approximation rates for functions defined on multi-sets where $N$ becomes a parameter of the input.