On Uniform Weighted Deep Polynomial approximation

TL;DR

This paper introduces weighted deep polynomial approximants that effectively approximate functions with asymmetric behavior, outperforming traditional methods by capturing local non-smoothness and global growth.

Contribution

It proposes a novel class of weighted deep polynomial models for asymmetric functions, with a stable graph-based parameterization for practical optimization.

Findings

Outperforms Taylor, Chebyshev, and standard deep polynomial approximants in numerical tests.

Effectively captures local non-smoothness and global growth of functions.

Provides a stable parameterization strategy for training deep polynomial models.

Abstract

It is a classical result in rational approximation theory that certain non-smooth or singular functions, such as $|x|$ and $x^{1/p}$, can be efficiently approximated using rational functions with root-exponential convergence in terms of degrees of freedom \cite{Sta, GN}. In contrast, polynomial approximations admit only algebraic convergence by Jackson's theorem \cite{Lub2}. Recent work shows that composite polynomial architectures can recover exponential approximation rates even without smoothness \cite{KY}. In this work, we introduce and analyze a class of weighted deep polynomial approximants tailored for functions with asymmetric behavior-growing unbounded on one side and decaying on the other. By multiplying a learnable deep polynomial with a one-sided weight, we capture both local non-smoothness and global growth. We show numerically that this framework outperforms Taylor, Chebyshev, and standard deep polynomial approximants, even when all use the same number of parameters. To optimize these approximants in practice, we propose a stable graph-based parameterization strategy building on \cite{Jar}.