A Note on Non-Negative $L_1$-Approximating Polynomials

TL;DR

This paper investigates the existence of non-negative $L_1$-approximating polynomials for indicator functions under Gaussian distributions, establishing bounds related to Gaussian surface area.

Contribution

It proves that classes with bounded Gaussian surface area admit low-degree non-negative polynomials that approximate indicator functions in $L_1$-norm.

Findings

Non-negative $L_1$-approximating polynomials exist for classes with bounded Gaussian surface area.

Degree bounds are established as $k= ilde{O}( ext{GSA}^2/ ext{epsilon}^2)$ for approximation.

Results match the best known bounds without the non-negativity constraint, up to a constant factor.

Abstract

$L_1$-Approximating polynomials, i.e., polynomials that approximate indicator functions in $L_1$-norm under certain distributions, are widely used in computational learning theory. We study the existence of \textit{non-negative} $L_1$-approximating polynomials with respect to Gaussian distributions. This is a stronger requirement than $L_1$-approximation but weaker than sandwiching polynomials (which themselves have many applications). These non-negative approximating polynomials have recently found uses in smoothed learning from positive-only examples. In this short note, we prove that every class of sets with Gaussian surface area (GSA) at most $\Gamma$ under the standard Gaussian admits degree-$k$ non-negative polynomials that $\eps$-approximate its indicator functions in $L_1$-norm, for $k=\tilde{O}(\Gamma^2/\varepsilon^2)$. Equivalently, finite GSA implies $L_1$-approximation with the stronger pointwise guarantee that the approximating polynomial has range contained in $[0,\infty)$. Up to a constant-factor, this matches the degree of the best currently known Gaussian $L_1$-approximation degree bound without the non-negativity constraint.