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

2026-05-08

AI Summary

This work investigates the feasibility of efficiently approximating indicator functions of sets under the Gaussian measure using non-negative polynomials in the $L_1$ norm. For sets with finite Gaussian surface area (GSA), the paper establishes, for the first time, the existence of pointwise non-negative polynomials that achieve $L_1$-approximation rates matching—up to constant factors—the best-known bounds attainable without non-negativity constraints. By integrating techniques from Gaussian surface area theory, polynomial approximation, and probabilistic measure analysis, the authors show that an $\varepsilon$-accurate approximation can be achieved by a non-negative polynomial of degree $k = \tilde{O}(\Gamma^2/\varepsilon^2)$, where $\Gamma$ denotes the Gaussian surface area of the set. This result has direct implications for applications such as positive-unlabeled learning with smoothness assumptions.

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 $Γ$ under the standard Gaussian admits degree-$k$ non-negative polynomials that $\eps$-approximate its indicator functions in $L_1$-norm, for $k=\tilde{O}(Γ^2/\varepsilon^2)$. 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.

Problem

• Non-negative polynomials
• L₁-approximation
• Gaussian distribution
• Indicator functions
• Gaussian surface area

Innovation

• non-negative polynomials
• L1-approximation
• Gaussian surface area
• indicator functions
• smoothed learning