A Note on Non-Negative $L_1$-Approximating Polynomials
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arXiv
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| Natura: | Preprint |
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2026
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| author | Lee, Jane H. Mehrotra, Anay Zampetakis, Manolis |
| author_facet | Lee, Jane H. Mehrotra, Anay Zampetakis, Manolis |
| contents | $L_1$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_08072 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Note on Non-Negative $L_1$-Approximating Polynomials Lee, Jane H. Mehrotra, Anay Zampetakis, Manolis Machine Learning Data Structures and Algorithms Statistics Theory $L_1$-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. |
| title | A Note on Non-Negative $L_1$-Approximating Polynomials |
| topic | Machine Learning Data Structures and Algorithms Statistics Theory |
| url | https://arxiv.org/abs/2605.08072 |