关于非负L1逼近多项式的一个注记
A Note on Non-Negative $L_1$-Approximating Polynomials
AI总结：
研究了在高斯分布下非负L1逼近多项式的存在性，证明了有限高斯表面面积的集合可通过特定次数的非负多项式实现L1逼近，结果与当前最佳的高斯L1逼近次数界相符。
摘要：
L1逼近多项式，即在某些分布下以L1范数逼近指示函数的多项式，广泛应用于计算学习理论。本文研究了在高斯分布下非负L1逼近多项式存在性。这比L1逼近要求更强，但比夹逼多项式（后者有广泛应用）弱。这些非负逼近多项式最近在平滑学习中被用于正例学习。本文证明，任何在标准高斯分布下具有高斯表面面积（GSA）至多Γ的集合，都允许次数为~O(Γ²/ε²)的非负多项式，以ε-精度逼近其指示函数。等价地，有限GSA意味着L1逼近，且逼近多项式范围被限制在[0,∞)内。此结果在不考虑非负性约束时，与当前最佳的高斯L1逼近次数界相差常数因子。
英文摘要：
$L_1$-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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
