A Note on Non-Negative L1-Approximating Polynomials

L1-Approximating polynomials, i.e., polynomials that approximate indicator functions in L1-norm under certain distributions, are widely used in computational learning theory. We study the existence of textitnon-negative L1-approximating polynomials with respect to Gaussian distributions. This is a stronger requirement than L1-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 L1-norm, for k=tildeO(Gamma2/varepsilon2). Equivalently, finite GSA implies L1-approximation with the stronger pointwise guarantee that the approximating polynomial has range contained in [0,∞). Up to a constant-factor, this matches the degree of the best currently known Gaussian L1-approximation degree bound without the non-negativity constraint.