Gaussian Approximation of Besov Functions for Logistic Classification on Manifolds

Guanhang Lei, Lei Shi, Xiaoming Yuan

We study Gaussian kernel approximation and regularized logistic classification on a connected compact smooth embedded manifold $X\subset\mathbb R^d$ without boundary and of intrinsic dimension $\varrho$. Two main challenges are to derive a fast oracle inequality without assuming a bounded Bayes function and to construct Gaussian approximants that exploit both the intrinsic dimension and the target's regularity without saturation. We address the first challenge through a Bernstein-type inequality whose constant depends linearly on the supremum norm of the predictor. Combined with localized complexity estimates, this yields an oracle inequality over separable normed function spaces. To address the second challenge, we establish a uniform Sobolev operator expansion and cancel its lower-order bias terms using a signed combination of Gaussian kernel integral operators at different bandwidths. Real interpolation then gives an $L_2(X)$ approximation error of order $γ^α$ for functions in $B^α_{2,\infty}(X)$, where $γ$ is the bandwidth. By constructing the approximants directly on the manifold without ambient extension, we obtain a squared Gaussian RKHS norm bound of order $γ^{-\varrho}$.…