Logistic classification on data lying on a low-dimensional manifold inside a high-dimensional space is a common but theoretically delicate problem. The authors study Gaussian kernel approximation and regularized logistic classification on a connected compact smooth embedded manifold of intrinsic dimension ρ. They address two main challenges: obtaining a fast oracle inequality without assuming a bounded Bayes function, and constructing Gaussian approximants that exploit both the intrinsic dimension and the target's regularity without suffering from saturation.
To handle the first challenge, they prove 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. For the second challenge, they establish a uniform Sobolev operator expansion and cancel lower-order bias terms using a signed combination of Gaussian kernel integral operators at different bandwidths. Real interpolation then gives an L₂(X) approximation error of order γ^α for functions in a Besov space, with γ the bandwidth.
By constructing approximants directly on the manifold, the squared Gaussian RKHS norm bound is of order γ^{-ρ}, overcoming second-order saturation while retaining norm control governed by the intrinsic dimension. When the marginal distribution has a density in L_p(X), the resulting logistic excess-risk rate is ((log n)^{ρ+1}/n)^{4α(1-1/p)/(4α(1-1/p)+3ρ)}. Clipping predictions yields a faster clipped logistic rate, and for bounded densities its polynomial exponent matches the minimax optimal rate for squared-loss regression; the corresponding excess misclassification risk also matches the minimax optimal rate.