Approximation by multivariate Bernstein–Durrmeyer operators and learning rates of least-squares regularized regression with multivariate polynomial kernels

In this paper, we establish error bounds for approximation by multivariate Bernstein–Durrmeyer operators in LρXp (1≤p<∞1≤p<∞) with respect to a general Borel probability measure ρXρX on a simplex X⊂RnX⊂Rn. By the error bounds, we provide convergence rates of type O(m−γ)O(m−γ) with some γ>0γ>0 for the least-squares regularized regression algorithm associated with a multivariate polynomial kernel (where mm is the sample size). The learning rates depend on the space dimension nn and the capacity of the reproducing kernel Hilbert space generated by the polynomial kernel.