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

摘要

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