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

Bing-Zheng Li

• In this paper, we establish error bounds for approximation by multivariate Bernstein.Durrmeyer operators in L p �ÏX (1 . p < ��) with respect to a general Borel probability measure �ÏX on a simplex X �¼ Rn.

  By the error bounds, we provide convergence rates of type O(m.�Á ) with some �Á > 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.