Learning by nonsymmetric kernels with data dependent spaces and l¹-regularizer

We study a learning algorithm for regression. The algorithm is a regularization scheme with l¹ regularizer stated in a hypothesis space trained from data or samples by a nonsymmetric kernel. The data dependent nature of the algorithm leads to an extra error term called hypothesis error, which is essentially different from regularization schemes with data independent hypothesis spaces. By dealing with regularization error, sample error and hypothesis error, we estimate the total error in terms of properties of the kernel, the input space, the marginal distribution, and the regression function of the regression problem. Learning rates are derived by choosing suitable values of the regularization parameter. An improved error decomposition approach is used in our data dependent setting.