SVM Learning Rates for Data with Low Intrinsic Dimension
share
We derive improved regression and classification rates for support vector machines using Gaussian kernels under the assumption that the data has some low-dimensional intrinsic structure. Our notion of intrinsic dimension is defined by the box-counting dimension of the support of the data generating distribution. Under some standard regularity assumptions for regression and classification we prove learning rates, where we essentially substitute in some well-established learning rates the dimension of the ambient space with the box-counting dimension of the support of the data generating distribution. Furthermore, we show that a training validation approach for choosing the hyperparameters of an SVM in a data dependent way achieves the same rates adaptively, i.e. without knowledge on the data generating distribution, in particular without knowing the box-counting dimension of the support of the data generating distribution.
READ FULL TEXT