.. _kernel_ridge: =========================== Kernel ridge regression =========================== .. currentmodule:: sklearn.kernel_ridge Kernel ridge regression (KRR) [M2012]_ combines :ref:`ridge_regression` (linear least squares with l2-norm regularization) with the kernel trick. It thus learns a linear function in the space induced by the respective kernel and the data. For non-linear kernels, this corresponds to a non-linear function in the original space. The form of the model learned by :class:`KernelRidge` is identical to support vector regression (:class:`SVR`). However, different loss functions are used: KRR uses squared error loss while support vector regression uses :math:`\epsilon`-insensitive loss, both combined with l2 regularization. In contrast to :class:`SVR`, fitting :class:`KernelRidge` can be done in closed-form and is typically faster for medium-sized datasets. On the other hand, the learned model is non-sparse and thus slower than SVR, which learns a sparse model for :math:`\epsilon > 0`, at prediction-time. The following figure compares :class:`KernelRidge` and :class:`SVR` on an artificial dataset, which consists of a sinusoidal target function and strong noise added to every fifth datapoint. The learned model of :class:`KernelRidge` and :class:`SVR` is plotted, where both complexity/regularization and bandwidth of the RBF kernel have been optimized using grid-search. The learned functions are very similar; however, fitting :class:`KernelRidge` is approx. seven times faster than fitting :class:`SVR` (both with grid-search). However, prediction of 100000 target values is more than three times faster with SVR since it has learned a sparse model using only approx. 1/3 of the 100 training datapoints as support vectors. .. figure:: ../auto_examples/images/sphx_glr_plot_kernel_ridge_regression_001.png :target: ../auto_examples/plot_kernel_ridge_regression.html :align: center The next figure compares the time for fitting and prediction of :class:`KernelRidge` and :class:`SVR` for different sizes of the training set. Fitting :class:`KernelRidge` is faster than :class:`SVR` for medium-sized training sets (less than 1000 samples); however, for larger training sets :class:`SVR` scales better. With regard to prediction time, :class:`SVR` is faster than :class:`KernelRidge` for all sizes of the training set because of the learned sparse solution. Note that the degree of sparsity and thus the prediction time depends on the parameters :math:`\epsilon` and :math:`C` of the :class:`SVR`; :math:`\epsilon = 0` would correspond to a dense model. .. figure:: ../auto_examples/images/sphx_glr_plot_kernel_ridge_regression_002.png :target: ../auto_examples/plot_kernel_ridge_regression.html :align: center .. topic:: References: .. [M2012] "Machine Learning: A Probabilistic Perspective" Murphy, K. P. - chapter 14.4.3, pp. 492-493, The MIT Press, 2012