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<div class="section" id="kernel-ridge-regression">
<span id="kernel-ridge"></span><h1>1.3. Kernel ridge regression<a class="headerlink" href="#kernel-ridge-regression" title="Permalink to this headline">¶</a></h1>
<p>Kernel ridge regression (KRR) <a class="reference internal" href="#m2012" id="id1">[M2012]</a> combines <a class="reference internal" href="linear_model.html#ridge-regression"><span>Ridge Regression</span></a>
(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.</p>
<p>The form of the model learned by <a class="reference internal" href="generated/sklearn.kernel_ridge.KernelRidge.html#sklearn.kernel_ridge.KernelRidge" title="sklearn.kernel_ridge.KernelRidge"><code class="xref py py-class docutils literal"><span class="pre">KernelRidge</span></code></a> is identical to support
vector regression (<code class="xref py py-class docutils literal"><span class="pre">SVR</span></code>). However, different loss functions are used:
KRR uses squared error loss while support vector regression uses
<img class="math" src="../_images/math/19bc0073dde1bcd1a8e6a32b251e80cced668f04.png" alt="\epsilon"/>-insensitive loss, both combined with l2 regularization. In
contrast to <code class="xref py py-class docutils literal"><span class="pre">SVR</span></code>, fitting <a class="reference internal" href="generated/sklearn.kernel_ridge.KernelRidge.html#sklearn.kernel_ridge.KernelRidge" title="sklearn.kernel_ridge.KernelRidge"><code class="xref py py-class docutils literal"><span class="pre">KernelRidge</span></code></a> 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 <img class="math" src="../_images/math/defc8dedc4e1c71aa65da56c385f2d8681f2ed4d.png" alt="\epsilon > 0"/>, at prediction-time.</p>
<p>The following figure compares <a class="reference internal" href="generated/sklearn.kernel_ridge.KernelRidge.html#sklearn.kernel_ridge.KernelRidge" title="sklearn.kernel_ridge.KernelRidge"><code class="xref py py-class docutils literal"><span class="pre">KernelRidge</span></code></a> and <code class="xref py py-class docutils literal"><span class="pre">SVR</span></code> on
an artificial dataset, which consists of a sinusoidal target function and
strong noise added to every fifth datapoint. The learned model of
<a class="reference internal" href="generated/sklearn.kernel_ridge.KernelRidge.html#sklearn.kernel_ridge.KernelRidge" title="sklearn.kernel_ridge.KernelRidge"><code class="xref py py-class docutils literal"><span class="pre">KernelRidge</span></code></a> and <code class="xref py py-class docutils literal"><span class="pre">SVR</span></code> 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
<a class="reference internal" href="generated/sklearn.kernel_ridge.KernelRidge.html#sklearn.kernel_ridge.KernelRidge" title="sklearn.kernel_ridge.KernelRidge"><code class="xref py py-class docutils literal"><span class="pre">KernelRidge</span></code></a> is approx. seven times faster than fitting <code class="xref py py-class docutils literal"><span class="pre">SVR</span></code>
(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.</p>
<div class="figure align-center">
<a class="reference external image-reference" href="../auto_examples/plot_kernel_ridge_regression.html"><img alt="../_images/plot_kernel_ridge_regression_0011.png" src="../_images/plot_kernel_ridge_regression_0011.png" /></a>
</div>
<p>The next figure compares the time for fitting and prediction of
<a class="reference internal" href="generated/sklearn.kernel_ridge.KernelRidge.html#sklearn.kernel_ridge.KernelRidge" title="sklearn.kernel_ridge.KernelRidge"><code class="xref py py-class docutils literal"><span class="pre">KernelRidge</span></code></a> and <code class="xref py py-class docutils literal"><span class="pre">SVR</span></code> for different sizes of the training set.
Fitting <a class="reference internal" href="generated/sklearn.kernel_ridge.KernelRidge.html#sklearn.kernel_ridge.KernelRidge" title="sklearn.kernel_ridge.KernelRidge"><code class="xref py py-class docutils literal"><span class="pre">KernelRidge</span></code></a> is faster than <code class="xref py py-class docutils literal"><span class="pre">SVR</span></code> for medium-sized
training sets (less than 1000 samples); however, for larger training sets
<code class="xref py py-class docutils literal"><span class="pre">SVR</span></code> scales better. With regard to prediction time, <code class="xref py py-class docutils literal"><span class="pre">SVR</span></code> is
faster than <a class="reference internal" href="generated/sklearn.kernel_ridge.KernelRidge.html#sklearn.kernel_ridge.KernelRidge" title="sklearn.kernel_ridge.KernelRidge"><code class="xref py py-class docutils literal"><span class="pre">KernelRidge</span></code></a> 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 <img class="math" src="../_images/math/19bc0073dde1bcd1a8e6a32b251e80cced668f04.png" alt="\epsilon"/> and <img class="math" src="../_images/math/2bcc65482aa8e15cd4c9e9f2542451fb4e971a91.png" alt="C"/> of the
<code class="xref py py-class docutils literal"><span class="pre">SVR</span></code>; <img class="math" src="../_images/math/51c3f634737f3900a622e86675e34390c43102a7.png" alt="\epsilon = 0"/> would correspond to a dense model.</p>
<div class="figure align-center">
<a class="reference external image-reference" href="../auto_examples/plot_kernel_ridge_regression.html"><img alt="../_images/plot_kernel_ridge_regression_0021.png" src="../_images/plot_kernel_ridge_regression_0021.png" /></a>
</div>
<div class="topic">
<p class="topic-title first">References:</p>
<table class="docutils citation" frame="void" id="m2012" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id1">[M2012]</a></td><td>“Machine Learning: A Probabilistic Perspective”
Murphy, K. P. - chapter 14.4.3, pp. 492-493, The MIT Press, 2012</td></tr>
</tbody>
</table>
</div>
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