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<ul>
<li><a class="reference internal" href="#">4.7. Kernel Approximation</a><ul>
<li><a class="reference internal" href="#nystroem-method-for-kernel-approximation">4.7.1. Nystroem Method for Kernel Approximation</a></li>
<li><a class="reference internal" href="#radial-basis-function-kernel">4.7.2. Radial Basis Function Kernel</a></li>
<li><a class="reference internal" href="#additive-chi-squared-kernel">4.7.3. Additive Chi Squared Kernel</a></li>
<li><a class="reference internal" href="#skewed-chi-squared-kernel">4.7.4. Skewed Chi Squared Kernel</a></li>
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<div class="section" id="kernel-approximation">
<span id="id1"></span><h1>4.7. Kernel Approximation<a class="headerlink" href="#kernel-approximation" title="Permalink to this headline">¶</a></h1>
<p>This submodule contains functions that approximate the feature mappings that
correspond to certain kernels, as they are used for example in support vector
machines (see <a class="reference internal" href="svm.html#svm"><span class="std std-ref">Support Vector Machines</span></a>).
The following feature functions perform non-linear transformations of the
input, which can serve as a basis for linear classification or other
algorithms.</p>
<p>The advantage of using approximate explicit feature maps compared to the
<a class="reference external" href="https://en.wikipedia.org/wiki/Kernel_trick">kernel trick</a>,
which makes use of feature maps implicitly, is that explicit mappings
can be better suited for online learning and can significantly reduce the cost
of learning with very large datasets.
Standard kernelized SVMs do not scale well to large datasets, but using an
approximate kernel map it is possible to use much more efficient linear SVMs.
In particular, the combination of kernel map approximations with
<a class="reference internal" href="generated/sklearn.linear_model.SGDClassifier.html#sklearn.linear_model.SGDClassifier" title="sklearn.linear_model.SGDClassifier"><code class="xref py py-class docutils literal"><span class="pre">SGDClassifier</span></code></a> can make non-linear learning on large datasets possible.</p>
<p>Since there has not been much empirical work using approximate embeddings, it
is advisable to compare results against exact kernel methods when possible.</p>
<div class="admonition seealso">
<p class="first admonition-title">See also</p>
<p class="last"><a class="reference internal" href="linear_model.html#polynomial-regression"><span class="std std-ref">Polynomial regression: extending linear models with basis functions</span></a> for an exact polynomial transformation.</p>
</div>
<div class="section" id="nystroem-method-for-kernel-approximation">
<span id="nystroem-kernel-approx"></span><h2>4.7.1. Nystroem Method for Kernel Approximation<a class="headerlink" href="#nystroem-method-for-kernel-approximation" title="Permalink to this headline">¶</a></h2>
<p>The Nystroem method, as implemented in <a class="reference internal" href="generated/sklearn.kernel_approximation.Nystroem.html#sklearn.kernel_approximation.Nystroem" title="sklearn.kernel_approximation.Nystroem"><code class="xref py py-class docutils literal"><span class="pre">Nystroem</span></code></a> is a general method
for low-rank approximations of kernels. It achieves this by essentially subsampling
the data on which the kernel is evaluated.
By default <a class="reference internal" href="generated/sklearn.kernel_approximation.Nystroem.html#sklearn.kernel_approximation.Nystroem" title="sklearn.kernel_approximation.Nystroem"><code class="xref py py-class docutils literal"><span class="pre">Nystroem</span></code></a> uses the <code class="docutils literal"><span class="pre">rbf</span></code> kernel, but it can use any
kernel function or a precomputed kernel matrix.
The number of samples used - which is also the dimensionality of the features computed -
is given by the parameter <code class="docutils literal"><span class="pre">n_components</span></code>.</p>
</div>
<div class="section" id="radial-basis-function-kernel">
<span id="rbf-kernel-approx"></span><h2>4.7.2. Radial Basis Function Kernel<a class="headerlink" href="#radial-basis-function-kernel" title="Permalink to this headline">¶</a></h2>
<p>The <a class="reference internal" href="generated/sklearn.kernel_approximation.RBFSampler.html#sklearn.kernel_approximation.RBFSampler" title="sklearn.kernel_approximation.RBFSampler"><code class="xref py py-class docutils literal"><span class="pre">RBFSampler</span></code></a> constructs an approximate mapping for the radial basis
function kernel, also known as <em>Random Kitchen Sinks</em> <a class="reference internal" href="#rr2007" id="id2">[RR2007]</a>. This
transformation can be used to explicitly model a kernel map, prior to applying
a linear algorithm, for example a linear SVM:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.kernel_approximation</span> <span class="k">import</span> <span class="n">RBFSampler</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="k">import</span> <span class="n">SGDClassifier</span>
<span class="gp">>>> </span><span class="n">X</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]]</span>
<span class="gp">>>> </span><span class="n">y</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">rbf_feature</span> <span class="o">=</span> <span class="n">RBFSampler</span><span class="p">(</span><span class="n">gamma</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">X_features</span> <span class="o">=</span> <span class="n">rbf_feature</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">clf</span> <span class="o">=</span> <span class="n">SGDClassifier</span><span class="p">(</span><span class="n">max_iter</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_features</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="go">SGDClassifier(alpha=0.0001, average=False, class_weight=None, epsilon=0.1,</span>
<span class="go"> eta0=0.0, fit_intercept=True, l1_ratio=0.15,</span>
<span class="go"> learning_rate='optimal', loss='hinge', max_iter=5, n_iter=None,</span>
<span class="go"> n_jobs=1, penalty='l2', power_t=0.5, random_state=None,</span>
<span class="go"> shuffle=True, tol=None, verbose=0, warm_start=False)</span>
<span class="gp">>>> </span><span class="n">clf</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_features</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="go">1.0</span>
</pre></div>
</div>
<p>The mapping relies on a Monte Carlo approximation to the
kernel values. The <code class="docutils literal"><span class="pre">fit</span></code> function performs the Monte Carlo sampling, whereas
the <code class="docutils literal"><span class="pre">transform</span></code> method performs the mapping of the data. Because of the
inherent randomness of the process, results may vary between different calls to
the <code class="docutils literal"><span class="pre">fit</span></code> function.</p>
<p>The <code class="docutils literal"><span class="pre">fit</span></code> function takes two arguments:
<code class="docutils literal"><span class="pre">n_components</span></code>, which is the target dimensionality of the feature transform,
and <code class="docutils literal"><span class="pre">gamma</span></code>, the parameter of the RBF-kernel. A higher <code class="docutils literal"><span class="pre">n_components</span></code> will
result in a better approximation of the kernel and will yield results more
similar to those produced by a kernel SVM. Note that “fitting” the feature
function does not actually depend on the data given to the <code class="docutils literal"><span class="pre">fit</span></code> function.
Only the dimensionality of the data is used.
Details on the method can be found in <a class="reference internal" href="#rr2007" id="id3">[RR2007]</a>.</p>
<p>For a given value of <code class="docutils literal"><span class="pre">n_components</span></code> <a class="reference internal" href="generated/sklearn.kernel_approximation.RBFSampler.html#sklearn.kernel_approximation.RBFSampler" title="sklearn.kernel_approximation.RBFSampler"><code class="xref py py-class docutils literal"><span class="pre">RBFSampler</span></code></a> is often less accurate
as <a class="reference internal" href="generated/sklearn.kernel_approximation.Nystroem.html#sklearn.kernel_approximation.Nystroem" title="sklearn.kernel_approximation.Nystroem"><code class="xref py py-class docutils literal"><span class="pre">Nystroem</span></code></a>. <a class="reference internal" href="generated/sklearn.kernel_approximation.RBFSampler.html#sklearn.kernel_approximation.RBFSampler" title="sklearn.kernel_approximation.RBFSampler"><code class="xref py py-class docutils literal"><span class="pre">RBFSampler</span></code></a> is cheaper to compute, though, making
use of larger feature spaces more efficient.</p>
<div class="figure align-center" id="id8">
<a class="reference external image-reference" href="../auto_examples/plot_kernel_approximation.html"><img alt="../_images/sphx_glr_plot_kernel_approximation_0021.png" src="../_images/sphx_glr_plot_kernel_approximation_0021.png" style="width: 600.0px; height: 250.0px;" /></a>
<p class="caption"><span class="caption-text">Comparing an exact RBF kernel (left) with the approximation (right)</span></p>
</div>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><a class="reference internal" href="../auto_examples/plot_kernel_approximation.html#sphx-glr-auto-examples-plot-kernel-approximation-py"><span class="std std-ref">Explicit feature map approximation for RBF kernels</span></a></li>
</ul>
</div>
</div>
<div class="section" id="additive-chi-squared-kernel">
<span id="additive-chi-kernel-approx"></span><h2>4.7.3. Additive Chi Squared Kernel<a class="headerlink" href="#additive-chi-squared-kernel" title="Permalink to this headline">¶</a></h2>
<p>The additive chi squared kernel is a kernel on histograms, often used in computer vision.</p>
<p>The additive chi squared kernel as used here is given by</p>
<div class="math">
\[k(x, y) = \sum_i \frac{2x_iy_i}{x_i+y_i}\]</div>
<p>This is not exactly the same as <code class="xref py py-func docutils literal"><span class="pre">sklearn.metrics.additive_chi2_kernel</span></code>.
The authors of <a class="reference internal" href="#vz2010" id="id4">[VZ2010]</a> prefer the version above as it is always positive
definite.
Since the kernel is additive, it is possible to treat all components
<span class="math">\(x_i\)</span> separately for embedding. This makes it possible to sample
the Fourier transform in regular intervals, instead of approximating
using Monte Carlo sampling.</p>
<p>The class <a class="reference internal" href="generated/sklearn.kernel_approximation.AdditiveChi2Sampler.html#sklearn.kernel_approximation.AdditiveChi2Sampler" title="sklearn.kernel_approximation.AdditiveChi2Sampler"><code class="xref py py-class docutils literal"><span class="pre">AdditiveChi2Sampler</span></code></a> implements this component wise
deterministic sampling. Each component is sampled <span class="math">\(n\)</span> times, yielding
<span class="math">\(2n+1\)</span> dimensions per input dimension (the multiple of two stems
from the real and complex part of the Fourier transform).
In the literature, <span class="math">\(n\)</span> is usually chosen to be 1 or 2, transforming
the dataset to size <code class="docutils literal"><span class="pre">n_samples</span> <span class="pre">*</span> <span class="pre">5</span> <span class="pre">*</span> <span class="pre">n_features</span></code> (in the case of <span class="math">\(n=2\)</span>).</p>
<p>The approximate feature map provided by <a class="reference internal" href="generated/sklearn.kernel_approximation.AdditiveChi2Sampler.html#sklearn.kernel_approximation.AdditiveChi2Sampler" title="sklearn.kernel_approximation.AdditiveChi2Sampler"><code class="xref py py-class docutils literal"><span class="pre">AdditiveChi2Sampler</span></code></a> can be combined
with the approximate feature map provided by <a class="reference internal" href="generated/sklearn.kernel_approximation.RBFSampler.html#sklearn.kernel_approximation.RBFSampler" title="sklearn.kernel_approximation.RBFSampler"><code class="xref py py-class docutils literal"><span class="pre">RBFSampler</span></code></a> to yield an approximate
feature map for the exponentiated chi squared kernel.
See the <a class="reference internal" href="#vz2010" id="id5">[VZ2010]</a> for details and <a class="reference internal" href="#vvz2010" id="id6">[VVZ2010]</a> for combination with the <a class="reference internal" href="generated/sklearn.kernel_approximation.RBFSampler.html#sklearn.kernel_approximation.RBFSampler" title="sklearn.kernel_approximation.RBFSampler"><code class="xref py py-class docutils literal"><span class="pre">RBFSampler</span></code></a>.</p>
</div>
<div class="section" id="skewed-chi-squared-kernel">
<span id="skewed-chi-kernel-approx"></span><h2>4.7.4. Skewed Chi Squared Kernel<a class="headerlink" href="#skewed-chi-squared-kernel" title="Permalink to this headline">¶</a></h2>
<p>The skewed chi squared kernel is given by:</p>
<div class="math">
\[k(x,y) = \prod_i \frac{2\sqrt{x_i+c}\sqrt{y_i+c}}{x_i + y_i + 2c}\]</div>
<p>It has properties that are similar to the exponentiated chi squared kernel
often used in computer vision, but allows for a simple Monte Carlo
approximation of the feature map.</p>
<p>The usage of the <a class="reference internal" href="generated/sklearn.kernel_approximation.SkewedChi2Sampler.html#sklearn.kernel_approximation.SkewedChi2Sampler" title="sklearn.kernel_approximation.SkewedChi2Sampler"><code class="xref py py-class docutils literal"><span class="pre">SkewedChi2Sampler</span></code></a> is the same as the usage described
above for the <a class="reference internal" href="generated/sklearn.kernel_approximation.RBFSampler.html#sklearn.kernel_approximation.RBFSampler" title="sklearn.kernel_approximation.RBFSampler"><code class="xref py py-class docutils literal"><span class="pre">RBFSampler</span></code></a>. The only difference is in the free
parameter, that is called <span class="math">\(c\)</span>.
For a motivation for this mapping and the mathematical details see <a class="reference internal" href="#ls2010" id="id7">[LS2010]</a>.</p>
</div>
<div class="section" id="mathematical-details">
<h2>4.7.5. Mathematical Details<a class="headerlink" href="#mathematical-details" title="Permalink to this headline">¶</a></h2>
<p>Kernel methods like support vector machines or kernelized
PCA rely on a property of reproducing kernel Hilbert spaces.
For any positive definite kernel function <span class="math">\(k\)</span> (a so called Mercer kernel),
it is guaranteed that there exists a mapping <span class="math">\(\phi\)</span>
into a Hilbert space <span class="math">\(\mathcal{H}\)</span>, such that</p>
<div class="math">
\[k(x,y) = \langle \phi(x), \phi(y) \rangle\]</div>
<p>Where <span class="math">\(\langle \cdot, \cdot \rangle\)</span> denotes the inner product in the
Hilbert space.</p>
<p>If an algorithm, such as a linear support vector machine or PCA,
relies only on the scalar product of data points <span class="math">\(x_i\)</span>, one may use
the value of <span class="math">\(k(x_i, x_j)\)</span>, which corresponds to applying the algorithm
to the mapped data points <span class="math">\(\phi(x_i)\)</span>.
The advantage of using <span class="math">\(k\)</span> is that the mapping <span class="math">\(\phi\)</span> never has
to be calculated explicitly, allowing for arbitrary large
features (even infinite).</p>
<p>One drawback of kernel methods is, that it might be necessary
to store many kernel values <span class="math">\(k(x_i, x_j)\)</span> during optimization.
If a kernelized classifier is applied to new data <span class="math">\(y_j\)</span>,
<span class="math">\(k(x_i, y_j)\)</span> needs to be computed to make predictions,
possibly for many different <span class="math">\(x_i\)</span> in the training set.</p>
<p>The classes in this submodule allow to approximate the embedding
<span class="math">\(\phi\)</span>, thereby working explicitly with the representations
<span class="math">\(\phi(x_i)\)</span>, which obviates the need to apply the kernel
or store training examples.</p>
<div class="topic">
<p class="topic-title first">References:</p>
<table class="docutils citation" frame="void" id="rr2007" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label">[RR2007]</td><td><em>(<a class="fn-backref" href="#id2">1</a>, <a class="fn-backref" href="#id3">2</a>)</em> <a class="reference external" href="http://www.robots.ox.ac.uk/~vgg/rg/papers/randomfeatures.pdf">“Random features for large-scale kernel machines”</a>
Rahimi, A. and Recht, B. - Advances in neural information processing 2007,</td></tr>
</tbody>
</table>
<table class="docutils citation" frame="void" id="ls2010" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id7">[LS2010]</a></td><td><a class="reference external" href="http://www.maths.lth.se/matematiklth/personal/sminchis/papers/lis_dagm10.pdf">“Random Fourier approximations for skewed multiplicative histogram kernels”</a>
Random Fourier approximations for skewed multiplicative histogram kernels
- Lecture Notes for Computer Sciencd (DAGM)</td></tr>
</tbody>
</table>
<table class="docutils citation" frame="void" id="vz2010" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label">[VZ2010]</td><td><em>(<a class="fn-backref" href="#id4">1</a>, <a class="fn-backref" href="#id5">2</a>)</em> <a class="reference external" href="https://www.robots.ox.ac.uk/~vgg/publications/2011/Vedaldi11/vedaldi11.pdf">“Efficient additive kernels via explicit feature maps”</a>
Vedaldi, A. and Zisserman, A. - Computer Vision and Pattern Recognition 2010</td></tr>
</tbody>
</table>
<table class="docutils citation" frame="void" id="vvz2010" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id6">[VVZ2010]</a></td><td><a class="reference external" href="https://www.robots.ox.ac.uk/~vgg/publications/2010/Sreekanth10/sreekanth10.pdf">“Generalized RBF feature maps for Efficient Detection”</a>
Vempati, S. and Vedaldi, A. and Zisserman, A. and Jawahar, CV - 2010</td></tr>
</tbody>
</table>
</div>
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