SherbyNet - /page/wikibook/entry.mu

Micron Document

<!DOCTYPE html>

<title>Polynomial kernel</title>

</head>

Polynomial kernel

</h1>

</header>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div role="note" class="hatnote navigation-not-searchable">This article is about machine learning. For polynomial kernels in complexity theory, see <a href="Kernelization" title="Kernelization">Kernelization</a>.</div>

In <a href="Machine_learning" title="Machine learning">machine learning</a>, the polynomial kernel is a <a href="Kernel_function" class="mw-redirect" title="Kernel function">kernel function</a> commonly used with <a href="Support_vector_machine" title="Support vector machine">support vector machines</a> (SVMs) and other <a href="Kernel_trick" class="mw-redirect" title="Kernel trick">kernelized</a> models, that represents the similarity of vectors (training samples) in a feature space over polynomials of the original variables, allowing learning of non-linear models.

Intuitively, the polynomial kernel looks not only at the given features of input samples to determine their similarity, but also combinations of these. In the context of <a href="Regression_analysis" title="Regression analysis">regression analysis</a>, such combinations are known as interaction features. The (implicit) feature space of a polynomial kernel is equivalent to that of <a href="Polynomial_regression" title="Polynomial regression">polynomial regression</a>, but without the combinatorial blowup in the number of parameters to be learned. When the input features are binary-valued (booleans), then the features correspond to <a href="Logical_conjunction" title="Logical conjunction">logical conjunctions</a> of input features.<a href="#cite_note-Goldberg2008-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

For degree-d polynomials, the polynomial kernel is defined as<a href="#cite_note-2">[2]</a>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(\mathbf {x} ,\mathbf {y} )=(\mathbf {x} ^{\mathsf {T}}\mathbf {y} +c)^{d}}</annotation>

</semantics>

</math><img src="./33d792a46139ba8d8aa3a1566bbdc4546ba8c452.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.752ex; height:3.176ex;" alt="{\displaystyle K(\mathbf {x} ,\mathbf {y} )=(\mathbf {x} ^{\mathsf {T}}\mathbf {y} +c)^{d}}" loading="lazy"></dd></dl>

where x and y are vectors of size n in the input space, i.e. vectors of features computed from training or test samples and c ≥ 0 is a free parameter trading off the influence of higher-order versus lower-order terms in the polynomial. When c = 0, the kernel is called homogeneous.<a href="#cite_note-3">[3]</a> (A further generalized polykernel divides xTy by a user-specified scalar parameter a.<a href="#cite_note-lin2012-4">[4]</a>)

As a kernel, K corresponds to an inner product in a feature space based on some mapping φ:

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(\mathbf {x} ,\mathbf {y} )=\langle \varphi (\mathbf {x} ),\varphi (\mathbf {y} )\rangle }</annotation>

</semantics>

</math><img src="./2b1ea28a28ee910cfbbd4983a2cabae11bd20a96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.154ex; height:2.843ex;" alt="{\displaystyle K(\mathbf {x} ,\mathbf {y} )=\langle \varphi (\mathbf {x} ),\varphi (\mathbf {y} )\rangle }" loading="lazy"></dd></dl>

The nature of φ can be seen from an example. Let d = 2, so we get the special case of the quadratic kernel. After using the <a href="Multinomial_theorem" title="Multinomial theorem">multinomial theorem</a> (twice—the outermost application is the <a href="Binomial_theorem" title="Binomial theorem">binomial theorem</a>) and regrouping,

</mrow>

</mrow>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</munderover>

<mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mrow>

</munderover>

<mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(\mathbf {x} ,\mathbf {y} )=\left(\sum _{i=1}^{n}x_{i}y_{i}+c\right)^{2}=\sum _{i=1}^{n}\left(x_{i}^{2}\right)\left(y_{i}^{2}\right)+\sum _{i=2}^{n}\sum _{j=1}^{i-1}\left({\sqrt {2}}x_{i}x_{j}\right)\left({\sqrt {2}}y_{i}y_{j}\right)+\sum _{i=1}^{n}\left({\sqrt {2c}}x_{i}\right)\left({\sqrt {2c}}y_{i}\right)+c^{2}}</annotation>

</semantics>

</math><img src="./f666eade816ad31661377b14d674ab4ca63ea991.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:101.242ex; height:8.176ex;" alt="{\displaystyle K(\mathbf {x} ,\mathbf {y} )=\left(\sum _{i=1}^{n}x_{i}y_{i}+c\right)^{2}=\sum _{i=1}^{n}\left(x_{i}^{2}\right)\left(y_{i}^{2}\right)+\sum _{i=2}^{n}\sum _{j=1}^{i-1}\left({\sqrt {2}}x_{i}x_{j}\right)\left({\sqrt {2}}y_{i}y_{j}\right)+\sum _{i=1}^{n}\left({\sqrt {2c}}x_{i}\right)\left({\sqrt {2c}}y_{i}\right)+c^{2}}" loading="lazy"></dd></dl>

From this it follows that the feature map is given by:

<mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msqrt>

</msqrt>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msqrt>

</msqrt>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msqrt>

</msqrt>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msqrt>

</msqrt>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msqrt>

</msqrt>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msqrt>

</msqrt>

</mrow>

<msub>

</mrow>

</msub>

<msqrt>

</msqrt>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (x)=\left(x_{n}^{2},\ldots ,x_{1}^{2},{\sqrt {2}}x_{n}x_{n-1},\ldots ,{\sqrt {2}}x_{n}x_{1},{\sqrt {2}}x_{n-1}x_{n-2},\ldots ,{\sqrt {2}}x_{n-1}x_{1},\ldots ,{\sqrt {2}}x_{2}x_{1},{\sqrt {2c}}x_{n},\ldots ,{\sqrt {2c}}x_{1},c\right)}</annotation>

</semantics>

</math><img src="./cdd9a3527cc0aac64a773dec1caa33557017cd18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:107.714ex; height:3.343ex;" alt="{\displaystyle \varphi (x)=\left(x_{n}^{2},\ldots ,x_{1}^{2},{\sqrt {2}}x_{n}x_{n-1},\ldots ,{\sqrt {2}}x_{n}x_{1},{\sqrt {2}}x_{n-1}x_{n-2},\ldots ,{\sqrt {2}}x_{n-1}x_{1},\ldots ,{\sqrt {2}}x_{2}x_{1},{\sqrt {2c}}x_{n},\ldots ,{\sqrt {2c}}x_{1},c\right)}" loading="lazy"></dd></dl>

generalizing for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathbf {x} ^{T}\mathbf {y} +c\right)^{d}}">

<msup>

<mrow>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(\mathbf {x} ^{T}\mathbf {y} +c\right)^{d}}</annotation>

</semantics>

</math><img src="./df9d3ae4c02b55bce1fd6f4ef2bfec71503ecd3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.28ex; height:3.843ex;" alt="{\displaystyle \left(\mathbf {x} ^{T}\mathbf {y} +c\right)^{d}}" loading="lazy">,

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} \in \mathbb {R} ^{n}}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} \in \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./f33c7feddfbe1e5c87647a55639651d9fb2f23de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.148ex; height:2.343ex;" alt="{\displaystyle \mathbf {x} \in \mathbb {R} ^{n}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} \in \mathbb {R} ^{n}}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} \in \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./78d65a808875b2392b1c67d1dcada12c42479d70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.148ex; height:2.676ex;" alt="{\displaystyle \mathbf {y} \in \mathbb {R} ^{n}}" loading="lazy"> and applying the <a href="Multinomial_theorem" title="Multinomial theorem">multinomial theorem</a>:

<mtr>

<mtd>

<msup>

<mrow>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<mfrac>

<msqrt>

</msqrt>

<msqrt>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</msqrt>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

<msup>

<msqrt>

</msqrt>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<mfrac>

<msqrt>

</msqrt>

<msqrt>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</msqrt>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

<msup>

<msqrt>

</msqrt>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}\left(\mathbf {x} ^{T}\mathbf {y} +c\right)^{d}&=\sum _{j_{1}+j_{2}+\dots +j_{n+1}=d}{\frac {\sqrt {d!}}{\sqrt {j_{1}!\cdots j_{n}!j_{n+1}!}}}x_{1}^{j_{1}}\cdots x_{n}^{j_{n}}{\sqrt {c}}^{j_{n+1}}{\frac {\sqrt {d!}}{\sqrt {j_{1}!\cdots j_{n}!j_{n+1}!}}}y_{1}^{j_{1}}\cdots y_{n}^{j_{n}}{\sqrt {c}}^{j_{n+1}}\\&=\varphi (\mathbf {x} )^{T}\varphi (\mathbf {y} )\end{alignedat}}}</annotation>

</semantics>

</math><img src="./c3bd5d9788560a9e2a89ac2f655acfd7052b8eac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:96.162ex; height:10.843ex;" alt="{\displaystyle {\begin{alignedat}{2}\left(\mathbf {x} ^{T}\mathbf {y} +c\right)^{d}&=\sum _{j_{1}+j_{2}+\dots +j_{n+1}=d}{\frac {\sqrt {d!}}{\sqrt {j_{1}!\cdots j_{n}!j_{n+1}!}}}x_{1}^{j_{1}}\cdots x_{n}^{j_{n}}{\sqrt {c}}^{j_{n+1}}{\frac {\sqrt {d!}}{\sqrt {j_{1}!\cdots j_{n}!j_{n+1}!}}}y_{1}^{j_{1}}\cdots y_{n}^{j_{n}}{\sqrt {c}}^{j_{n+1}}\\&=\varphi (\mathbf {x} )^{T}\varphi (\mathbf {y} )\end{alignedat}}}" loading="lazy">

The last summation has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l_{d}={\tbinom {n+d}{d}}}">

<msub>

</mrow>

</msub>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle l_{d}={\tbinom {n+d}{d}}}</annotation>

</semantics>

</math><img src="./45b0d439520d9e3e4d008f765eb5915784701345.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.138ex; height:3.509ex;" alt="{\displaystyle l_{d}={\tbinom {n+d}{d}}}" loading="lazy"> elements, so that:

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (\mathbf {x} )=\left(a_{1},\dots ,a_{l},\dots ,a_{l_{d}}\right)}</annotation>

</semantics>

</math><img src="./2113807461ac3aa6120c86808c6600c7c64c8057.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.376ex; height:3.176ex;" alt="{\displaystyle \varphi (\mathbf {x} )=\left(a_{1},\dots ,a_{l},\dots ,a_{l_{d}}\right)}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l=(j_{1},j_{2},...,j_{n},j_{n+1})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle l=(j_{1},j_{2},...,j_{n},j_{n+1})}</annotation>

</semantics>

</math><img src="./f46d5365fd3d1465db511c09d879f6102aecb3d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.317ex; height:2.843ex;" alt="{\displaystyle l=(j_{1},j_{2},...,j_{n},j_{n+1})}" loading="lazy"> and

<msub>

</mrow>

</msub>

<mfrac>

<msqrt>

</msqrt>

<msqrt>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</msqrt>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

<msup>

<msqrt>

</msqrt>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{l}={\frac {\sqrt {d!}}{\sqrt {j_{1}!\cdots j_{n}!j_{n+1}!}}}x_{1}^{j_{1}}\cdots x_{n}^{j_{n}}{\sqrt {c}}^{j_{n+1}}\quad |\quad j_{1}+j_{2}+\dots +j_{n}+j_{n+1}=d}</annotation>

</semantics>

</math><img src="./733a3c864565c1e8341986b1f54925eb7d43a14d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:72.579ex; height:7.176ex;" alt="{\displaystyle a_{l}={\frac {\sqrt {d!}}{\sqrt {j_{1}!\cdots j_{n}!j_{n+1}!}}}x_{1}^{j_{1}}\cdots x_{n}^{j_{n}}{\sqrt {c}}^{j_{n+1}}\quad |\quad j_{1}+j_{2}+\dots +j_{n}+j_{n+1}=d}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Practical_use">Practical use</h2></div>

Although the <a href="RBF_kernel" class="mw-redirect" title="RBF kernel">RBF kernel</a> is more popular in SVM classification than the polynomial kernel, the latter is quite popular in <a href="Natural_language_processing" title="Natural language processing">natural language processing</a> (NLP).<a href="#cite_note-Goldberg2008-1">[1]</a><a href="#cite_note-Chang2010-5">[5]</a>

The most common degree is d = 2 (quadratic), since larger degrees tend to <a href="Overfitting" title="Overfitting">overfit</a> on NLP problems.

Various ways of computing the polynomial kernel (both exact and approximate) have been devised as alternatives to the usual non-linear SVM training algorithms, including:

<ul><li>full expansion of the kernel prior to training/testing with a linear SVM,<a href="#cite_note-Chang2010-5">[5]</a> i.e. full computation of the mapping φ as in polynomial regression;</li>

<li><a href="Association_rule_learning" title="Association rule learning">basket mining</a> (using a variant of the <a href="Apriori_algorithm" title="Apriori algorithm">apriori algorithm</a>) for the most commonly occurring feature conjunctions in a training set to produce an approximate expansion;<a href="#cite_note-Kudo2003-6">[6]</a></li>

<li><a href="Inverted_index" title="Inverted index">inverted indexing</a> of support vectors.<a href="#cite_note-Kudo2003-6">[6]</a><a href="#cite_note-Goldberg2008-1">[1]</a></li></ul>

One problem with the polynomial kernel is that it may suffer from <a href="Numerical_stability" title="Numerical stability">numerical instability</a>: when xTy + c < 1, K(x, y) = (xTy + c)d tends to zero with increasing d, whereas when xTy + c > 1, K(x, y) tends to infinity.<a href="#cite_note-lin2012-4">[4]</a>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<li id="cite_note-Goldberg2008-1">^ <a href="#cite_ref-Goldberg2008_1-0">a</a> <a href="#cite_ref-Goldberg2008_1-1">b</a> <a href="#cite_ref-Goldberg2008_1-2">c</a> Yoav Goldberg and Michael Elhadad (2008). splitSVM: Fast, Space-Efficient, non-Heuristic, Polynomial Kernel Computation for NLP Applications. Proc. ACL-08: HLT.

</li>

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20130415231446/http://www.cs.tufts.edu/~roni/Teaching/CLT/LN/lecture18.pdf">"Archived copy"</a> (PDF). Archived from <a rel="nofollow" class="external text" href="https://www.cs.tufts.edu/~roni/Teaching/CLT/LN/lecture18.pdf">the original</a> (PDF) on 2013-04-15. Retrieved 2012-11-12.</cite><code class="cs1-code">{{cite web}}</code>: CS1 maint: archived copy as title (link)

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFShashua2009" class="citation arxiv cs1">Shashua, Amnon (2009). "Introduction to Machine Learning: Class Notes 67577". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0904.3664v1">0904.3664v1</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cs.LG">cs.LG</a>].</cite>

</li>

<li id="cite_note-lin2012-4">^ <a href="#cite_ref-lin2012_4-0">a</a> <a href="#cite_ref-lin2012_4-1">b</a> <cite id="CITEREFLin2012" class="citation conference cs1">Lin, Chih-Jen (2012). <a rel="nofollow" class="external text" href="http://www.csie.ntu.edu.tw/~cjlin/talks/mlss_kyoto.pdf">Machine learning software: design and practical use</a> (PDF). Machine Learning Summer School. Kyoto.</cite>

</li>

<li id="cite_note-Chang2010-5">^ <a href="#cite_ref-Chang2010_5-0">a</a> <a href="#cite_ref-Chang2010_5-1">b</a> <cite id="CITEREFChangHsiehChangRinggaard2010" class="citation journal cs1">Chang, Yin-Wen; Hsieh, Cho-Jui; Chang, Kai-Wei; Ringgaard, Michael; Lin, Chih-Jen (2010). <a rel="nofollow" class="external text" href="http://jmlr.csail.mit.edu/papers/v11/chang10a.html">"Training and testing low-degree polynomial data mappings via linear SVM"</a>. <a href="Journal_of_Machine_Learning_Research" title="Journal of Machine Learning Research">Journal of Machine Learning Research</a>. 11: 1471–1490.</cite>

</li>

<li id="cite_note-Kudo2003-6">^ <a href="#cite_ref-Kudo2003_6-0">a</a> <a href="#cite_ref-Kudo2003_6-1">b</a> <cite id="CITEREFKudoMatsumoto2003" class="citation conference cs1">Kudo, T.; Matsumoto, Y. (2003). Fast methods for kernel-based text analysis. Proc. ACL.</cite>

</li>

</ol></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2024-09-07" href="https://en.wikipedia.org/wiki/?title=Polynomial_kernel&oldid=1244553685">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>