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String kernel

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In <a href="Machine_learning" title="Machine learning">machine learning</a> and <a href="Data_mining" title="Data mining">data mining</a>, a string kernel is a <a href="Positive-definite_kernel" title="Positive-definite kernel">kernel function</a> that operates on <a href="String_(computer_science)" title="String (computer science)">strings</a>, i.e. finite sequences of symbols that need not be of the same length. String kernels can be intuitively understood as functions measuring the <a href="Similarity_measure" title="Similarity measure">similarity</a> of pairs of strings: the more similar two strings a and b are, the higher the value of a string kernel K(a, b) will be.

Using string kernels with <a href="Kernel_trick" class="mw-redirect" title="Kernel trick">kernelized</a> learning algorithms such as <a href="Support_vector_machine" title="Support vector machine">support vector machines</a> allow such algorithms to work with strings, without having to translate these to fixed-length, real-valued <a href="Feature_vector" class="mw-redirect" title="Feature vector">feature vectors</a>.<a href="#cite_note-Lodhi-1">[1]</a> String kernels are used in domains where sequence data are to be <a href="Cluster_analysis" title="Cluster analysis">clustered</a> or <a href="Statistical_classification" title="Statistical classification">classified</a>, e.g. in <a href="Text_mining" title="Text mining">text mining</a> and <a href="Bioinformatics" title="Bioinformatics">gene analysis</a>.<a href="#cite_note-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="Informal_introduction">Informal introduction</h2></div>

Suppose one wants to compare some text passages automatically and indicate their relative similarity.

For many applications, it might be sufficient to find some keywords which match exactly.

One example where exact matching is not always enough is found in <a href="Spam_detection" class="mw-redirect" title="Spam detection">spam detection</a>.<a href="#cite_note-3">[3]</a>

Another would be in computational gene analysis, where <a href="Homology_(biology)" title="Homology (biology)">homologous</a> <a href="Genes" class="mw-redirect" title="Genes">genes</a> have <a href="Mutated" class="mw-redirect" title="Mutated">mutated</a>, resulting in common subsequences along with deleted, inserted or replaced symbols.

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

Since several well-proven data clustering, classification and information retrieval

methods (for example support vector machines) are designed to work on vectors

(i.e. data are elements of a vector space), using a string kernel allows the extension of these methods to handle sequence data.

The string kernel method is to be contrasted with earlier approaches for text classification where feature vectors only indicated

the presence or absence of a word.

Not only does it improve on these approaches, but it is an example for a whole class of kernels adapted to data structures, which

began to appear at the turn of the 21st century. A survey of such methods has been compiled by Gärtner.<a href="#cite_note-4">[4]</a>

In bioinformatics string kernels are used especially to transform biological sequences such as proteins or DNA into vectors for further use in machine learning models. An example of a string kernel used for that purpose is the profile kernel.<a href="#cite_note-5">[5]</a>

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

A <a href="Kernel_trick" class="mw-redirect" title="Kernel trick">kernel</a> on a domain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

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<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

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</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> is a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K:D\times D\rightarrow \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle K:D\times D\rightarrow \mathbb {R} }</annotation>

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</math><img src="./576e8af9ffe507de9555ec125edef461a1a68833.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.984ex; height:2.176ex;" alt="{\displaystyle K:D\times D\rightarrow \mathbb {R} }" loading="lazy">

satisfying some conditions (being <a href="Symmetric" class="mw-redirect" title="Symmetric">symmetric</a> in the arguments, <a href="Continuous_function" title="Continuous function">continuous</a> and <a href="Positive-semidefinite_function" class="mw-redirect" title="Positive-semidefinite function">positive semidefinite</a> in a certain sense).

<a href="Mercer's_theorem" title="Mercer's theorem">Mercer's theorem</a> asserts that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> can then be expressed as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(x,y)=\varphi (x)\cdot \varphi (y)}">

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

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</math><img src="./4e8f01e182019fb298fdb92c89e089b62c52e2e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.316ex; height:2.843ex;" alt="{\displaystyle K(x,y)=\varphi (x)\cdot \varphi (y)}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

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</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> mapping the arguments into an <a href="Inner_product_space" title="Inner product space">inner product space</a>.

We can now reproduce the definition of a string subsequence kernel<a href="#cite_note-Lodhi-1">[1]</a>

on strings over an <a href="Alphabet_(computer_science)" class="mw-redirect" title="Alphabet (computer science)">alphabet</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

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<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

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</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy">. Coordinate-wise, the mapping is defined as follows:

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{u}:\left\{{\begin{array}{l}\Sigma ^{n}\rightarrow \mathbb {R} ^{\Sigma ^{n}}\\s\mapsto \sum _{\mathbf {i} :u=s_{\mathbf {i} }}\lambda ^{l(\mathbf {i} )}\end{array}}\right.}</annotation>

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</math><img src="./8c4e6ae403fcf4c1603f2e5c79cca3668e2c0061.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.476ex; height:7.509ex;" alt="{\displaystyle \varphi _{u}:\left\{{\begin{array}{l}\Sigma ^{n}\rightarrow \mathbb {R} ^{\Sigma ^{n}}\\s\mapsto \sum _{\mathbf {i} :u=s_{\mathbf {i} }}\lambda ^{l(\mathbf {i} )}\end{array}}\right.}" loading="lazy"></dd></dl>

The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {i} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {i} }</annotation>

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</math><img src="./dc909dd0c91a68cdb9cda4ae133fa1f82c987c01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.742ex; height:2.176ex;" alt="{\displaystyle \mathbf {i} }" loading="lazy"> are <a href="Multiindices" class="mw-redirect" title="Multiindices">multiindices</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

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</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> is a string of length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">:

subsequences can occur in a non-contiguous manner, but gaps are penalized.

The multiindex <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {i} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {i} }</annotation>

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</math><img src="./dc909dd0c91a68cdb9cda4ae133fa1f82c987c01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.742ex; height:2.176ex;" alt="{\displaystyle \mathbf {i} }" loading="lazy"> gives the positions of the characters matching <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

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</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

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<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

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</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l(\mathbf {i} )}">

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<annotation encoding="application/x-tex">{\displaystyle l(\mathbf {i} )}</annotation>

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</math><img src="./82ab14a1e48042da1c8f9b9fb32268ff1ba6bfce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.245ex; height:2.843ex;" alt="{\displaystyle l(\mathbf {i} )}" loading="lazy"> is the difference between the first and last entry in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {i} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {i} }</annotation>

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</math><img src="./dc909dd0c91a68cdb9cda4ae133fa1f82c987c01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.742ex; height:2.176ex;" alt="{\displaystyle \mathbf {i} }" loading="lazy">, that is: how far apart in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

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<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

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</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> the subsequence matching <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

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The parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

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<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

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</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> may be set to any value between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

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<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

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</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> (gaps are not allowed, as only <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{0}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle 0^{0}}</annotation>

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</math><img src="./106f0c4e1cbccbfcbb61001a8c17b8427c65366d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.676ex;" alt="{\displaystyle 0^{0}}" loading="lazy"> is not <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

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<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

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</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

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<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

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</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy">) and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

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<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

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(even widely-spread "occurrences" are weighted the same as appearances as a contiguous substring, as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1^{l(\mathbf {i} )}=1}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle 1^{l(\mathbf {i} )}=1}</annotation>

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</math><img src="./dec54abd517509949f036ba6acee91942ec894e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.95ex; height:2.843ex;" alt="{\displaystyle 1^{l(\mathbf {i} )}=1}" loading="lazy">).

For several relevant algorithms, data enters into the algorithm only in expressions involving an inner product of feature vectors,

hence the name <a href="Kernel_methods" class="mw-redirect" title="Kernel methods">kernel methods</a>. A desirable consequence of this is that one does not need to explicitly calculate the transformation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (x)}">

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<annotation encoding="application/x-tex">{\displaystyle \phi (x)}</annotation>

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</math><img src="./546b660b2f3cfb5f34be7b3ed8371d54f5c74227.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.524ex; height:2.843ex;" alt="{\displaystyle \phi (x)}" loading="lazy">, only the inner product via the kernel, which may be a lot quicker, especially when <a href="Approximation" title="Approximation">approximated</a>.<a href="#cite_note-Lodhi-1">[1]</a>

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

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</style><cite id="CITEREFLodhiSaundersShawe-TaylorCristianini2002" class="citation journal cs1">Lodhi, Huma; Saunders, Craig; Shawe-Taylor, John; Cristianini, Nello; Watkins, Chris (2002). "Text classification using string kernels". <a href="Journal_of_Machine_Learning_Research" title="Journal of Machine Learning Research">Journal of Machine Learning Research</a>: 419–444.</cite>

</li>

<cite id="CITEREFLeslieEskinNoble2002" class="citation cs2">Leslie, C.; Eskin, E.; Noble, W.S. (2002), "The spectrum kernel: A string kernel for SVM protein classification", Proceedings of the Pacific Symposium on Biocomputing, vol. 7, pp. 566–575, <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/11928508">11928508</a></cite>

</li>

<cite id="CITEREFAmayri2009" class="citation cs2">Amayri, O. (2009), "Improved Online Support Vector Machines Spam Filtering Using String Kernels", Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications, Lecture Notes in Computer Science, vol. 5856, p. 621, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009LNCS.5856..621A">2009LNCS.5856..621A</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-10268-4_73">10.1007/978-3-642-10268-4_73</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-642-10267-7</bdi></cite>

</li>

<cite id="CITEREFGärtner2003" class="citation cs2">Gärtner, T. (2003), "A survey of kernels for structured data", ACM SIGKDD Explorations Newsletter, 5 (1), <a href="Association_for_Computing_Machinery" title="Association for Computing Machinery">ACM</a>: 58, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F959242.959248">10.1145/959242.959248</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:4471326">4471326</a></cite>

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<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFKuangIeWangWang2005" class="citation journal cs1">Kuang, Rui; Ie, Eugene; Wang, Ke; Wang, Kai; Siddiqi, Mahira; Freund, Yoav; Leslie, Christina (2005-06-01). "Profile-based string kernels for remote homology detection and motif extraction". Journal of Bioinformatics and Computational Biology. 3 (3): 527–550. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2Fs021972000500120x">10.1142/s021972000500120x</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0219-7200">0219-7200</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16108083">16108083</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14032548">14032548</a>.</cite>

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