{"id":118,"date":"2017-02-02T14:54:09","date_gmt":"2017-02-02T14:54:09","guid":{"rendered":"http:\/\/sag.art.uniroma2.it\/kelp_wordpress\/?page_id=118"},"modified":"2017-02-14T16:22:09","modified_gmt":"2017-02-14T16:22:09","slug":"kernel-functions","status":"publish","type":"page","link":"http:\/\/www.kelp-ml.org\/?page_id=118","title":{"rendered":"Kernel Functions"},"content":{"rendered":"<p>Kernel methods (Shawe-Taylor and Cristianini, 2004) are a powerful class of algorithms for pattern analysis that, exploiting the so called kernel functions, can operate in an implicit high-dimensional feature space without explicitly computing the coordinates of the data in that space. Most of the existing machine learning platforms provide kernel methods that operate only on vectorial data. On the contrary, KeLP\u00a0has the fundamental advantage that there is no restriction on a specific data structure, and kernels directly operating on vectors, sequences, trees, graphs, or other structured data can be defined.<\/p>\n<p>Furthermore, another appealing characteristic of KeLP\u00a0is that kernels can be composed and combined in order to create richer similarity metrics in which different information from different <strong>Representation<\/strong>s can be simultaneously exploited.<\/p>\n<p>As shown in the figure below, KeLP completely supports the composition and combination mechanisms providing three abstractions of the <strong>Kernel<\/strong> class:<\/p>\n<ul>\n<li><strong>DirectKernel<\/strong>: in computing the kernel similarity it operates directly on a specific <strong>Representation<\/strong> that it automatically extracts from the <strong>Example<\/strong>s to be compared. For instance,\u00a0KeLP implements <strong>LinearKernel<\/strong>\u00a0on Vector representations, and\u00a0<strong>SubTreeKernel<\/strong>\u00a0(Collins and Duffy, 2001) and <strong>PartialTreeKernel<\/strong>\u00a0(Moschitti, 2006) on\u00a0<strong>TreeRepresentation<\/strong>s.<\/li>\n<li><strong>KernelComposition<\/strong>: it enriches the kernel similarity provided by any another kernel. Some KeLP implementations are\u00a0<strong>PolynomialKernel<\/strong>, <strong>RBFKernel and<\/strong>\u00a0<strong>NormalizationKernel<\/strong>.<\/li>\n<li><strong>KernelCombination<\/strong>: it combines different kernels in a specific function. Some KeLP implementations are\u00a0<strong>LinearKernelCombination<\/strong> and\u00a0<strong>KernelMultiplication<\/strong>.<\/li>\n<\/ul>\n<p>Moreover, the class <strong>KernelOnPairs<\/strong> models kernels operating on <strong>ExamplePair<\/strong>s: all kernels discussed in (Filice et al., 2015)\u00a0and the <strong>PreferenceKernel<\/strong>\u00a0(Shen and Joshi, 2003) implement this class.<\/p>\n<div style=\"width: 1858px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/sag.art.uniroma2.it\/wp-content\/uploads\/2015\/12\/kernel.png\" width=\"1848\" height=\"1556\" \/><p class=\"wp-caption-text\">A simplified class diagram of the kernel taxonomy in KeLP.<\/p><\/div>\n<hr \/>\n<h2><span style=\"color: #000000;\">Existing Kernels<\/span><\/h2>\n<p><strong>Kernels on Vectors<\/strong>: they are <strong>DirectKernel<\/strong>s operating on <strong>Vector<\/strong> representations (both <strong>DenseVector<\/strong> and <strong>SparseVector<\/strong>).<\/p>\n<ul>\n<li><strong>LinearKernel<\/strong>: it executes the dot product between two <strong>Vector<\/strong>\u00a0representations.<\/li>\n<\/ul>\n<hr \/>\n<p><strong>Kernels on Sequences<\/strong>:\u00a0they are\u00a0<strong>DirectKernel<\/strong>s operating on\u00a0<strong>SequenceRepresention<\/strong>s.<\/p>\n<ul>\n<li><strong>SequenceKernel<\/strong>:\u00a0it is a convolution kernel between sequences. The algorithm corresponds to the recursive computation presented in (Bunescu and Mooney, 2005).<\/li>\n<\/ul>\n<hr \/>\n<p><strong>Kernels on Trees<\/strong>:\u00a0they are\u00a0<strong>DirectKernel<\/strong>s operating on\u00a0<strong>TreeRepresention<\/strong>s.<\/p>\n<div id=\"attachment_273\" style=\"width: 359px\" class=\"wp-caption aligncenter\"><img aria-describedby=\"caption-attachment-273\" decoding=\"async\" loading=\"lazy\" class=\"wp-image-273\" src=\"http:\/\/sag.art.uniroma2.it\/kelp_wordpress\/wp-content\/uploads\/2017\/02\/tree_and_fragments.png\" width=\"349\" height=\"426\" srcset=\"http:\/\/www.kelp-ml.org\/wp-content\/uploads\/2017\/02\/tree_and_fragments.png 808w, http:\/\/www.kelp-ml.org\/wp-content\/uploads\/2017\/02\/tree_and_fragments-246x300.png 246w, http:\/\/www.kelp-ml.org\/wp-content\/uploads\/2017\/02\/tree_and_fragments-768x937.png 768w\" sizes=\"(max-width: 349px) 100vw, 349px\" \/><p id=\"caption-attachment-273\" class=\"wp-caption-text\">a) Constituency parse tree for the sentence Federer won against Nadal, b) some subtrees, c) some subset trees, d) some partial trees.<\/p><\/div>\n<ul>\n<li><strong>SubTreeKernel<\/strong>:\u00a0it\u00a0is a convolution kernel that evaluates the tree fragments shared between two trees. The considered fragments are subtrees, i.e.,\u00a0a node and its complete descendancy. For more details see (Vishwanathan and Smola, 2003).<\/li>\n<li><strong>SubSetTreeKernel<\/strong>: the SubSetTree Kernel, a.k.a. Syntactic Tree Kernel, is a convolution kernel that evaluates the tree fragments shared between two trees. The considered fragments are subset-trees, i.e.,\u00a0a node and its partial descendancy (the descendancy can be incomplete in depth, but no partial productions are allowed; in other words, given a node either all its children or none of them must be considered). For further details see (Collins and Duffy, 2001).<\/li>\n<li><strong>PartialTreeKernel<\/strong>: it is a convolution kernel that evaluates the tree fragments shared between two trees. The considered fragments are partial trees, i.e.,\u00a0a node and its partial descendancy (the descendancy can be incomplete, i.e.,\u00a0a partial production is allowed). For further details see (Moschitti, 2006).<\/li>\n<li><strong>SmoothedPartialTreeKernel<\/strong>:\u00a0it\u00a0is a convolution kernel that evaluates the tree fragments shared between two trees (Croce et al., 2011). The considered fragments are partial trees (as for the <strong>PartialTreeKernel<\/strong>) whose nodes are identical or similar according to a node similarity function: the contribution of the fragment pairs in the overall kernel thus depends on the number of shared substructures, whose nodes contribute according to such a metrics.<br \/>\nThis kernel is very flexible as the adoption of node similarity functions allows the definition of more expressive kernels, such as the Compositionally Smoothed Partial Tree Kernel (Annesi et al, 2014).<\/li>\n<\/ul>\n<hr \/>\n<p><strong>Kernels on Graphs:\u00a0<\/strong>they are\u00a0<strong>DirectKernel<\/strong>s operating on\u00a0<strong>DirectedGraphRepresention<\/strong>s<strong>.<\/strong><\/p>\n<ul>\n<li><strong>ShortestPathKernel<\/strong>: it\u00a0associates a feature to each pair of nodes of one graph. The feature name corresponds to pair of node labels while the value is the length of the shortest path between the nodes in the graph. Further details can be found in (Borgwardt\u00a0and\u00a0Kriegel, 2005).<\/li>\n<li><strong>WLSubtreeMapper<\/strong>: it is actually an explicit feature extractor that output vectors containing the features of the <span class=\"s1\">Weisfeiler<\/span><span class=\"s2\">&#8211;<\/span><span class=\"s1\">Lehman<\/span> Subtree Kernel for Graphs (Shervashidze, 2011). A <strong>LinearKernel<\/strong>\u00a0can be used to exploit the produced vectors.<\/li>\n<\/ul>\n<hr \/>\n<p><strong>Kernel Compositions<\/strong>: they are kernels that enriches the computation of another kernel.<\/p>\n<ul>\n<li><strong>PolynomialKernel<\/strong>: it\u00a0implicitly works in a features space where all the polynomials of the original features are available. As an example, a <img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-9980056f03e6bbf28811681e26353914_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#94;&#123;&#110;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"25\" style=\"vertical-align: 0px;\"\/> degree polynomial kernel applied over a linear kernel on\u00a0vector representations will automatically consider pairs of features in its similarity evaluation. Given a base kernel <em>K<\/em>, \u00a0the\u00a0<strong>PolynomialKernel<\/strong> applies the following\u00a0formula: <img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-57509d0f8a877348502421921223f96c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#40;&#112;&#111;&#108;&#121;&#95;&#75;&#40;&#120;&#44;&#121;&#41;&#61;&#92;&#98;&#105;&#103;&#32;&#40;&#32;&#97;&#75;&#40;&#120;&#44;&#121;&#41;&#43;&#98;&#32;&#92;&#98;&#105;&#103;&#32;&#41;&#32;&#94;&#100;&#92;&#41;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"231\" style=\"vertical-align: -7px;\"\/>, where\u00a0<img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-d3882ad7682ea2ab89dbf5201930e7b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#44;&#32;&#97;&#44;&#32;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"42\" style=\"vertical-align: -4px;\"\/> are kernel parameters. Common values are\u00a0<img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-90899c4debe310841a2f67893781f68c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: 0px;\"\/>,\u00a0<img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-44db0db81bdc2def0aa21b7e350d2324_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: -1px;\"\/> and\u00a0<img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-597ae6c01abdebf6816c77f3ce1fd9de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: 0px;\"\/>.<\/li>\n<li><strong>RbfKernel<\/strong>:\u00a0the Radial Basis Function (RBF) Kernel, a.k.a. Gaussian Kernel, enriches another kernel according to the following formula <img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-94e0be71697a6b2270aa722ce53db044_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#66;&#70;&#95;&#75;&#40;&#120;&#44;&#121;&#41;&#32;&#61;&#32;&#101;&#94;&#123;&#45;&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#32;&#92;&#108;&#101;&#102;&#116;&#32;&#92;&#108;&#86;&#101;&#114;&#116;&#32;&#120;&#45;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#32;&#92;&#114;&#86;&#101;&#114;&#116;&#95;&#123;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#72;&#125;&#95;&#75;&#125;&#32;&#94;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"204\" style=\"vertical-align: -4px;\"\/>, where:<img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-66a12293e5adb7ef09eb4a308f5414fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#108;&#86;&#101;&#114;&#116;&#32;&#97;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#114;&#86;&#101;&#114;&#116;&#95;&#123;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#72;&#125;&#95;&#75;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"47\" style=\"vertical-align: -7px;\"\/> is the norm of <img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"\/> in the kernel space <img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-42d3cf5d308c38b9302bcd954e449dbb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#72;&#125;&#95;&#75;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"28\" style=\"vertical-align: -3px;\"\/> generated by a base kernel <img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-ea9c87a513e4a72624155d392fae86e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#75;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\"\/>.<img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-d3ecad389066804f9bfb5f24a5090869_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#108;&#86;&#101;&#114;&#116;&#32;&#120;&#45;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#114;&#86;&#101;&#114;&#116;&#95;&#123;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#72;&#125;&#95;&#75;&#125;&#32;&#94;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"79\" style=\"vertical-align: -7px;\"\/> can be computed as <img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-69cf94ac3c1047b1feacf5099168c861_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#108;&#86;&#101;&#114;&#116;&#32;&#120;&#45;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#114;&#86;&#101;&#114;&#116;&#95;&#123;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#72;&#125;&#95;&#75;&#125;&#32;&#94;&#50;&#32;&#61;&#32;&#92;&#108;&#101;&#102;&#116;&#92;&#108;&#86;&#101;&#114;&#116;&#32;&#120;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#114;&#86;&#101;&#114;&#116;&#95;&#123;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#72;&#125;&#95;&#75;&#125;&#32;&#94;&#50;&#32;&#43;&#32;&#92;&#108;&#101;&#102;&#116;&#92;&#108;&#86;&#101;&#114;&#116;&#32;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#114;&#86;&#101;&#114;&#116;&#95;&#123;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#72;&#125;&#95;&#75;&#125;&#32;&#94;&#50;&#32;&#45;&#32;&#50;&#75;&#40;&#120;&#44;&#121;&#41;&#32;&#61;&#32;&#75;&#40;&#120;&#44;&#120;&#41;&#32;&#43;&#32;&#75;&#40;&#121;&#44;&#121;&#41;&#32;&#45;&#32;&#50;&#75;&#40;&#120;&#44;&#121;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"564\" style=\"vertical-align: -7px;\"\/>. This allows to apply the RBF operation to any kernel base kernel <em><em>K. <\/em><\/em>It depends on a width\u00a0 parameter\u00a0<img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-4de02fc502ed5dbd15f371728ea270a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#97;&#109;&#109;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\"\/> which regulates how fast the\u00a0<strong>RbfKernel<\/strong> similarity decays w.r.t. the distance of the input objects in<img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-42d3cf5d308c38b9302bcd954e449dbb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#72;&#125;&#95;&#75;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"28\" style=\"vertical-align: -3px;\"\/>. It can be proven that the Gaussian Kernel produces an infinite dimensional RKHS.<\/li>\n<li><strong>NormalizationKernel<\/strong>: it normalizes another kernel <em>K<\/em> according to the following formula:\u00a0<img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-d74727203f95114555984846fae8ef9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;&#95;&#123;&#75;&#125;&#40;&#120;&#44;&#121;&#41;&#32;&#61;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#75;&#40;&#120;&#44;&#121;&#41;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#75;&#40;&#120;&#44;&#120;&#41;&#75;&#40;&#121;&#44;&#121;&#41;&#125;&#125;&#32;&#61;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#112;&#104;&#105;&#40;&#120;&#41;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#112;&#104;&#105;&#40;&#121;&#41;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#32;&#92;&#108;&#101;&#102;&#116;&#32;&#92;&#124;&#32;&#92;&#112;&#104;&#105;&#40;&#120;&#41;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#32;&#92;&#124;&#94;&#50;&#32;&#92;&#108;&#101;&#102;&#116;&#32;&#92;&#124;&#32;&#92;&#112;&#104;&#105;&#40;&#121;&#41;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#32;&#92;&#124;&#94;&#50;&#125;&#125;&#32;&#61;&#32;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#112;&#104;&#105;&#40;&#120;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#32;&#92;&#124;&#32;&#92;&#112;&#104;&#105;&#40;&#120;&#41;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#32;&#92;&#124;&#125;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#112;&#104;&#105;&#40;&#121;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#32;&#92;&#124;&#32;&#92;&#112;&#104;&#105;&#40;&#121;&#41;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#32;&#92;&#124;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"454\" style=\"vertical-align: -15px;\"\/>, where <img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-006bb587ec9036c2bc1773ffca785be7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#105;&#40;&#92;&#99;&#100;&#111;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"29\" style=\"vertical-align: -4px;\"\/> is the implicit projection function operated by the kernel <em>K<\/em>. The normalization operation corresponds to a dot product in the RKHS of the normalized projections of the input instances. When <em>K<\/em>\u00a0is <strong>LinearKernel\u00a0<\/strong>on two vectors, the\u00a0<strong>NormalizationKernel<\/strong> equals to the cosine similarity between the two vectors. The normalization operation is required when the instances to be compared are very different in size, in order to avoid that large instances (for instance long texts) are associated with larger similarities. For instance it is usually applied to tree kernels, in order properly compare trees having very different sizes.<\/li>\n<\/ul>\n<hr \/>\n<p><strong>Kernel Combinations<\/strong>: they combine different kernels, allowing the possibility to simultaneously exploit different data representations.<\/p>\n<ul>\n<li><strong>LinearKernelCombination<\/strong>: given a set <em>n<\/em> of <strong>Kernel<\/strong>s <img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-5bb088ff56cd34f038bad5523c4aae08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#75;&#95;&#49;&#44;&#32;&#92;&#108;&#100;&#111;&#116;&#115;&#44;&#32;&#75;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"85\" style=\"vertical-align: -4px;\"\/>, it computes the weighted sum of the kernel similarities: <img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-87c62e56a64f32f46a6eb03ce4ee485e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#75;&#40;&#120;&#44;&#121;&#41;&#92;&#115;&#117;&#109;&#95;&#123;&#105;&#92;&#108;&#101;&#113;&#32;&#110;&#125;&#99;&#95;&#105;&#75;&#95;&#105;&#40;&#120;&#44;&#121;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"181\" style=\"vertical-align: -7px;\"\/> where\u00a0<img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-73f745a16381e58d8cc648594702fab4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#95;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"13\" style=\"vertical-align: -3px;\"\/> are the parametrizable weights of the combination.<\/li>\n<li><strong>KernelMultiplication<\/strong>: given a set <em>n<\/em> of <strong>Kernel<\/strong>s <img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-5bb088ff56cd34f038bad5523c4aae08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#75;&#95;&#49;&#44;&#32;&#92;&#108;&#100;&#111;&#116;&#115;&#44;&#32;&#75;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"85\" style=\"vertical-align: -4px;\"\/>, it computes the product\u00a0of the kernel similarities: <img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-6deb20e4e6b9715c4f9bf946c8d1fbcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#75;&#40;&#120;&#44;&#32;&#121;&#41;&#32;&#61;&#32;&#92;&#112;&#114;&#111;&#100;&#95;&#123;&#105;&#92;&#108;&#101;&#113;&#32;&#110;&#125;&#75;&#95;&#105;&#40;&#120;&#44;&#121;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"186\" style=\"vertical-align: -7px;\"\/>.<\/li>\n<\/ul>\n<hr \/>\n<p><strong>Kernels on Pairs<\/strong>: They operate on instances of <strong>ExamplePair<\/strong>.<\/p>\n<ul>\n<li><strong>PreferenceKernel<\/strong>: In the learning to rank scenario, the preference kernel (Shen and Joshi, 2003) compares two pairs of ordered objects <img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-529bbcec1a3a3b03df6fd9a63f5cb91e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#95;&#97;&#32;&#61;&#32;&#92;&#108;&#97;&#110;&#103;&#108;&#101;&#32;&#97;&#95;&#49;&#44;&#32;&#97;&#95;&#50;&#32;&#92;&#114;&#97;&#110;&#103;&#108;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"\/> and<img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-e6bd51843c4087eaafdbbe0022aa4376_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#95;&#98;&#32;&#61;&#32;&#92;&#108;&#97;&#110;&#103;&#108;&#101;&#32;&#98;&#95;&#49;&#44;&#32;&#98;&#95;&#50;&#32;&#92;&#114;&#97;&#110;&#103;&#108;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"91\" style=\"vertical-align: -5px;\"\/>: <img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-d708dbc909608368fd2450e2754eb4bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#75;&#32;&#40;&#32;&#112;&#95;&#97;&#44;&#32;&#112;&#95;&#98;&#32;&#41;&#32;&#61;&#32;&#66;&#75;&#40;&#97;&#95;&#49;&#44;&#32;&#98;&#95;&#49;&#41;&#32;&#43;&#32;&#66;&#75;&#40;&#97;&#95;&#50;&#44;&#32;&#98;&#95;&#50;&#41;&#32;&#45;&#32;&#66;&#75;&#40;&#97;&#95;&#49;&#44;&#32;&#98;&#95;&#50;&#41;&#32;&#45;&#32;&#66;&#75;&#40;&#97;&#95;&#50;&#44;&#32;&#98;&#95;&#49;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"497\" style=\"vertical-align: -4px;\"\/>, where <em>BK<\/em> is a generic kernel operating on the elements of the pairs. The underlying idea is to evaluate whether the first pair<img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-8fc28837bec807e2fd0e358038df0e5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#110;&#103;&#108;&#101;&#32;&#97;&#95;&#49;&#44;&#32;&#97;&#95;&#50;&#32;&#92;&#114;&#97;&#110;&#103;&#108;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"53\" style=\"vertical-align: -5px;\"\/> aligns better to the second pair in its regular order<img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-022d6ac7058ad27f9b6579ad5af717b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#110;&#103;&#108;&#101;&#32;&#98;&#95;&#49;&#44;&#32;&#98;&#95;&#50;&#32;&#92;&#114;&#97;&#110;&#103;&#108;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"49\" style=\"vertical-align: -5px;\"\/> rather than to its inverted order<img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-642909b4805517437596fd08fce78184_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#110;&#103;&#108;&#101;&#32;&#98;&#95;&#50;&#44;&#32;&#98;&#95;&#49;&#32;&#92;&#114;&#97;&#110;&#103;&#108;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"49\" style=\"vertical-align: -5px;\"\/>.<\/li>\n<li><strong>UncrossedPairwiseSumKernel<\/strong>: it\u00a0compares two pairs of ordered objects <img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-529bbcec1a3a3b03df6fd9a63f5cb91e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#95;&#97;&#32;&#61;&#32;&#92;&#108;&#97;&#110;&#103;&#108;&#101;&#32;&#97;&#95;&#49;&#44;&#32;&#97;&#95;&#50;&#32;&#92;&#114;&#97;&#110;&#103;&#108;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"\/> and <img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-e6bd51843c4087eaafdbbe0022aa4376_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#95;&#98;&#32;&#61;&#32;&#92;&#108;&#97;&#110;&#103;&#108;&#101;&#32;&#98;&#95;&#49;&#44;&#32;&#98;&#95;&#50;&#32;&#92;&#114;&#97;&#110;&#103;&#108;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"91\" style=\"vertical-align: -5px;\"\/>, summing the contributions of the single element similarities:\u00a0<img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-85e63376de0372b828df5611c74d08d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#75;&#32;&#40;&#32;&#112;&#95;&#97;&#44;&#32;&#112;&#95;&#98;&#32;&#41;&#32;&#61;&#32;&#66;&#75;&#40;&#97;&#95;&#49;&#44;&#32;&#98;&#95;&#49;&#41;&#32;&#43;&#32;&#66;&#75;&#40;&#97;&#95;&#50;&#44;&#32;&#98;&#95;&#50;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"284\" style=\"vertical-align: -4px;\"\/>, where <em>BK<\/em> is a generic kernel operating on the elements of the pairs. It has been used in learning scenarios where the elements within a pair have different roles, such as <em>text<\/em> and <em>hypothesis<\/em> in Recognizing Textual Entailment (Filice\u00a0et al., 2015), or <em>question<\/em> and <em>answer<\/em> in Question Answering\u00a0(Filice et al., 2016).<\/li>\n<li><strong>UncrossedPairwiseProductKernel<\/strong>: it\u00a0compares two pairs of ordered objects <img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-529bbcec1a3a3b03df6fd9a63f5cb91e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#95;&#97;&#32;&#61;&#32;&#92;&#108;&#97;&#110;&#103;&#108;&#101;&#32;&#97;&#95;&#49;&#44;&#32;&#97;&#95;&#50;&#32;&#92;&#114;&#97;&#110;&#103;&#108;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"\/> and <img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-e6bd51843c4087eaafdbbe0022aa4376_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#95;&#98;&#32;&#61;&#32;&#92;&#108;&#97;&#110;&#103;&#108;&#101;&#32;&#98;&#95;&#49;&#44;&#32;&#98;&#95;&#50;&#32;&#92;&#114;&#97;&#110;&#103;&#108;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"91\" style=\"vertical-align: -5px;\"\/>, multiplying\u00a0the contributions of the single element similarities:\u00a0<img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-a5ee8b4d7b39d22e113dba177c8c58f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#75;&#32;&#40;&#32;&#112;&#95;&#97;&#44;&#32;&#112;&#95;&#98;&#32;&#41;&#32;&#61;&#32;&#66;&#75;&#40;&#97;&#95;&#49;&#44;&#32;&#98;&#95;&#49;&#41;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#66;&#75;&#40;&#97;&#95;&#50;&#44;&#32;&#98;&#95;&#50;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"276\" style=\"vertical-align: -4px;\"\/>, where <em>BK<\/em> is a generic kernel operating on the elements of the pairs. As for the\u00a0<strong>UncrossedPairwiseSumKernel<\/strong>, it has been used in learning scenarios where the elements within a pair have different roles, such as <em>text<\/em> and <em>hypothesis<\/em> in Recognizing Textual Entailment (Filice\u00a0et al., 2015), or <em>question<\/em> and <em>answer<\/em> in Question Answering\u00a0(Filice et al., 2016). The product operation inherently applies a sort of logic and between the\u00a0<img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-4ff25c2cefaf4e188031a8b3fc7457be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;&#75;&#40;&#97;&#95;&#49;&#44;&#32;&#98;&#95;&#49;&#41;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"84\" style=\"vertical-align: -4px;\"\/> and\u00a0<img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-bcf3b0974365dc8c92010ec40b5d4b82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;&#75;&#40;&#97;&#95;&#50;&#44;&#32;&#98;&#95;&#50;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"84\" style=\"vertical-align: -4px;\"\/>.<\/li>\n<li><strong>PairwiseSumKernel<\/strong>: it\u00a0compares two pairs of objects <img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-529bbcec1a3a3b03df6fd9a63f5cb91e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#95;&#97;&#32;&#61;&#32;&#92;&#108;&#97;&#110;&#103;&#108;&#101;&#32;&#97;&#95;&#49;&#44;&#32;&#97;&#95;&#50;&#32;&#92;&#114;&#97;&#110;&#103;&#108;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"\/> and <img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-e6bd51843c4087eaafdbbe0022aa4376_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#95;&#98;&#32;&#61;&#32;&#92;&#108;&#97;&#110;&#103;&#108;&#101;&#32;&#98;&#95;&#49;&#44;&#32;&#98;&#95;&#50;&#32;&#92;&#114;&#97;&#110;&#103;&#108;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"91\" style=\"vertical-align: -5px;\"\/>, summing the contributions of all pairwise similarities between the single elements:\u00a0<img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-0714e713fb6b4e4dda471a1d1ba87159_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#75;&#32;&#40;&#32;&#112;&#95;&#97;&#44;&#32;&#112;&#95;&#98;&#32;&#41;&#32;&#61;&#32;&#66;&#75;&#40;&#97;&#95;&#49;&#44;&#32;&#98;&#95;&#49;&#41;&#32;&#43;&#32;&#66;&#75;&#40;&#97;&#95;&#50;&#44;&#32;&#98;&#95;&#50;&#41;&#32;&#43;&#32;&#66;&#75;&#40;&#97;&#95;&#49;&#44;&#32;&#98;&#95;&#50;&#41;&#32;&#43;&#32;&#66;&#75;&#40;&#97;&#95;&#50;&#44;&#32;&#98;&#95;&#49;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"497\" style=\"vertical-align: -4px;\"\/>, where <em>BK<\/em> is a generic kernel operating on the elements of the pairs. It has been used in symmetric tasks, such as Paraphrase Identification, where the elements within a pair are interchangeable (Filice\u00a0et al., 2015).<\/li>\n<li><strong>PairwiseProductKernel<\/strong>: it\u00a0compares two pairs of objects <img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-529bbcec1a3a3b03df6fd9a63f5cb91e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#95;&#97;&#32;&#61;&#32;&#92;&#108;&#97;&#110;&#103;&#108;&#101;&#32;&#97;&#95;&#49;&#44;&#32;&#97;&#95;&#50;&#32;&#92;&#114;&#97;&#110;&#103;&#108;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"\/> and <img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-e6bd51843c4087eaafdbbe0022aa4376_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#95;&#98;&#32;&#61;&#32;&#92;&#108;&#97;&#110;&#103;&#108;&#101;&#32;&#98;&#95;&#49;&#44;&#32;&#98;&#95;&#50;&#32;&#92;&#114;&#97;&#110;&#103;&#108;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"91\" style=\"vertical-align: -5px;\"\/>, summing the contributions of the two possible\u00a0pairwise alignments:\u00a0<img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-f8cc00e109b1e1fb3df32bcb723d583f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#75;&#32;&#40;&#32;&#112;&#95;&#97;&#44;&#32;&#112;&#95;&#98;&#32;&#41;&#32;&#61;&#32;&#66;&#75;&#40;&#97;&#95;&#49;&#44;&#32;&#98;&#95;&#49;&#41;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#66;&#75;&#40;&#97;&#95;&#50;&#44;&#32;&#98;&#95;&#50;&#41;&#32;&#43;&#32;&#66;&#75;&#40;&#97;&#95;&#49;&#44;&#32;&#98;&#95;&#50;&#41;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#66;&#75;&#40;&#97;&#95;&#50;&#44;&#32;&#98;&#95;&#49;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"479\" style=\"vertical-align: -4px;\"\/>, where <em>BK<\/em> is a generic kernel operating on the elements of the pairs. It has been used in symmetric tasks, such as Paraphrase Identification, where the elements within a pair are interchangeable (Filice\u00a0et al., 2015).<\/li>\n<li><strong>BestPairwiseAlignmentKernel<\/strong>: it compares two pairs of objects<img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-529bbcec1a3a3b03df6fd9a63f5cb91e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#95;&#97;&#32;&#61;&#32;&#92;&#108;&#97;&#110;&#103;&#108;&#101;&#32;&#97;&#95;&#49;&#44;&#32;&#97;&#95;&#50;&#32;&#92;&#114;&#97;&#110;&#103;&#108;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"\/> and <img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-e6bd51843c4087eaafdbbe0022aa4376_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#95;&#98;&#32;&#61;&#32;&#92;&#108;&#97;&#110;&#103;&#108;&#101;&#32;&#98;&#95;&#49;&#44;&#32;&#98;&#95;&#50;&#32;&#92;&#114;&#97;&#110;&#103;&#108;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"91\" style=\"vertical-align: -5px;\"\/>, evaluating the best pairwise alignment:<img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-5115e15c5ccfaadd12c39f6171260b25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#75;&#40;&#32;&#112;&#95;&#97;&#44;&#32;&#112;&#95;&#98;&#32;&#41;&#32;&#61;&#32;&#115;&#111;&#102;&#116;&#109;&#97;&#120;&#32;&#92;&#66;&#105;&#103;&#32;&#40;&#32;&#66;&#75;&#40;&#97;&#95;&#49;&#44;&#98;&#95;&#49;&#41;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#66;&#75;&#40;&#97;&#95;&#50;&#44;&#32;&#98;&#95;&#50;&#41;&#44;&#32;&#66;&#75;&#40;&#97;&#95;&#49;&#44;&#98;&#95;&#50;&#41;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#66;&#75;&#40;&#97;&#95;&#50;&#44;&#98;&#95;&#49;&#41;&#32;&#92;&#66;&#105;&#103;&#32;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"554\" style=\"vertical-align: -12px;\"\/>, where <em>BK<\/em> is a generic kernel operating on the elements of the pairs, and <em>softmax<\/em> is a function put in place of the max operation, which would cause <em>K<\/em>\u00a0not to be a valid kernel function (i.e.,\u00a0the resulting Gram matrix can violate the Mercer&#8217;s conditions). In particular,<img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.kelp-ml.org\/wp-content\/ql-cache\/quicklatex.com-dfda441e94cdbafac5d3f51b3609eed3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#111;&#102;&#116;&#109;&#97;&#120;&#36;&#40;&#120;&#95;&#49;&#44;&#120;&#95;&#50;&#41;&#61;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#99;&#125;&#32;&#92;&#108;&#111;&#103;&#40;&#92;&#101;&#120;&#112;&#40;&#99;&#32;&#120;&#95;&#49;&#41;&#32;&#43;&#32;&#92;&#101;&#120;&#112;&#40;&#99;&#32;&#120;&#95;&#50;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"346\" style=\"vertical-align: -6px;\"\/> (c=100 is accurate enough). The\u00a0<strong>BestPairwiseAlignmentKernel<\/strong> has been used in symmetric tasks, such as Paraphrase Identification (Filice et al., 2015), where the elements within a pair are interchangeable.<\/li>\n<\/ul>\n<hr \/>\n<h3>References<\/h3>\n<p>Paolo Annesi, Danilo Croce, and Roberto Basili.\u00a0<em>Semantic compositionality in tree kernels<\/em>. In Proceedings of the 23rd ACM International Conference on Conference on In- formation and Knowledge Management, CIKM \u201914, pages 1029\u20131038, New York, NY, USA, 2014. ACM.<\/p>\n<p class=\"p1\">K. M. <span class=\"s1\">Borgwardt<\/span> and H.<span class=\"s2\">&#8211;<\/span>P. <span class=\"s1\">Kriegel.\u00a0<\/span><em>Shortest<span class=\"s2\">&#8211;<\/span>Path Kernels on Graphs<\/em>. in Proceedings of the Fifth IEEE International Conference on Data Mining, 2005, <span class=\"s1\">pp<\/span>. 74\u201381.<\/p>\n<p>Razvan C. Bunescu and Raymond J. Mooney.\u00a0<em>Subsequence kernels for relation extraction<\/em>. 2005. In NIPS.<\/p>\n<p>Michael Collins and Nigel Duffy.\u00a0<em>Convolution kernels for natural language<\/em>. In Proceedings of the 14th Conference on Neural Information Processing Systems, 2001.<\/p>\n<p>Danilo Croce, Alessandro Moschitti, and Roberto Basili.\u00a0<em>Structured lexical similarity via convolution kernels on dependency trees<\/em>. In Proceedings of EMNLP, Edinburgh, Scotland, UK., 2011.<\/p>\n<p>Simone Filice, Giovanni Da San Martino and Alessandro Moschitti<em>. Relational Information for Learning from Structured Text Pairs. <\/em>In Proceedings of the 53<sup>rd<\/sup> Annual Meeting of the Association for Computational Linguistics, ACL 2015.<\/p>\n<p>Simone Filice, Danilo Croce, Alessandro Moschitti and Roberto Basili<em>. KeLP at SemEval-2016 Task 3: Learning Semantic Relations between Questions and Answers. <\/em>In Proceedings of the 10th International Workshop on Semantic Evaluation (SemEval 2016), Association for Computational Linguistics. (Best system @ SemEval-2016 task 3)<\/p>\n<p>Alessandro Moschitti.\u00a0<em>Efficient convolution kernels for dependency and constituent syntactic trees<\/em>. In ECML, Berlin, Germany, September 2006.<\/p>\n<p>Libin Shen and Aravind K. Joshi.\u00a0<em>An svm based voting algorithm with application to parse reranking<\/em>. In In Proc. of CoNLL 2003, pages 9\u201316, 2003<\/p>\n<p>John Shawe-Taylor and Nello Cristianini.\u00a0<em>Kernel Methods for Pattern Analysis<\/em>. Cambridge University Press, New York, NY, USA, 2004. ISBN 0521813972.<\/p>\n<p>N. <span class=\"s1\">Shervashidze<\/span>, <em><span class=\"s1\">Weisfeiler<\/span><span class=\"s2\">&#8211;<\/span><span class=\"s1\">lehman<\/span> graph kernels<\/em>, JMLR, <span class=\"s1\">vol<\/span>. 12, <span class=\"s1\">pp<\/span>. 2539\u20132561, 2011<\/p>\n<p class=\"p1\">S.V.N. <span class=\"s1\">Vishwanathan<\/span> and A.J. <span class=\"s1\">Smola<\/span>. Fast\u00a0kernels on strings and trees. In Proceedings of Neural Information Processing\u00a0Systems (NIPS), 2003.<\/p>\n<p>Fabio massimo Zanzotto, Marco Pennacchiotti, and Alessandro Moschitti.\u00a0<em>A machine learning approach to textual entailment recognition<\/em>. Nat. Lang. Eng., 15(4): 551\u2013582, October 2009. ISSN 1351-3249.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Kernel methods (Shawe-Taylor and Cristianini, 2004) are a powerful class of algorithms for pattern analysis that, exploiting the so called kernel functions, can operate in an implicit high-dimensional feature space without explicitly computing the coordinates of the data in that space. Most of the existing machine learning platforms provide kernel methods that operate only on <a href=\"http:\/\/www.kelp-ml.org\/?page_id=118\" rel=\"nofollow\"><span class=\"sr-only\">Read more about Kernel Functions<\/span>[&hellip;]<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":2,"menu_order":15,"comment_status":"closed","ping_status":"closed","template":"","meta":[],"_links":{"self":[{"href":"http:\/\/www.kelp-ml.org\/index.php?rest_route=\/wp\/v2\/pages\/118"}],"collection":[{"href":"http:\/\/www.kelp-ml.org\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"http:\/\/www.kelp-ml.org\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"http:\/\/www.kelp-ml.org\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.kelp-ml.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=118"}],"version-history":[{"count":34,"href":"http:\/\/www.kelp-ml.org\/index.php?rest_route=\/wp\/v2\/pages\/118\/revisions"}],"predecessor-version":[{"id":310,"href":"http:\/\/www.kelp-ml.org\/index.php?rest_route=\/wp\/v2\/pages\/118\/revisions\/310"}],"up":[{"embeddable":true,"href":"http:\/\/www.kelp-ml.org\/index.php?rest_route=\/wp\/v2\/pages\/2"}],"wp:attachment":[{"href":"http:\/\/www.kelp-ml.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=118"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}