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<title>Kernel method</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Kernel method</span></span>

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="Machine_learning" title="Machine learning">Machine learning</a><br>and <a href="Data_mining" title="Data mining">data mining</a></th></tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Paradigms</div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Supervised_learning" title="Supervised learning">Supervised learning</a></li>

<li><a href="Unsupervised_learning" title="Unsupervised learning">Unsupervised learning</a></li>

<li><a href="Semi-supervised_learning" class="mw-redirect" title="Semi-supervised learning">Semi-supervised learning</a></li>

<li><a href="Self-supervised_learning" title="Self-supervised learning">Self-supervised learning</a></li>

<li><a href="Reinforcement_learning" title="Reinforcement learning">Reinforcement learning</a></li>

<li><a href="Meta-learning_(computer_science)" title="Meta-learning (computer science)">Meta-learning</a></li>

<li><a href="Online_machine_learning" title="Online machine learning">Online learning</a></li>

<li><a href="Batch_learning" class="mw-redirect" title="Batch learning">Batch learning</a></li>

<li><a href="Curriculum_learning" title="Curriculum learning">Curriculum learning</a></li>

<li><a href="Rule-based_machine_learning" title="Rule-based machine learning">Rule-based learning</a></li>

<li><a href="Neuro-symbolic_AI" title="Neuro-symbolic AI">Neuro-symbolic AI</a></li>

<li><a href="Neuromorphic_engineering" class="mw-redirect" title="Neuromorphic engineering">Neuromorphic engineering</a></li>

<li><a href="Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Problems</div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Statistical_classification" title="Statistical classification">Classification</a></li>

<li><a href="Generative_model" title="Generative model">Generative modeling</a></li>

<li><a href="Regression_analysis" title="Regression analysis">Regression</a></li>

<li><a href="Cluster_analysis" title="Cluster analysis">Clustering</a></li>

<li><a href="Dimensionality_reduction" title="Dimensionality reduction">Dimensionality reduction</a></li>

<li><a href="Density_estimation" title="Density estimation">Density estimation</a></li>

<li><a href="Anomaly_detection" title="Anomaly detection">Anomaly detection</a></li>

<li><a href="Data_cleaning" class="mw-redirect" title="Data cleaning">Data cleaning</a></li>

<li><a href="Automated_machine_learning" title="Automated machine learning">AutoML</a></li>

<li><a href="Association_rule_learning" title="Association rule learning">Association rules</a></li>

<li><a href="Semantic_analysis_(machine_learning)" title="Semantic analysis (machine learning)">Semantic analysis</a></li>

<li><a href="Structured_prediction" title="Structured prediction">Structured prediction</a></li>

<li><a href="Feature_engineering" title="Feature engineering">Feature engineering</a></li>

<li><a href="Feature_learning" title="Feature learning">Feature learning</a></li>

<li><a href="Learning_to_rank" title="Learning to rank">Learning to rank</a></li>

<li><a href="Grammar_induction" title="Grammar induction">Grammar induction</a></li>

<li><a href="Ontology_learning" title="Ontology learning">Ontology learning</a></li>

<li><a href="Multimodal_learning" title="Multimodal learning">Multimodal learning</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="Supervised_learning" title="Supervised learning">Supervised learning</a><br><span class="nobold"><span style="font-size: 85%;">(<b><a href="Statistical_classification" title="Statistical classification">classification</a></b> • <b><a href="Regression_analysis" title="Regression analysis">regression</a></b>)</span></span> </div></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Apprenticeship_learning" title="Apprenticeship learning">Apprenticeship learning</a></li>

<li><a href="Decision_tree_learning" title="Decision tree learning">Decision trees</a></li>

<li><a href="Ensemble_learning" title="Ensemble learning">Ensembles</a>

<ul><li><a href="Bootstrap_aggregating" title="Bootstrap aggregating">Bagging</a></li>

<li><a href="Boosting_(machine_learning)" title="Boosting (machine learning)">Boosting</a></li>

<li><a href="Random_forest" title="Random forest">Random forest</a></li></ul></li>

<li><a href="Linear_regression" title="Linear regression">Linear regression</a></li>

<li><a href="Naive_Bayes_classifier" title="Naive Bayes classifier">Naive Bayes</a></li>

<li><a href="Artificial_neural_network" class="mw-redirect" title="Artificial neural network">Artificial neural networks</a></li>

<li><a href="Logistic_regression" title="Logistic regression">Logistic regression</a></li>

<li><a href="Perceptron" title="Perceptron">Perceptron</a></li>

<li><a href="Relevance_vector_machine" title="Relevance vector machine">Relevance vector machine (RVM)</a></li>

<li><a href="Support_vector_machine" title="Support vector machine">Support vector machine (SVM)</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="Cluster_analysis" title="Cluster analysis">Clustering</a></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="BIRCH" title="BIRCH">BIRCH</a></li>

<li><a href="Hierarchical_clustering" title="Hierarchical clustering">Hierarchical</a></li>

<li><a href="K-means_clustering" title="K-means clustering"><i>k</i>-means</a></li>

<li><a href="Fuzzy_clustering" title="Fuzzy clustering">Fuzzy</a></li>

<li><a href="Expectation%E2%80%93maximization_algorithm" title="Expectation–maximization algorithm">Expectation–maximization (EM)</a></li>

<li><br><a href="DBSCAN" title="DBSCAN">DBSCAN</a></li>

<li><a href="OPTICS_algorithm" title="OPTICS algorithm">OPTICS</a></li>

<li><a href="Mean_shift" title="Mean shift">Mean shift</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="Dimensionality_reduction" title="Dimensionality reduction">Dimensionality reduction</a></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Factor_analysis" title="Factor analysis">Factor analysis</a></li>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="Structured_prediction" title="Structured prediction">Structured prediction</a></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Graphical_model" title="Graphical model">Graphical models</a>

<ul><li><a href="Bayesian_network" title="Bayesian network">Bayes net</a></li>

<li><a href="Conditional_random_field" title="Conditional random field">Conditional random field</a></li>

<li><a href="Hidden_Markov_model" title="Hidden Markov model">Hidden Markov</a></li></ul></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="Anomaly_detection" title="Anomaly detection">Anomaly detection</a></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Random_sample_consensus" title="Random sample consensus">RANSAC</a></li>

<li><a href="Local_outlier_factor" title="Local outlier factor">Local outlier factor</a></li>

<li><a href="Isolation_forest" title="Isolation forest">Isolation forest</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="Neural_network_(machine_learning)" title="Neural network (machine learning)">Neural networks</a></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Autoencoder" title="Autoencoder">Autoencoder</a></li>

<li><a href="Deep_learning" title="Deep learning">Deep learning</a></li>

<li><a href="Feedforward_neural_network" title="Feedforward neural network">Feedforward neural network</a></li>

<li><a href="Recurrent_neural_network" title="Recurrent neural network">Recurrent neural network</a>

<li><a href="Reservoir_computing" title="Reservoir computing">reservoir computing</a></li></ul></li>

<li><a href="Boltzmann_machine" title="Boltzmann machine">Boltzmann machine</a>

<ul><li><a href="Restricted_Boltzmann_machine" title="Restricted Boltzmann machine">Restricted</a></li></ul></li>

<li><a href="Diffusion_model" title="Diffusion model">Diffusion model</a></li>

<li><a href="Convolutional_neural_network" title="Convolutional neural network">Convolutional neural network</a>

<li><a href="LeNet" title="LeNet">LeNet</a></li>

<li><a href="AlexNet" title="AlexNet">AlexNet</a></li>

<li><a href="DeepDream" title="DeepDream">DeepDream</a></li></ul></li>

<li><a href="Neural_field" title="Neural field">Neural field</a>

<ul><li><a href="Neural_radiance_field" title="Neural radiance field">Neural radiance field</a></li>

<li><a href="Physics-informed_neural_networks" title="Physics-informed neural networks">Physics-informed neural networks</a></li></ul></li>

<li><a href="Transformer_(deep_learning_architecture)" title="Transformer (deep learning architecture)">Transformer</a>

<ul><li><a href="Vision_transformer" title="Vision transformer">Vision</a></li></ul></li>

<li><a href="Mamba_(deep_learning_architecture)" title="Mamba (deep learning architecture)">Mamba</a></li>

<li><a href="Spiking_neural_network" title="Spiking neural network">Spiking neural network</a></li>

<li><a href="Memtransistor" title="Memtransistor">Memtransistor</a></li>

<li><a href="Electrochemical_RAM" title="Electrochemical RAM">Electrochemical RAM</a> (ECRAM)</li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="Reinforcement_learning" title="Reinforcement learning">Reinforcement learning</a></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Q-learning" title="Q-learning">Q-learning</a></li>

<li><a href="Policy_gradient_method" title="Policy gradient method">Policy gradient</a></li>

<li><a href="State%E2%80%93action%E2%80%93reward%E2%80%93state%E2%80%93action" title="State–action–reward–state–action">SARSA</a></li>

<li><a href="Temporal_difference_learning" title="Temporal difference learning">Temporal difference (TD)</a></li>

<li><a href="Multi-agent_reinforcement_learning" title="Multi-agent reinforcement learning">Multi-agent</a>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Learning with humans</div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Active_learning_(machine_learning)" title="Active learning (machine learning)">Active learning</a></li>

<li><a href="Crowdsourcing" title="Crowdsourcing">Crowdsourcing</a></li>

<li><a href="Human-in-the-loop" title="Human-in-the-loop">Human-in-the-loop</a></li>

<li><a href="Mechanistic_interpretability" title="Mechanistic interpretability">Mechanistic interpretability</a></li>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Model diagnostics</div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li>

<li><a href="Confusion_matrix" title="Confusion matrix">Confusion matrix</a></li>

<li><a href="Learning_curve_(machine_learning)" title="Learning curve (machine learning)">Learning curve</a></li>

<li><a href="Receiver_operating_characteristic" title="Receiver operating characteristic">ROC curve</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Mathematical foundations</div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Kernel_machines" class="mw-redirect" title="Kernel machines">Kernel machines</a></li>

<li><a href="Bias%E2%80%93variance_tradeoff" title="Bias–variance tradeoff">Bias–variance tradeoff</a></li>

<li><a href="Computational_learning_theory" title="Computational learning theory">Computational learning theory</a></li>

<li><a href="Empirical_risk_minimization" title="Empirical risk minimization">Empirical risk minimization</a></li>

<li><a href="Occam_learning" title="Occam learning">Occam learning</a></li>

<li><a href="Probably_approximately_correct_learning" title="Probably approximately correct learning">PAC learning</a></li>

<li><a href="Statistical_learning_theory" title="Statistical learning theory">Statistical learning</a></li>

<li><a href="Vapnik%E2%80%93Chervonenkis_theory" title="Vapnik–Chervonenkis theory">VC theory</a></li>

<li><a href="Topological_deep_learning" title="Topological deep learning">Topological deep learning</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Journals and conferences</div><div class="sidebar-list-content mw-collapsible-content hlist">

<li><a href="Conference_on_Neural_Information_Processing_Systems" title="Conference on Neural Information Processing Systems">NeurIPS</a></li>

<li><a href="International_Joint_Conference_on_Artificial_Intelligence" title="International Joint Conference on Artificial Intelligence">IJCAI</a></li>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Related articles</div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Glossary_of_artificial_intelligence" title="Glossary of artificial intelligence">Glossary of artificial intelligence</a></li>

<li><a href="List_of_datasets_for_machine-learning_research" title="List of datasets for machine-learning research">List of datasets for machine-learning research</a>

<ul><li><a href="List_of_datasets_in_computer_vision_and_image_processing" title="List of datasets in computer vision and image processing">List of datasets in computer vision and image processing</a></li></ul></li>

<li><a href="Outline_of_machine_learning" title="Outline of machine learning">Outline of machine learning</a></li></ul></div></div></td>

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<p>In <a href="Machine_learning" title="Machine learning">machine learning</a>, <b>kernel machines</b> are a class of algorithms for <a href="Pattern_analysis" class="mw-redirect" title="Pattern analysis">pattern analysis</a>, whose best known member is the <a href="Support-vector_machine" class="mw-redirect" title="Support-vector machine">support-vector machine</a> (SVM). These methods involve using linear classifiers to solve nonlinear problems.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The general task of <a href="Pattern_analysis" class="mw-redirect" title="Pattern analysis">pattern analysis</a> is to find and study general types of relations (for example <a href="Cluster_analysis" title="Cluster analysis">clusters</a>, <a href="Ranking" title="Ranking">rankings</a>, <a href="Principal_components" class="mw-redirect" title="Principal components">principal components</a>, <a href="Correlation" title="Correlation">correlations</a>, <a href="Statistical_classification" title="Statistical classification">classifications</a>) in datasets. For many algorithms that solve these tasks, the data in raw representation have to be explicitly transformed into <a href="Feature_vector" class="mw-redirect" title="Feature vector">feature vector</a> representations via a user-specified <i>feature map</i>: in contrast, kernel methods require only a user-specified <i>kernel</i>, i.e., a <a href="Similarity_function" class="mw-redirect" title="Similarity function">similarity function</a> over all pairs of data points computed using <a href="Inner_products" class="mw-redirect" title="Inner products">inner products</a>. The feature map in kernel machines is infinite dimensional but only requires a finite dimensional matrix from user-input according to the <a href="Representer_theorem" title="Representer theorem">representer theorem</a>. Kernel machines are slow to compute for datasets larger than a couple of thousand examples without parallel processing.

</p><p>Kernel methods owe their name to the use of <a href="Positive-definite_kernel" title="Positive-definite kernel">kernel functions</a>, which enable them to operate in a high-dimensional, <i>implicit</i> <a href="Feature_space" class="mw-redirect" title="Feature space">feature space</a> without ever computing the coordinates of the data in that space, but rather by simply computing the <a href="Inner_product" class="mw-redirect" title="Inner product">inner products</a> between the <a href="Image_(mathematics)" title="Image (mathematics)">images</a> of all pairs of data in the feature space. This operation is often computationally cheaper than the explicit computation of the coordinates. This approach is called the "<b>kernel trick</b>".<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Kernel functions have been introduced for sequence data, <a href="Graph_kernel" title="Graph kernel">graphs</a>, text, images, as well as vectors.

</p><p>Algorithms capable of operating with kernels include the <a href="Kernel_perceptron" title="Kernel perceptron">kernel perceptron</a>, support-vector machines (SVM), <a href="Gaussian_process" title="Gaussian process">Gaussian processes</a>, <a href="Principal_components_analysis" class="mw-redirect" title="Principal components analysis">principal components analysis</a> (PCA), <a href="Canonical_correlation_analysis" class="mw-redirect" title="Canonical correlation analysis">canonical correlation analysis</a>, <a href="Ridge_regression" title="Ridge regression">ridge regression</a>, <a href="Spectral_clustering" title="Spectral clustering">spectral clustering</a>, <a href="Adaptive_filter" title="Adaptive filter">linear adaptive filters</a> and many others.

</p><p>Most kernel algorithms are based on <a href="Convex_optimization" title="Convex optimization">convex optimization</a> or <a href="Eigenvalue%2C_eigenvector_and_eigenspace" class="mw-redirect" title="Eigenvalue, eigenvector and eigenspace">eigenproblems</a> and are statistically well-founded. Typically, their statistical properties are analyzed using <a href="Statistical_learning_theory" title="Statistical learning theory">statistical learning theory</a> (for example, using <a href="Rademacher_complexity" title="Rademacher complexity">Rademacher complexity</a>).

</p>

<div class="mw-heading mw-heading2"><h2 id="Motivation_and_informal_explanation">Motivation and informal explanation</h2></div>

<p>Kernel methods can be thought of as <a href="Instance-based_learning" title="Instance-based learning">instance-based learners</a>: rather than learning some fixed set of parameters corresponding to the features of their inputs, they instead "remember" the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

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

</semantics>

</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span>-th training example <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {x} _{i},y_{i})}">

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

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</math></span><img src="./b1391d404dfff362732744fbced84d474a44e1c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.993ex; height:2.843ex;" alt="{\displaystyle (\mathbf {x} _{i},y_{i})}" loading="lazy"></span> and learn for it a corresponding weight <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{i}}">

<msub>

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

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</math></span><img src="./fe22f0329d3ecb2e1880d44d191aba0e5475db68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.464ex; height:2.009ex;" alt="{\displaystyle w_{i}}" loading="lazy"></span>. Prediction for unlabeled inputs, i.e., those not in the training set, is treated by the application of a <a href="Similarity_function" class="mw-redirect" title="Similarity function">similarity function</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

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</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>, called a <b>kernel</b>, between the unlabeled input <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x'} }">

<msup>

</msup>

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</mstyle>

</mrow>

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

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</math></span><img src="./7d14ab6186e99346cb608a30858c3e1580f760e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.096ex; height:2.509ex;" alt="{\displaystyle \mathbf {x'} }" loading="lazy"></span> and each of the training inputs <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} _{i}}">

<msub>

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</mrow>

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</mrow>

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

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</math></span><img src="./57d2ef3df60acdb53bdf90535264041fea7231cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.211ex; height:2.009ex;" alt="{\displaystyle \mathbf {x} _{i}}" loading="lazy"></span>. For instance, a kernelized <a href="Binary_classifier" class="mw-redirect" title="Binary classifier">binary classifier</a> typically computes a weighted sum of similarities

<mover>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\hat {y}}=\operatorname {sgn} \sum _{i=1}^{n}w_{i}y_{i}k(\mathbf {x} _{i},\mathbf {x'} ),}</annotation>

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where

</p>

<mover>

</mover>

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<annotation encoding="application/x-tex">{\displaystyle {\hat {y}}\in \{-1,+1\}}</annotation>

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</math></span><img src="./3ff919c29b83be4191b91d3067b89a217401341d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.443ex; height:2.843ex;" alt="{\displaystyle {\hat {y}}\in \{-1,+1\}}" loading="lazy"></span> is the kernelized binary classifier's predicted label for the unlabeled input <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x'} }">

<msup>

</msup>

</mrow>

</mstyle>

</mrow>

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

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</math></span><img src="./7d14ab6186e99346cb608a30858c3e1580f760e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.096ex; height:2.509ex;" alt="{\displaystyle \mathbf {x'} }" loading="lazy"></span> whose hidden true label <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

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

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</math></span><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span> is of interest;</li>

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<annotation encoding="application/x-tex">{\displaystyle k\colon {\mathcal {X}}\times {\mathcal {X}}\to \mathbb {R} }</annotation>

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</math></span><img src="./ecc3c27f7e04f4ce7c0088a69e0b414a74869e3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.129ex; height:2.176ex;" alt="{\displaystyle k\colon {\mathcal {X}}\times {\mathcal {X}}\to \mathbb {R} }" loading="lazy"></span> is the kernel function that measures similarity between any pair of inputs <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} ,\mathbf {x'} \in {\mathcal {X}}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} ,\mathbf {x'} \in {\mathcal {X}}}</annotation>

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</math></span><img src="./f029197c8bd56f41a73dca35dc1afccc8baaa665.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.257ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} ,\mathbf {x'} \in {\mathcal {X}}}" loading="lazy"></span>;</li>

<li>the sum ranges over the <span class="texhtml mvar" style="font-style:italic;">n</span> labeled examples <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(\mathbf {x} _{i},y_{i})\}_{i=1}^{n}}">

<msub>

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

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</math></span><img src="./fa50701446bc8bb883c6aa304144373d240c1c85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.218ex; height:3.009ex;" alt="{\displaystyle \{(\mathbf {x} _{i},y_{i})\}_{i=1}^{n}}" loading="lazy"></span> in the classifier's training set, with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}\in \{-1,+1\}}">

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{i}\in \{-1,+1\}}</annotation>

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</math></span><img src="./287b80f546eb07ca26f570566d3a0f71bf237e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.08ex; height:2.843ex;" alt="{\displaystyle y_{i}\in \{-1,+1\}}" loading="lazy"></span>;</li>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle w_{i}\in \mathbb {R} }</annotation>

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</math></span><img src="./bd3678bb35c0b03cbbd918b869c2461838a1c98f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.983ex; height:2.509ex;" alt="{\displaystyle w_{i}\in \mathbb {R} }" loading="lazy"></span> are the weights for the training examples, as determined by the learning algorithm;</li>

<li>the <a href="Sign_function" title="Sign function">sign function</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn} }">

</mstyle>

</mrow>

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

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</math></span><img src="./ec838dfd8a4a659b2877f93a6b53f22fc7777d07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.009ex;" alt="{\displaystyle \operatorname {sgn} }" loading="lazy"></span> determines whether the predicted classification <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {y}}}">

<mover>

</mover>

</mrow>

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

</semantics>

</math></span><img src="./3dc8de3d8ea01304329ef9518fad7a6d196c4c01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.302ex; height:2.509ex;" alt="{\displaystyle {\hat {y}}}" loading="lazy"></span> comes out positive or negative.</li></ul>

<p>Kernel classifiers were described as early as the 1960s, with the invention of the <a href="Kernel_perceptron" title="Kernel perceptron">kernel perceptron</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> They rose to great prominence with the popularity of the <a href="Support-vector_machine" class="mw-redirect" title="Support-vector machine">support-vector machine</a> (SVM) in the 1990s, when the SVM was found to be competitive with <a href="Artificial_neural_network" class="mw-redirect" title="Artificial neural network">neural networks</a> on tasks such as <a href="Handwriting_recognition" title="Handwriting recognition">handwriting recognition</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Mathematics:_the_kernel_trick">Mathematics: the kernel trick</h2></div>

<p>The kernel trick avoids the explicit mapping that is needed to get linear <a href="Learning_algorithms" class="mw-redirect" title="Learning algorithms">learning algorithms</a> to learn a nonlinear function or <a href="Decision_boundary" title="Decision boundary">decision boundary</a>. For all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x'} }">

<msup>

</msup>

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</mstyle>

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

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</math></span><img src="./7d14ab6186e99346cb608a30858c3e1580f760e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.096ex; height:2.509ex;" alt="{\displaystyle \mathbf {x'} }" loading="lazy"></span> in the input space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {X}}}">

</mrow>

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

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</math></span><img src="./8c7e5461c5286852df4ef652fca7e4b0b63030e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.875ex; height:2.176ex;" alt="{\displaystyle {\mathcal {X}}}" loading="lazy"></span>, certain functions <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k(\mathbf {x} ,\mathbf {x'} )}">

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<annotation encoding="application/x-tex">{\displaystyle k(\mathbf {x} ,\mathbf {x'} )}</annotation>

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</math></span><img src="./7d02f87329f893c16295074bcfe9d974fb72c4eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.561ex; height:3.009ex;" alt="{\displaystyle k(\mathbf {x} ,\mathbf {x'} )}" loading="lazy"></span> can be expressed as an <a href="Inner_product" class="mw-redirect" title="Inner product">inner product</a> in another space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {V}}}">

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

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</math></span><img src="./47d69f309b6deb2e5008f6130ee11e09bbabd7b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.529ex; height:2.176ex;" alt="{\displaystyle {\mathcal {V}}}" loading="lazy"></span>. The function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\colon {\mathcal {X}}\times {\mathcal {X}}\to \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle k\colon {\mathcal {X}}\times {\mathcal {X}}\to \mathbb {R} }</annotation>

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</math></span><img src="./ecc3c27f7e04f4ce7c0088a69e0b414a74869e3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.129ex; height:2.176ex;" alt="{\displaystyle k\colon {\mathcal {X}}\times {\mathcal {X}}\to \mathbb {R} }" loading="lazy"></span> is often referred to as a <i>kernel</i> or a <i><a href="Kernel_function" class="mw-redirect" title="Kernel function">kernel function</a></i>. The word "kernel" is used in mathematics to denote a weighting function for a weighted sum or <a href="Integral" title="Integral">integral</a>.

</p><p>Certain problems in machine learning have more structure than an arbitrary weighting function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>. The computation is made much simpler if the kernel can be written in the form of a "feature map" <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \colon {\mathcal {X}}\to {\mathcal {V}}}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi \colon {\mathcal {X}}\to {\mathcal {V}}}</annotation>

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</math></span><img src="./b8484d2e5a2fb0ed38f151079dceaca4a395eca5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.573ex; height:2.676ex;" alt="{\displaystyle \varphi \colon {\mathcal {X}}\to {\mathcal {V}}}" loading="lazy"></span> which satisfies

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</msup>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k(\mathbf {x} ,\mathbf {x'} )=\langle \varphi (\mathbf {x} ),\varphi (\mathbf {x'} )\rangle _{\mathcal {V}}.}</annotation>

</semantics>

</math></span></span>The key restriction is that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \cdot ,\cdot \rangle _{\mathcal {V}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle \cdot ,\cdot \rangle _{\mathcal {V}}}</annotation>

</semantics>

</math></span><img src="./cfaf5152d8c0788782803dc35974e62634f7a635.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.451ex; height:2.843ex;" alt="{\displaystyle \langle \cdot ,\cdot \rangle _{\mathcal {V}}}" loading="lazy"></span> must be a proper inner product. On the other hand, an explicit representation for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> is not necessary, as long as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {V}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./47d69f309b6deb2e5008f6130ee11e09bbabd7b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.529ex; height:2.176ex;" alt="{\displaystyle {\mathcal {V}}}" loading="lazy"></span> is an <a href="Inner_product_space" title="Inner product space">inner product space</a>. The alternative follows from <a href="Mercer's_theorem" title="Mercer's theorem">Mercer's theorem</a>: an implicitly defined function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> exists whenever the space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {X}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./8c7e5461c5286852df4ef652fca7e4b0b63030e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.875ex; height:2.176ex;" alt="{\displaystyle {\mathcal {X}}}" loading="lazy"></span> can be equipped with a suitable <a href="Measure_(mathematics)" title="Measure (mathematics)">measure</a> ensuring the function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> satisfies <a href="Mercer's_condition" class="mw-redirect" title="Mercer's condition">Mercer's condition</a>.

</p><p>Mercer's theorem is similar to a generalization of the result from linear algebra that <a href="Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">associates an inner product to any positive-definite matrix</a>. In fact, Mercer's condition can be reduced to this simpler case. If we choose as our measure the <a href="Counting_measure" title="Counting measure">counting measure</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (T)=|T|}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./a44af909e4eab9daa53ba7b9e6901c8df5bf2bc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.876ex; height:2.843ex;" alt="{\displaystyle \mu (T)=|T|}" loading="lazy"></span> for all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\subset X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T\subset X}</annotation>

</semantics>

</math></span><img src="./28eb68af8d21de80992dc26e6ee9b6a99f5c54f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.715ex; height:2.176ex;" alt="{\displaystyle T\subset X}" loading="lazy"></span>, which counts the number of points inside the set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"></span>, then the integral in Mercer's theorem reduces to a summation<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}\sum _{j=1}^{n}k(\mathbf {x} _{i},\mathbf {x} _{j})c_{i}c_{j}\geq 0.}">

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}\sum _{j=1}^{n}k(\mathbf {x} _{i},\mathbf {x} _{j})c_{i}c_{j}\geq 0.}</annotation>

</semantics>

</math></span></span>If this summation holds for all finite sequences of points <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {x} _{1},\dotsc ,\mathbf {x} _{n})}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbf {x} _{1},\dotsc ,\mathbf {x} _{n})}</annotation>

</semantics>

</math></span><img src="./5b4e4c8cc45e704f262c34fda4e3a3fa52754d0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.082ex; height:2.843ex;" alt="{\displaystyle (\mathbf {x} _{1},\dotsc ,\mathbf {x} _{n})}" loading="lazy"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {X}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./8c7e5461c5286852df4ef652fca7e4b0b63030e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.875ex; height:2.176ex;" alt="{\displaystyle {\mathcal {X}}}" loading="lazy"></span> and all choices of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> real-valued coefficients <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c_{1},\dots ,c_{n})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (c_{1},\dots ,c_{n})}</annotation>

</semantics>

</math></span><img src="./bd731fbc6215cae64c53bf0120c4ebfee01d3f96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.274ex; height:2.843ex;" alt="{\displaystyle (c_{1},\dots ,c_{n})}" loading="lazy"></span> (cf. <a href="Positive_definite_kernel" class="mw-redirect" title="Positive definite kernel">positive definite kernel</a>), then the function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> satisfies Mercer's condition.

</p><p>Some algorithms that depend on arbitrary relationships in the native space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {X}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./8c7e5461c5286852df4ef652fca7e4b0b63030e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.875ex; height:2.176ex;" alt="{\displaystyle {\mathcal {X}}}" loading="lazy"></span> would, in fact, have a linear interpretation in a different setting: the range space of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span>. The linear interpretation gives us insight about the algorithm. Furthermore, there is often no need to compute <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> directly during computation, as is the case with <a href="Support-vector_machine" class="mw-redirect" title="Support-vector machine">support-vector machines</a>. Some cite this running time shortcut as the primary benefit. Researchers also use it to justify the meanings and properties of existing algorithms.

</p><p>Theoretically, a <a href="Gram_matrix" title="Gram matrix">Gram matrix</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {K} \in \mathbb {R} ^{n\times n}}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./5ddf49f743a8541a3c6812638951cc6d13015d07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.096ex; height:2.343ex;" alt="{\displaystyle \mathbf {K} \in \mathbb {R} ^{n\times n}}" loading="lazy"></span> with respect to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\mathbf {x} _{1},\dotsc ,\mathbf {x} _{n}\}}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{\mathbf {x} _{1},\dotsc ,\mathbf {x} _{n}\}}</annotation>

</semantics>

</math></span><img src="./8246b26ae9914d260265dfbea03c55be5e8d00b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.598ex; height:2.843ex;" alt="{\displaystyle \{\mathbf {x} _{1},\dotsc ,\mathbf {x} _{n}\}}" loading="lazy"></span> (sometimes also called a "kernel matrix"<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>), where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{ij}=k(\mathbf {x} _{i},\mathbf {x} _{j})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{ij}=k(\mathbf {x} _{i},\mathbf {x} _{j})}</annotation>

</semantics>

</math></span><img src="./bd50a73f3c68c1ec4fad86ce50b4c413b22b075e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.134ex; height:3.009ex;" alt="{\displaystyle K_{ij}=k(\mathbf {x} _{i},\mathbf {x} _{j})}" loading="lazy"></span>, must be <a href="Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">positive semi-definite (PSD)</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Empirically, for machine learning heuristics, choices of a function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> that do not satisfy Mercer's condition may still perform reasonably if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> at least approximates the intuitive idea of similarity.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Regardless of whether <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> is a Mercer kernel, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> may still be referred to as a "kernel".

</p><p>If the kernel function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> is also a <a href="Covariance_function" title="Covariance function">covariance function</a> as used in <a href="Gaussian_processes" class="mw-redirect" title="Gaussian processes">Gaussian processes</a>, then the Gram matrix <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {K} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./368b3827262016c64b340a761d9b95e0b031d6dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.094ex; height:2.176ex;" alt="{\displaystyle \mathbf {K} }" loading="lazy"></span> can also be called a <a href="Covariance_matrix" title="Covariance matrix">covariance matrix</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p>

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

<p>Application areas of kernel methods are diverse and include <a href="Geostatistics" title="Geostatistics">geostatistics</a>,<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> <a href="Kriging" title="Kriging">kriging</a>, <a href="Inverse_distance_weighting" title="Inverse distance weighting">inverse distance weighting</a>, <a href="3D_reconstruction" title="3D reconstruction">3D reconstruction</a>, <a href="Bioinformatics" title="Bioinformatics">bioinformatics</a>, <a href="Cheminformatics" title="Cheminformatics">cheminformatics</a>, <a href="Information_extraction" title="Information extraction">information extraction</a> and <a href="Handwriting_recognition" title="Handwriting recognition">handwriting recognition</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Popular_kernels">Popular kernels</h2></div>

<ul><li><a href="Fisher_kernel" title="Fisher kernel">Fisher kernel</a></li>

<li><a href="Graph_kernel" title="Graph kernel">Graph kernels</a></li>

<li><a href="Kernel_smoother" title="Kernel smoother">Kernel smoother</a></li>

<li><a href="Polynomial_kernel" title="Polynomial kernel">Polynomial kernel</a></li>

<li><a href="Radial_basis_function_kernel" title="Radial basis function kernel">Radial basis function kernel</a> (RBF)</li>

<li><a href="String_kernel" title="String kernel">String kernels</a></li>

<li><a href="Neural_tangent_kernel" title="Neural tangent kernel">Neural tangent kernel</a></li>

<li><a href="Neural_network_Gaussian_process" title="Neural network Gaussian process">Neural network Gaussian process</a> (NNGP) kernel</li></ul>

<ul><li><a href="Kernel_methods_for_vector_output" title="Kernel methods for vector output">Kernel methods for vector output</a></li>

<li><a href="Kernel_density_estimation" title="Kernel density estimation">Kernel density estimation</a></li>

<li><a href="Representer_theorem" title="Representer theorem">Representer theorem</a></li>

<li><a href="Similarity_learning" title="Similarity learning">Similarity learning</a></li>

<li><a href="Cover's_theorem" title="Cover's theorem">Cover's theorem</a></li></ul>

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

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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.engati.com/glossary/kernel-method">"Kernel method"</a>. <i>Engati</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-04-04</span></span>.</cite></span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFTheodoridis2008" class="citation book cs1">Theodoridis, Sergios (2008). <i>Pattern Recognition</i>. Elsevier B.V. p. 203. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780080949123</bdi>.</cite></span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFAizermanBravermanRozonoer1964" class="citation journal cs1">Aizerman, M. A.; Braverman, Emmanuel M.; Rozonoer, L. I. (1964). "Theoretical foundations of the potential function method in pattern recognition learning". <i>Automation and Remote Control</i>. <b>25</b>: <span class="nowrap">821–</span>837.</cite> Cited in <cite id="CITEREFGuyonBoserVapnik1993" class="citation conference cs1">Guyon, Isabelle; Boser, B.; Vapnik, Vladimir (1993). <i>Automatic capacity tuning of very large VC-dimension classifiers</i>. Advances in neural information processing systems. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.17.7215">10.1.1.17.7215</a></span>.</cite></span>

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<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFHofmannSchölkopfSmola2008" class="citation journal cs1">Hofmann, Thomas; Schölkopf, Bernhard; Smola, Alexander J. (2008). <a rel="nofollow" class="external text" href="https://projecteuclid.org/download/pdfview_1/euclid.aos/1211819561">"Kernel Methods in Machine Learning"</a>. <i>The Annals of Statistics</i>. <b>36</b> (3). <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0701907">math/0701907</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2F009053607000000677">10.1214/009053607000000677</a></span>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:88516979">88516979</a>.</cite></span>

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<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFMohriRostamizadehTalwalkar2012" class="citation book cs1"><a href="Mehryar_Mohri" title="Mehryar Mohri">Mohri, Mehryar</a>; Rostamizadeh, Afshin; Talwalkar, Ameet (2012). <i>Foundations of Machine Learning</i>. US, Massachusetts: MIT Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780262018258</bdi>.</cite></span>

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<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFSewell" class="citation web cs1">Sewell, Martin. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20181015031456/http://www.svms.org/mercer/">"Support Vector Machines: Mercer's Condition"</a>. Support Vector Machines. Archived from <a rel="nofollow" class="external text" href="http://www.svms.org/mercer/">the original</a> on 2018-10-15<span class="reference-accessdate">. Retrieved <span class="nowrap">2014-05-30</span></span>.</cite></span>

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<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFRasmussenWilliams2006" class="citation book cs1">Rasmussen, Carl Edward; Williams, Christopher K. I. (2006). <i>Gaussian Processes for Machine Learning</i>. MIT Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-262-18253-X</bdi>.</cite> </span>

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<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFHonarkhahCaers2010" class="citation journal cs1">Honarkhah, M.; Caers, J. (2010). "Stochastic Simulation of Patterns Using Distance-Based Pattern Modeling". <i><a href="Mathematical_Geosciences" title="Mathematical Geosciences">Mathematical Geosciences</a></i>. <b>42</b> (5): <span class="nowrap">487–</span>517. <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/2010MaGeo..42..487H">2010MaGeo..42..487H</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%2Fs11004-010-9276-7">10.1007/s11004-010-9276-7</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:73657847">73657847</a>.</cite></span>

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<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFShawe-TaylorCristianini2004" class="citation book cs1"><a href="John_Shawe-Taylor" title="John Shawe-Taylor">Shawe-Taylor, J.</a>; <a href="Nello_Cristianini" title="Nello Cristianini">Cristianini, N.</a> (2004). <i>Kernel Methods for Pattern Analysis</i>. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780511809682</bdi>.</cite></li>

<li><cite id="CITEREFLiuPrincipeHaykin2010" class="citation book cs1">Liu, W.; Principe, J.; Haykin, S. (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=eWUwB_P5pW0C"><i>Kernel Adaptive Filtering: A Comprehensive Introduction</i></a>. Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781118211212</bdi>.</cite></li>

<li><cite id="CITEREFSchölkopfSmolaBach2018" class="citation book cs1"><a href="Bernhard_Sch%C3%B6lkopf" title="Bernhard Schölkopf">Schölkopf, B.</a>; Smola, A. J.; Bach, F. (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZQxiuAEACAAJ"><i>Learning with Kernels : Support Vector Machines, Regularization, Optimization, and Beyond</i></a>. MIT Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-262-53657-8</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.kernel-machines.org">Kernel-Machines Org</a>—community website</li>

<li><a rel="nofollow" class="external text" href="http://onlineprediction.net/?n=Main.KernelMethods">onlineprediction.net Kernel Methods Article</a></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-08-03" href="https://en.wikipedia.org/wiki/?title=Kernel_method&oldid=1304048486">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.

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