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<h1 id="firstHeading" class="firstHeading" lang="en">t-distributed stochastic neighbor embedding</h1>
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<div id="mw-content-text" lang="en" dir="ltr" class="mw-content-ltr"><div class="mw-parser-output"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Technique for dimensionality reduction</div>
<div role="note" class="hatnote navigation-not-searchable">"TSNE" redirects here. For the Boston-based organization, see <a href="/wiki/Third_Sector_New_England" title="Third Sector New England">Third Sector New England</a>.</div>
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<table class="vertical-navbox nowraplinks" style="float:right;clear:right;width:22.0em;margin:0 0 1.0em 1.0em;background:#f9f9f9;border:1px solid #aaa;padding:0.2em;border-spacing:0.4em 0;text-align:center;line-height:1.4em;font-size:88%"><tbody><tr><th style="padding:0.2em 0.4em 0.2em;font-size:145%;line-height:1.2em"><a href="/wiki/Machine_learning" title="Machine learning">Machine learning</a> and<br /><a href="/wiki/Data_mining" title="Data mining">data mining</a></th></tr><tr><td style="padding:0.2em 0 0.4em;padding:0.25em 0.25em 0.75em;"><a href="/wiki/File:Kernel_Machine.svg" class="image"><img alt="Kernel Machine.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Kernel_Machine.svg/220px-Kernel_Machine.svg.png" decoding="async" width="220" height="100" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Kernel_Machine.svg/330px-Kernel_Machine.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Kernel_Machine.svg/440px-Kernel_Machine.svg.png 2x" data-file-width="512" data-file-height="233" /></a></td></tr><tr><td style="padding:0 0.1em 0.4em">
<div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left">Problems</div><div class="NavContent" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"><div class="hlist">
<ul><li><a href="/wiki/Statistical_classification" title="Statistical classification">Classification</a></li>
<li><a href="/wiki/Cluster_analysis" title="Cluster analysis">Clustering</a></li>
<li><a href="/wiki/Regression_analysis" title="Regression analysis">Regression</a></li>
<li><a href="/wiki/Anomaly_detection" title="Anomaly detection">Anomaly detection</a></li>
<li><a href="/wiki/Automated_machine_learning" title="Automated machine learning">AutoML</a></li>
<li><a href="/wiki/Association_rule_learning" title="Association rule learning">Association rules</a></li>
<li><a href="/wiki/Reinforcement_learning" title="Reinforcement learning">Reinforcement learning</a></li>
<li><a href="/wiki/Structured_prediction" title="Structured prediction">Structured prediction</a></li>
<li><a href="/wiki/Feature_engineering" title="Feature engineering">Feature engineering</a></li>
<li><a href="/wiki/Feature_learning" title="Feature learning">Feature learning</a></li>
<li><a href="/wiki/Online_machine_learning" title="Online machine learning">Online learning</a></li>
<li><a href="/wiki/Semi-supervised_learning" title="Semi-supervised learning">Semi-supervised learning</a></li>
<li><a href="/wiki/Unsupervised_learning" title="Unsupervised learning">Unsupervised learning</a></li>
<li><a href="/wiki/Learning_to_rank" title="Learning to rank">Learning to rank</a></li>
<li><a href="/wiki/Grammar_induction" title="Grammar induction">Grammar induction</a></li></ul>
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</tr><tr><td style="padding:0 0.1em 0.4em">
<div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left"><div style="padding:0.1em 0;line-height:1.2em;"><a href="/wiki/Supervised_learning" title="Supervised learning">Supervised learning</a><br /><style data-mw-deduplicate="TemplateStyles:r886047488">.mw-parser-output .nobold{font-weight:normal}</style><span class="nobold"><span style="font-size:85%;">(<b><a href="/wiki/Statistical_classification" title="Statistical classification">classification</a></b>&#160;&#8226;&#32;<b><a href="/wiki/Regression_analysis" title="Regression analysis">regression</a></b>)</span></span> </div></div><div class="NavContent" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"><div class="hlist">
<ul><li><a href="/wiki/Decision_tree_learning" title="Decision tree learning">Decision trees</a></li>
<li><a href="/wiki/Ensemble_learning" title="Ensemble learning">Ensembles</a>
<ul><li><a href="/wiki/Bootstrap_aggregating" title="Bootstrap aggregating">Bagging</a></li>
<li><a href="/wiki/Boosting_(machine_learning)" title="Boosting (machine learning)">Boosting</a></li>
<li><a href="/wiki/Random_forest" title="Random forest">Random forest</a></li></ul></li>
<li><a href="/wiki/K-nearest_neighbors_algorithm" title="K-nearest neighbors algorithm"><i>k</i>-NN</a></li>
<li><a href="/wiki/Linear_regression" title="Linear regression">Linear regression</a></li>
<li><a href="/wiki/Naive_Bayes_classifier" title="Naive Bayes classifier">Naive Bayes</a></li>
<li><a href="/wiki/Artificial_neural_network" title="Artificial neural network">Artificial neural networks</a></li>
<li><a href="/wiki/Logistic_regression" title="Logistic regression">Logistic regression</a></li>
<li><a href="/wiki/Perceptron" title="Perceptron">Perceptron</a></li>
<li><a href="/wiki/Relevance_vector_machine" title="Relevance vector machine">Relevance vector machine (RVM)</a></li>
<li><a href="/wiki/Support-vector_machine" title="Support-vector machine">Support vector machine (SVM)</a></li></ul>
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<div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left"><a href="/wiki/Cluster_analysis" title="Cluster analysis">Clustering</a></div><div class="NavContent" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"><div class="hlist">
<ul><li><a href="/wiki/BIRCH" title="BIRCH">BIRCH</a></li>
<li><a href="/wiki/CURE_data_clustering_algorithm" class="mw-redirect" title="CURE data clustering algorithm">CURE</a></li>
<li><a href="/wiki/Hierarchical_clustering" title="Hierarchical clustering">Hierarchical</a></li>
<li><a href="/wiki/K-means_clustering" title="K-means clustering"><i>k</i>-means</a></li>
<li><a href="/wiki/Expectation%E2%80%93maximization_algorithm" title="Expectation–maximization algorithm">Expectation–maximization (EM)</a></li>
<li><br /><a href="/wiki/DBSCAN" title="DBSCAN">DBSCAN</a></li>
<li><a href="/wiki/OPTICS_algorithm" title="OPTICS algorithm">OPTICS</a></li>
<li><a href="/wiki/Mean-shift" class="mw-redirect" title="Mean-shift">Mean-shift</a></li></ul>
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<div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left"><a href="/wiki/Dimensionality_reduction" title="Dimensionality reduction">Dimensionality reduction</a></div><div class="NavContent" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"><div class="hlist">
<ul><li><a href="/wiki/Factor_analysis" title="Factor analysis">Factor analysis</a></li>
<li><a href="/wiki/Canonical_correlation_analysis" class="mw-redirect" title="Canonical correlation analysis">CCA</a></li>
<li><a href="/wiki/Independent_component_analysis" title="Independent component analysis">ICA</a></li>
<li><a href="/wiki/Linear_discriminant_analysis" title="Linear discriminant analysis">LDA</a></li>
<li><a href="/wiki/Non-negative_matrix_factorization" title="Non-negative matrix factorization">NMF</a></li>
<li><a href="/wiki/Principal_component_analysis" title="Principal component analysis">PCA</a></li>
<li><a class="mw-selflink selflink">t-SNE</a></li></ul>
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<div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left"><a href="/wiki/Structured_prediction" title="Structured prediction">Structured prediction</a></div><div class="NavContent" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"><div class="hlist">
<ul><li><a href="/wiki/Graphical_model" title="Graphical model">Graphical models</a>
<ul><li><a href="/wiki/Bayesian_network" title="Bayesian network">Bayes net</a></li>
<li><a href="/wiki/Conditional_random_field" title="Conditional random field">Conditional random field</a></li>
<li><a href="/wiki/Hidden_Markov_model" title="Hidden Markov model">Hidden Markov</a></li></ul></li></ul>
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<div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left"><a href="/wiki/Anomaly_detection" title="Anomaly detection">Anomaly detection</a></div><div class="NavContent" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"><div class="hlist">
<ul><li><a href="/wiki/K-nearest_neighbors_classification" class="mw-redirect" title="K-nearest neighbors classification"><i>k</i>-NN</a></li>
<li><a href="/wiki/Local_outlier_factor" title="Local outlier factor">Local outlier factor</a></li></ul>
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<div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left"><a href="/wiki/Artificial_neural_networks" class="mw-redirect" title="Artificial neural networks">Artificial neural networks</a></div><div class="NavContent" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"><div class="hlist">
<ul><li><a href="/wiki/Autoencoder" title="Autoencoder">Autoencoder</a></li>
<li><a href="/wiki/Deep_learning" title="Deep learning">Deep learning</a></li>
<li><a href="/wiki/DeepDream" title="DeepDream">DeepDream</a></li>
<li><a href="/wiki/Multilayer_perceptron" title="Multilayer perceptron">Multilayer perceptron</a></li>
<li><a href="/wiki/Recurrent_neural_network" title="Recurrent neural network">RNN</a>
<ul><li><a href="/wiki/Long_short-term_memory" title="Long short-term memory">LSTM</a></li>
<li><a href="/wiki/Gated_recurrent_unit" title="Gated recurrent unit">GRU</a></li></ul></li>
<li><a href="/wiki/Restricted_Boltzmann_machine" title="Restricted Boltzmann machine">Restricted Boltzmann machine</a></li>
<li><a href="/wiki/Generative_adversarial_network" title="Generative adversarial network">GAN</a></li>
<li><a href="/wiki/Self-organizing_map" title="Self-organizing map">SOM</a></li>
<li><a href="/wiki/Convolutional_neural_network" title="Convolutional neural network">Convolutional neural network</a>
<ul><li><a href="/wiki/U-Net" title="U-Net">U-Net</a></li></ul></li></ul>
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<div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left"><a href="/wiki/Reinforcement_learning" title="Reinforcement learning">Reinforcement learning</a></div><div class="NavContent" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"><div class="hlist">
<ul><li><a href="/wiki/Q-learning" title="Q-learning">Q-learning</a></li>
<li><a href="/wiki/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="/wiki/Temporal_difference_learning" title="Temporal difference learning">Temporal difference (TD)</a></li></ul>
</div></div></div></td>
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<div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left">Theory</div><div class="NavContent" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"><div class="hlist">
<ul><li><a href="/wiki/Bias%E2%80%93variance_dilemma" class="mw-redirect" title="Bias–variance dilemma">Bias–variance dilemma</a></li>
<li><a href="/wiki/Computational_learning_theory" title="Computational learning theory">Computational learning theory</a></li>
<li><a href="/wiki/Empirical_risk_minimization" title="Empirical risk minimization">Empirical risk minimization</a></li>
<li><a href="/wiki/Occam_learning" title="Occam learning">Occam learning</a></li>
<li><a href="/wiki/Probably_approximately_correct_learning" title="Probably approximately correct learning">PAC learning</a></li>
<li><a href="/wiki/Statistical_learning_theory" title="Statistical learning theory">Statistical learning</a></li>
<li><a href="/wiki/Vapnik%E2%80%93Chervonenkis_theory" title="Vapnik–Chervonenkis theory">VC theory</a></li></ul>
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<div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left">Machine-learning venues</div><div class="NavContent" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"><div class="hlist">
<ul><li><a href="/wiki/Conference_on_Neural_Information_Processing_Systems" title="Conference on Neural Information Processing Systems">NeurIPS</a></li>
<li><a href="/wiki/International_Conference_on_Machine_Learning" title="International Conference on Machine Learning">ICML</a></li>
<li><a href="/wiki/Machine_Learning_(journal)" title="Machine Learning (journal)">ML</a></li>
<li><a href="/wiki/Journal_of_Machine_Learning_Research" title="Journal of Machine Learning Research">JMLR</a></li>
<li><a rel="nofollow" class="external text" href="https://arxiv.org/list/cs.LG/recent">ArXiv:cs.LG</a></li></ul>
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<div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left"><a href="/wiki/Glossary_of_artificial_intelligence" title="Glossary of artificial intelligence">Glossary of artificial intelligence</a></div><div class="NavContent" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"><div class="hlist">
<ul><li><a href="/wiki/Glossary_of_artificial_intelligence" title="Glossary of artificial intelligence">Glossary of artificial intelligence</a></li></ul>
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<div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left">Related articles</div><div class="NavContent" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"><div class="hlist">
<ul><li><a href="/wiki/List_of_datasets_for_machine-learning_research" title="List of datasets for machine-learning research">List of datasets for machine-learning research</a></li>
<li><a href="/wiki/Outline_of_machine_learning" title="Outline of machine learning">Outline of machine learning</a></li></ul>
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<p><b>T-distributed Stochastic Neighbor Embedding (t-SNE)</b> is a <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a> algorithm for <a href="/wiki/Data_visualization" title="Data visualization">visualization</a> developed by <a href="/w/index.php?title=Laurens_van_der_Maaten&amp;action=edit&amp;redlink=1" class="new" title="Laurens van der Maaten (page does not exist)">Laurens van der Maaten</a> and <a href="/wiki/Geoffrey_Hinton" title="Geoffrey Hinton">Geoffrey Hinton</a>.<sup id="cite_ref-MaatenHinton_1-0" class="reference"><a href="#cite_note-MaatenHinton-1">&#91;1&#93;</a></sup> It is a <a href="/wiki/Nonlinear_dimensionality_reduction" title="Nonlinear dimensionality reduction">nonlinear dimensionality reduction</a> technique well-suited for embedding high-dimensional data for visualization in a low-dimensional space of two or three dimensions. Specifically, it models each high-dimensional object by a two- or three-dimensional point in such a way that similar objects are modeled by nearby points and dissimilar objects are modeled by distant points with high probability.
</p><p>The t-SNE algorithm comprises two main stages. First, t-SNE constructs a <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> over pairs of high-dimensional objects in such a way that similar objects have a high probability of being picked while dissimilar points have an extremely small probability of being picked. Second, t-SNE defines a similar probability distribution over the points in the low-dimensional map, and it minimizes the <a href="/wiki/Kullback%E2%80%93Leibler_divergence" title="Kullback–Leibler divergence">Kullback–Leibler divergence</a> between the two distributions with respect to the locations of the points in the map. Note that while the original algorithm uses the <a href="/wiki/Euclidean_distance" title="Euclidean distance">Euclidean distance</a> between objects as the base of its similarity metric, this should be changed as appropriate.
</p><p>t-SNE has been used for visualization in a wide range of applications, including <a href="/wiki/Computer_security" title="Computer security">computer security</a> research,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2">&#91;2&#93;</a></sup> <a href="/wiki/Music_analysis" class="mw-redirect" title="Music analysis">music analysis</a>,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3">&#91;3&#93;</a></sup> <a href="/wiki/Cancer_research" title="Cancer research">cancer research</a>,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4">&#91;4&#93;</a></sup> <a href="/wiki/Bioinformatics" title="Bioinformatics">bioinformatics</a>,<sup id="cite_ref-5" class="reference"><a href="#cite_note-5">&#91;5&#93;</a></sup> and biomedical signal processing.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6">&#91;6&#93;</a></sup> It is often used to visualize high-level representations learned by an <a href="/wiki/Artificial_neural_network" title="Artificial neural network">artificial neural network</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7">&#91;7&#93;</a></sup>
</p><p>While t-SNE plots often seem to display <a href="/wiki/Cluster_analysis" title="Cluster analysis">clusters</a>, the visual clusters can be influenced strongly by the chosen parameterization and therefore a good understanding of the parameters for t-SNE is necessary. Such "clusters" can be shown to even appear in non-clustered data,<sup id="cite_ref-8" class="reference"><a href="#cite_note-8">&#91;8&#93;</a></sup> and thus may be false findings. Interactive exploration may thus be necessary to choose parameters and validate results.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9">&#91;9&#93;</a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10">&#91;10&#93;</a></sup> It has been demonstrated that t-SNE is often able to recover well-separated clusters, and with special parameter choices, approximates a simple form of <a href="/wiki/Spectral_clustering" title="Spectral clustering">spectral clustering</a>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11">&#91;11&#93;</a></sup>
</p>
<div id="toc" class="toc"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="en" dir="ltr"><h2>Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#Details"><span class="tocnumber">1</span> <span class="toctext">Details</span></a></li>
<li class="toclevel-1 tocsection-2"><a href="#Software"><span class="tocnumber">2</span> <span class="toctext">Software</span></a></li>
<li class="toclevel-1 tocsection-3"><a href="#References"><span class="tocnumber">3</span> <span class="toctext">References</span></a></li>
<li class="toclevel-1 tocsection-4"><a href="#External_links"><span class="tocnumber">4</span> <span class="toctext">External links</span></a></li>
</ul>
</div>
<h2><span class="mw-headline" id="Details">Details</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-distributed_stochastic_neighbor_embedding&amp;action=edit&amp;section=1" title="Edit section: Details">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<p>Given a set of <span class="mwe-math-element"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>N</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle N}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="N"/></span> high-dimensional objects <span class="mwe-math-element"><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},\dots ,\mathbf {x} _{N}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>&#x2026;<!-- … --></mo>
<mo>,</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>N</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{1},\dots ,\mathbf {x} _{N}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f1a4c9aea89b8fc822c278914f91d9fc4e4aa26" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:10.746ex; height:2.009ex;" alt="\mathbf {x} _{1},\dots ,\mathbf {x} _{N}"/></span>, t-SNE first computes probabilities <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p_{ij}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca46e6d560ac4e615adcd6d053cd476f4aadfcbd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:2.736ex; height:2.343ex;" alt="p_{ij}"/></span> that are proportional to the similarity of objects <span class="mwe-math-element"><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}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{i}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57d2ef3df60acdb53bdf90535264041fea7231cd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.211ex; height:2.009ex;" alt="\mathbf {x} _{i}"/></span> and <span class="mwe-math-element"><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} _{j}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{j}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da7e57d3f8c537992b45488f9586aec0c35a85f0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:2.321ex; height:2.343ex;" alt="\mathbf {x} _{j}"/></span>, as follows:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{j\mid i}={\frac {\exp(-\lVert \mathbf {x} _{i}-\mathbf {x} _{j}\rVert ^{2}/2\sigma _{i}^{2})}{\sum _{k\neq i}\exp(-\lVert \mathbf {x} _{i}-\mathbf {x} _{k}\rVert ^{2}/2\sigma _{i}^{2})}},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mo>&#x2223;<!-- ∣ --></mo>
<mi>i</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>exp</mi>
<mo>&#x2061;<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mo>&#x2212;<!-- − --></mo>
<mo fence="false" stretchy="false">&#x2016;<!-- ‖ --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
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<mi>i</mi>
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<mo>&#x2212;<!-- − --></mo>
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<mi>j</mi>
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<msup>
<mo fence="false" stretchy="false">&#x2016;<!-- ‖ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<mo>/</mo>
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<mn>2</mn>
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<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
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<mo stretchy="false">)</mo>
</mrow>
<mrow>
<munder>
<mo>&#x2211;<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>&#x2260;<!-- ≠ --></mo>
<mi>i</mi>
</mrow>
</munder>
<mi>exp</mi>
<mo>&#x2061;<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mo>&#x2212;<!-- − --></mo>
<mo fence="false" stretchy="false">&#x2016;<!-- ‖ --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>&#x2212;<!-- − --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<msup>
<mo fence="false" stretchy="false">&#x2016;<!-- ‖ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p_{j\mid i}={\frac {\exp(-\lVert \mathbf {x} _{i}-\mathbf {x} _{j}\rVert ^{2}/2\sigma _{i}^{2})}{\sum _{k\neq i}\exp(-\lVert \mathbf {x} _{i}-\mathbf {x} _{k}\rVert ^{2}/2\sigma _{i}^{2})}},}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cc3ef3b4d237787cd82e5ef638d96d642a1e43d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.171ex; margin-left: -0.089ex; width:36.36ex; height:7.343ex;" alt="{\displaystyle p_{j\mid i}={\frac {\exp(-\lVert \mathbf {x} _{i}-\mathbf {x} _{j}\rVert ^{2}/2\sigma _{i}^{2})}{\sum _{k\neq i}\exp(-\lVert \mathbf {x} _{i}-\mathbf {x} _{k}\rVert ^{2}/2\sigma _{i}^{2})}},}"/></span></dd></dl>
<p>As Van der Maaten and Hinton explained: "The similarity of datapoint <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{j}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5db47cb3d2f9496205a17a6856c91c1d3d363ccd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:2.239ex; height:2.343ex;" alt="x_{j}"/></span> to datapoint <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="x_{i}"/></span> is the conditional probability, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{j|i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p_{j|i}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/350d4978c797110eff8a6a67d6bd4a905a22cf27" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.171ex; margin-left: -0.089ex; width:3.193ex; height:2.509ex;" alt="{\displaystyle p_{j|i}}"/></span>, that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="x_{i}"/></span> would pick <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{j}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5db47cb3d2f9496205a17a6856c91c1d3d363ccd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:2.239ex; height:2.343ex;" alt="x_{j}"/></span> as its neighbor if neighbors were picked in proportion to their probability density under a Gaussian centered at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="x_{i}"/></span>."<sup id="cite_ref-MaatenHinton_1-1" class="reference"><a href="#cite_note-MaatenHinton-1">&#91;1&#93;</a></sup>
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{ij}={\frac {p_{j\mid i}+p_{i\mid j}}{2N}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mo>&#x2223;<!-- ∣ --></mo>
<mi>i</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>&#x2223;<!-- ∣ --></mo>
<mi>j</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mn>2</mn>
<mi>N</mi>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p_{ij}={\frac {p_{j\mid i}+p_{i\mid j}}{2N}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53cc5533bb4b3b8f18231c58df4e4215546a0fc" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:15.719ex; height:5.509ex;" alt="{\displaystyle p_{ij}={\frac {p_{j\mid i}+p_{i\mid j}}{2N}}}"/></span></dd></dl>
<p>Moreover, the probabilities with <span class="mwe-math-element"><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=j}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
<mo>=</mo>
<mi>j</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i=j}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/706e0928b2bf0f24076b0c90bb20616ff2068343" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:4.859ex; height:2.509ex;" alt="{\displaystyle i=j}"/></span> are set to zero&#160;: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{ii}=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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<mn>0</mn>
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<annotation encoding="application/x-tex">{\displaystyle p_{ii}=0}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8921493a1f5861118bb75df75ecf2e712ca5b48c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.887ex; height:2.509ex;" alt="{\displaystyle p_{ii}=0}"/></span>
</p><p>The bandwidth of the <a href="/wiki/Gaussian_kernel" class="mw-redirect" title="Gaussian kernel">Gaussian kernels</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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<annotation encoding="application/x-tex">{\displaystyle \sigma _{i}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ab3208a7d0c634ef720e03ff5a9949e8310edc4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.127ex; height:2.009ex;" alt="\sigma _{i}"/></span> is set in such a way that the <a href="/wiki/Perplexity" title="Perplexity">perplexity</a> of the conditional distribution equals a predefined perplexity using the <a href="/wiki/Bisection_method" title="Bisection method">bisection method</a>. As a result, the bandwidth is adapted to the <a href="/wiki/Density" title="Density">density</a> of the data: smaller values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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<annotation encoding="application/x-tex">{\displaystyle \sigma _{i}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ab3208a7d0c634ef720e03ff5a9949e8310edc4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.127ex; height:2.009ex;" alt="\sigma _{i}"/></span> are used in denser parts of the data space.
</p><p>Since the Gaussian kernel uses the Euclidean distance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lVert x_{i}-x_{j}\rVert }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">&#x2016;<!-- ‖ --></mo>
<msub>
<mi>x</mi>
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<mi>i</mi>
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<mo>&#x2212;<!-- − --></mo>
<msub>
<mi>x</mi>
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<mi>j</mi>
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<mo fence="false" stretchy="false">&#x2016;<!-- ‖ --></mo>
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<annotation encoding="application/x-tex">{\displaystyle \lVert x_{i}-x_{j}\rVert }</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/629c7171b13d2c65964333970b68e9294e4a12b3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:9.534ex; height:3.009ex;" alt="{\displaystyle \lVert x_{i}-x_{j}\rVert }"/></span>, it is affected by the <a href="/wiki/Curse_of_dimensionality" title="Curse of dimensionality">curse of dimensionality</a>, and in high dimensional data when distances lose the ability to discriminate, the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
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<annotation encoding="application/x-tex">{\displaystyle p_{ij}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca46e6d560ac4e615adcd6d053cd476f4aadfcbd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:2.736ex; height:2.343ex;" alt="p_{ij}"/></span> become too similar (asymptotically, they would converge to a constant). It has been proposed to adjust the distances with a power transform, based on the <a href="/wiki/Intrinsic_dimension" title="Intrinsic dimension">intrinsic dimension</a> of each point, to alleviate this.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12">&#91;12&#93;</a></sup>
</p><p>t-SNE aims to learn a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
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<annotation encoding="application/x-tex">{\displaystyle d}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="d"/></span>-dimensional map <span class="mwe-math-element"><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 {y} _{1},\dots ,\mathbf {y} _{N}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
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<mrow class="MJX-TeXAtom-ORD">
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<mo>,</mo>
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<mo>,</mo>
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<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>N</mi>
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<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{1},\dots ,\mathbf {y} _{N}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cda5b5d378bff5bd2bed385a0ee4a96aa6fe4e5b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:10.746ex; height:2.176ex;" alt="\mathbf {y} _{1},\dots ,\mathbf {y} _{N}"/></span> (with <span class="mwe-math-element"><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 {y} _{i}\in \mathbb {R} ^{d}}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
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<mrow class="MJX-TeXAtom-ORD">
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<mo>&#x2208;<!-- ∈ --></mo>
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<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{i}\in \mathbb {R} ^{d}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cb42ba93cd364e9ea009717f7214633b24e05e4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:7.821ex; height:3.176ex;" alt="\mathbf {y} _{i}\in \mathbb {R} ^{d}"/></span>) that reflects the similarities <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
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<annotation encoding="application/x-tex">{\displaystyle p_{ij}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca46e6d560ac4e615adcd6d053cd476f4aadfcbd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:2.736ex; height:2.343ex;" alt="p_{ij}"/></span> as well as possible. To this end, it measures similarities <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
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</msub>
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<annotation encoding="application/x-tex">{\displaystyle q_{ij}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b08ec83005828a8789b639e4944b11905e9b18b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:2.514ex; height:2.343ex;" alt="q_{ij}"/></span> between two points in the map <span class="mwe-math-element"><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 {y} _{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{i}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a762b3bf7b8e1b988c736ec7cbee2e81e3e04cf" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:2.211ex; height:2.176ex;" alt="\mathbf {y} _{i}"/></span> and <span class="mwe-math-element"><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 {y} _{j}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{j}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f85b86e3d6099153c81ce7101473fc1caad1634f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.171ex; width:2.321ex; height:2.509ex;" alt="\mathbf {y} _{j}"/></span>, using a very similar approach. Specifically, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle q_{ij}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b08ec83005828a8789b639e4944b11905e9b18b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:2.514ex; height:2.343ex;" alt="q_{ij}"/></span> is defined as:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{ij}={\frac {(1+\lVert \mathbf {y} _{i}-\mathbf {y} _{j}\rVert ^{2})^{-1}}{\sum _{k\neq i}(1+\lVert \mathbf {y} _{i}-\mathbf {y} _{k}\rVert ^{2})^{-1}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
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<mo>=</mo>
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<mfrac>
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<mi mathvariant="bold">y</mi>
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<mrow class="MJX-TeXAtom-ORD">
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<mo stretchy="false">)</mo>
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<mo>&#x2212;<!-- − --></mo>
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<mrow>
<munder>
<mo>&#x2211;<!-- ∑ --></mo>
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<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
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<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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<mo>&#x2212;<!-- − --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
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<msup>
<mo fence="false" stretchy="false">&#x2016;<!-- ‖ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- − --></mo>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle q_{ij}={\frac {(1+\lVert \mathbf {y} _{i}-\mathbf {y} _{j}\rVert ^{2})^{-1}}{\sum _{k\neq i}(1+\lVert \mathbf {y} _{i}-\mathbf {y} _{k}\rVert ^{2})^{-1}}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/332b46963d03a1fa12b1d6524652c43efc60930e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:30.912ex; height:7.343ex;" alt="{\displaystyle q_{ij}={\frac {(1+\lVert \mathbf {y} _{i}-\mathbf {y} _{j}\rVert ^{2})^{-1}}{\sum _{k\neq i}(1+\lVert \mathbf {y} _{i}-\mathbf {y} _{k}\rVert ^{2})^{-1}}}}"/></span></dd></dl>
<p>Herein a heavy-tailed <a href="/wiki/Student_t-distribution" class="mw-redirect" title="Student t-distribution">Student t-distribution</a> (with one-degree of freedom, which is the same as a <a href="/wiki/Cauchy_distribution" title="Cauchy distribution">Cauchy distribution</a>) is used to measure similarities between low-dimensional points in order to allow dissimilar objects to be modeled far apart in the map. Note that also in this case we set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{ii}=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>i</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle q_{ii}=0}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ec871c16d7076c85d166c4916ba916bb29ca1b1" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:6.665ex; height:2.509ex;" alt="{\displaystyle q_{ii}=0}"/></span>
</p><p>The locations of the points <span class="mwe-math-element"><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 {y} _{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{i}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a762b3bf7b8e1b988c736ec7cbee2e81e3e04cf" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:2.211ex; height:2.176ex;" alt="\mathbf {y} _{i}"/></span> in the map are determined by minimizing the (non-symmetric) <a href="/wiki/Kullback%E2%80%93Leibler_divergence" title="Kullback–Leibler divergence">Kullback–Leibler divergence</a> of the distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="Q"/></span> from the distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="P"/></span>, that is:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle KL(P||Q)=\sum _{i\neq j}p_{ij}\log {\frac {p_{ij}}{q_{ij}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>K</mi>
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<annotation encoding="application/x-tex">{\displaystyle KL(P||Q)=\sum _{i\neq j}p_{ij}\log {\frac {p_{ij}}{q_{ij}}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cae779cfc3a41b382e68850f0381b6a6b7fdede7" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.505ex; width:27.051ex; height:6.676ex;" alt="{\displaystyle KL(P||Q)=\sum _{i\neq j}p_{ij}\log {\frac {p_{ij}}{q_{ij}}}}"/></span></dd></dl>
<p>The minimization of the Kullback–Leibler divergence with respect to the points <span class="mwe-math-element"><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 {y} _{i}}">
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<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{i}}</annotation>
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</p>
<h2><span class="mw-headline" id="Software">Software</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-distributed_stochastic_neighbor_embedding&amp;action=edit&amp;section=2" title="Edit section: Software">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<ul><li>Laurens van der Maaten's t-Distributed Stochastic Neighbor Embedding <a rel="nofollow" class="external free" href="https://lvdmaaten.github.io/tsne/">https://lvdmaaten.github.io/tsne/</a></li>
<li><a href="/wiki/ELKI" title="ELKI">ELKI</a> contains tSNE, also with Barnes-Hut approximation. <a rel="nofollow" class="external free" href="https://github.com/elki-project/elki/blob/master/elki/src/main/java/de/lmu/ifi/dbs/elki/algorithm/projection/TSNE.java">https://github.com/elki-project/elki/blob/master/elki/src/main/java/de/lmu/ifi/dbs/elki/algorithm/projection/TSNE.java</a></li></ul>
<h2><span class="mw-headline" id="References">References</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-distributed_stochastic_neighbor_embedding&amp;action=edit&amp;section=3" title="Edit section: References">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<div class="reflist" style="list-style-type: decimal;">
<div class="mw-references-wrap mw-references-columns"><ol class="references">
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<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><cite class="citation arxiv">Linderman, George C.; Steinerberger, Stefan (2017-06-08). "Clustering with t-SNE, provably". <a href="/wiki/ArXiv" title="ArXiv">arXiv</a>:<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="//arxiv.org/abs/1706.02582">1706.02582</a></span> [<a rel="nofollow" class="external text" href="//arxiv.org/archive/cs.LG">cs.LG</a>].</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=preprint&amp;rft.jtitle=arXiv&amp;rft.atitle=Clustering+with+t-SNE%2C+provably&amp;rft.date=2017-06-08&amp;rft_id=info%3Aarxiv%2F1706.02582&amp;rft.aulast=Linderman&amp;rft.aufirst=George+C.&amp;rft.au=Steinerberger%2C+Stefan&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-distributed+stochastic+neighbor+embedding" class="Z3988"></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886058088"/></span>
</li>
<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><cite class="citation conference">Schubert, Erich; Gertz, Michael (2017-10-04). <i>Intrinsic t-Stochastic Neighbor Embedding for Visualization and Outlier Detection</i>. SISAP 2017 – 10th International Conference on Similarity Search and Applications. pp.&#160;188–203. <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="//doi.org/10.1007%2F978-3-319-68474-1_13">10.1007/978-3-319-68474-1_13</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=conference&amp;rft.btitle=Intrinsic+t-Stochastic+Neighbor+Embedding+for+Visualization+and+Outlier+Detection&amp;rft.pages=188-203&amp;rft.date=2017-10-04&amp;rft_id=info%3Adoi%2F10.1007%2F978-3-319-68474-1_13&amp;rft.aulast=Schubert&amp;rft.aufirst=Erich&amp;rft.au=Gertz%2C+Michael&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-distributed+stochastic+neighbor+embedding" class="Z3988"></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886058088"/></span>
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<h2><span class="mw-headline" id="External_links">External links</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-distributed_stochastic_neighbor_embedding&amp;action=edit&amp;section=4" title="Edit section: External links">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<ul><li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=RJVL80Gg3lA">Visualizing Data Using t-SNE</a>, Google Tech Talk about t-SNE</li></ul>
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