t-distributed stochastic neighbor embedding

<h2 id="details">Details</h2>
Given a set of 
 
 
 
 N
 
 
 {\displaystyle N}
 
 high-dimensional objects 
 
 
 
 
 
 x
 
 
 1
 
 
 ,
 …
 ,
 
 
 x
 
 
 N
 
 
 
 
 {\displaystyle \mathbf {x} _{1},\dots ,\mathbf {x} _{N}}
 
, t-SNE first computes probabilities 
 
 
 
 
 p
 
 i
 j
 
 
 
 
 {\displaystyle p_{ij}}
 
 that are proportional to the similarity of objects 
 
 
 
 
 
 x
 
 
 i
 
 
 
 
 {\displaystyle \mathbf {x} _{i}}
 
 and 
 
 
 
 
 
 x
 
 
 j
 
 
 
 
 {\displaystyle \mathbf {x} _{j}}
 
, as follows.
For 
 
 
 
 i
 ≠
 j
 
 
 {\displaystyle i\neq j}
 
, define

p
          
            j
            ∣
            i
          
        
        =
        
          
            
              exp
              ⁡
              (
              −
              ‖
              
                
                  x
                
                
                  i
                
              
              −
              
                
                  x
                
                
                  j
                
              
              
                ‖
                
                  2
                
              
              
                /
              
              2
              
                σ
                
                  i
                
                
                  2
                
              
              )
            
            
              
                ∑
                
                  k
                  ≠
                  i
                
              
              exp
              ⁡
              (
              −
              ‖
              
                
                  x
                
                
                  i
                
              
              −
              
                
                  x
                
                
                  k
                
              
              
                ‖
                
                  2
                
              
              
                /
              
              2
              
                σ
                
                  i
                
                
                  2
                
              
              )
            
          
        
      
    
    {\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})}}}

and set 
 
 
 
 
 p
 
 i
 ∣
 i
 
 
 =
 0
 
 
 {\displaystyle p_{i\mid i}=0}
 
. 
Note the above denominator ensures 
 
 
 
 
 ∑
 
 j
 
 
 
 p
 
 j
 ∣
 i
 
 
 =
 1
 
 
 {\displaystyle \sum _{j}p_{j\mid i}=1}
 
 for all 
 
 
 
 i
 
 
 {\displaystyle i}
 
.
As van der Maaten and Hinton explained: "The similarity of datapoint 
 
 
 
 
 x
 
 j
 
 
 
 
 {\displaystyle x_{j}}
 
 to datapoint 
 
 
 
 
 x
 
 i
 
 
 
 
 {\displaystyle x_{i}}
 
 is the conditional probability, 
 
 
 
 
 p
 
 j
 
 |
 
 i
 
 
 
 
 {\displaystyle p_{j|i}}
 
, that 
 
 
 
 
 x
 
 i
 
 
 
 
 {\displaystyle x_{i}}
 
 would pick 
 
 
 
 
 x
 
 j
 
 
 
 
 {\displaystyle x_{j}}
 
 as its neighbor if neighbors were picked in proportion to their probability density under a Gaussian centered at 
 
 
 
 
 x
 
 i
 
 
 
 
 {\displaystyle x_{i}}
 
."<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>
Now define

p
          
            i
            j
          
        
        =
        
          
            
              
                p
                
                  j
                  ∣
                  i
                
              
              +
              
                p
                
                  i
                  ∣
                  j
                
              
            
            
              2
              N
            
          
        
      
    
    {\displaystyle p_{ij}={\frac {p_{j\mid i}+p_{i\mid j}}{2N}}}

This is motivated because 
 
 
 
 
 p
 
 i
 
 
 
 
 {\displaystyle p_{i}}
 
 and 
 
 
 
 
 p
 
 j
 
 
 
 
 {\displaystyle p_{j}}
 
 from the N samples are estimated as 1/N, so the conditional probability can be written as 
 
 
 
 
 p
 
 i
 ∣
 j
 
 
 =
 N
 
 p
 
 i
 j
 
 
 
 
 {\displaystyle p_{i\mid j}=Np_{ij}}
 
 and 
 
 
 
 
 p
 
 j
 ∣
 i
 
 
 =
 N
 
 p
 
 j
 i
 
 
 
 
 {\displaystyle p_{j\mid i}=Np_{ji}}
 
 . Since 
 
 
 
 
 p
 
 i
 j
 
 
 =
 
 p
 
 j
 i
 
 
 
 
 {\displaystyle p_{ij}=p_{ji}}
 
, you can obtain previous formula.
Also note that 
 
 
 
 
 p
 
 i
 i
 
 
 =
 0
 
 
 {\displaystyle p_{ii}=0}
 
 and 
 
 
 
 
 ∑
 
 i
 ,
 j
 
 
 
 p
 
 i
 j
 
 
 =
 1
 
 
 {\displaystyle \sum _{i,j}p_{ij}=1}
 
.
The bandwidth of the <a href="/facts/Gaussian_kernel/5bG8Kpp9">Gaussian kernels</a> 
 
 
 
 
 σ
 
 i
 
 
 
 
 {\displaystyle \sigma _{i}}
 
 is set in such a way that the <a href="/facts/Entropy_(information_theory)/NLg4NLvt">entropy</a> of the conditional distribution equals a predefined entropy using the <a href="/facts/Bisection_method/3tWmtoIp">bisection method</a>. As a result, the bandwidth is adapted to the <a href="/facts/Density/92p6hh9I">density</a> of the data: smaller values of 
 
 
 
 
 σ
 
 i
 
 
 
 
 {\displaystyle \sigma _{i}}
 
 are used in denser parts of the data space. The entropy increases with the <a href="/facts/Perplexity/VfmbcSMg">perplexity</a> of this distribution 
 
 
 
 
 P
 
 i
 
 
 
 
 {\displaystyle P_{i}}
 
; this relation is seen as

P
        e
        r
        p
        (
        
          P
          
            i
          
        
        )
        =
        
          2
          
            H
            (
            
              P
              
                i
              
            
            )
          
        
      
    
    {\displaystyle Perp(P_{i})=2^{H(P_{i})}}

where 
 
 
 
 H
 (
 
 P
 
 i
 
 
 )
 
 
 {\displaystyle H(P_{i})}
 
 is the Shannon entropy 
 
 
 
 H
 (
 
 P
 
 i
 
 
 )
 =
 −
 
 ∑
 
 j
 
 
 
 p
 
 j
 
 |
 
 i
 
 
 
 log
 
 2
 
 
 ⁡
 
 p
 
 j
 
 |
 
 i
 
 
 .
 
 
 {\displaystyle H(P_{i})=-\sum _{j}p_{j|i}\log _{2}p_{j|i}.}

The perplexity is a hand-chosen parameter of t-SNE, and as the authors state, "perplexity can be interpreted as a smooth measure of the effective number of neighbors. The performance of SNE is fairly robust to changes in the perplexity, and typical values are between 5 and 50.".<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a>
Since the Gaussian kernel uses the <a href="/facts/Euclidean_distance/9qDoQKQe">Euclidean distance</a> 
 
 
 
 ‖
 
 x
 
 i
 
 
 −
 
 x
 
 j
 
 
 ‖
 
 
 {\displaystyle \lVert x_{i}-x_{j}\rVert }
 
, it is affected by the <a href="/facts/Curse_of_dimensionality/dk7SnY6L">curse of dimensionality</a>, and in high dimensional data when distances lose the ability to discriminate, the 
 
 
 
 
 p
 
 i
 j
 
 
 
 
 {\displaystyle p_{ij}}
 
 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="/facts/Intrinsic_dimension/nEVb4KUt">intrinsic dimension</a> of each point, to alleviate this.<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>
t-SNE aims to learn a 
 
 
 
 d
 
 
 {\displaystyle d}
 
-dimensional map 
 
 
 
 
 
 y
 
 
 1
 
 
 ,
 …
 ,
 
 
 y
 
 
 N
 
 
 
 
 {\displaystyle \mathbf {y} _{1},\dots ,\mathbf {y} _{N}}
 
 (with 
 
 
 
 
 
 y
 
 
 i
 
 
 ∈
 
 
 R
 
 
 d
 
 
 
 
 {\displaystyle \mathbf {y} _{i}\in \mathbb {R} ^{d}}
 
 and 
 
 
 
 d
 
 
 {\displaystyle d}
 
 typically chosen as 2 or 3) that reflects the similarities 
 
 
 
 
 p
 
 i
 j
 
 
 
 
 {\displaystyle p_{ij}}
 
 as well as possible. To this end, it measures similarities 
 
 
 
 
 q
 
 i
 j
 
 
 
 
 {\displaystyle q_{ij}}
 
 between two points in the map 
 
 
 
 
 
 y
 
 
 i
 
 
 
 
 {\displaystyle \mathbf {y} _{i}}
 
 and 
 
 
 
 
 
 y
 
 
 j
 
 
 
 
 {\displaystyle \mathbf {y} _{j}}
 
, using a very similar approach. 
Specifically, for 
 
 
 
 i
 ≠
 j
 
 
 {\displaystyle i\neq j}
 
, define 
 
 
 
 
 q
 
 i
 j
 
 
 
 
 {\displaystyle q_{ij}}
 
 as

q
          
            i
            j
          
        
        =
        
          
            
              (
              1
              +
              ‖
              
                
                  y
                
                
                  i
                
              
              −
              
                
                  y
                
                
                  j
                
              
              
                ‖
                
                  2
                
              
              
                )
                
                  −
                  1
                
              
            
            
              
                ∑
                
                  k
                
              
              
                ∑
                
                  l
                  ≠
                  k
                
              
              (
              1
              +
              ‖
              
                
                  y
                
                
                  k
                
              
              −
              
                
                  y
                
                
                  l
                
              
              
                ‖
                
                  2
                
              
              
                )
                
                  −
                  1
                
              
            
          
        
      
    
    {\displaystyle q_{ij}={\frac {(1+\lVert \mathbf {y} _{i}-\mathbf {y} _{j}\rVert ^{2})^{-1}}{\sum _{k}\sum _{l\neq k}(1+\lVert \mathbf {y} _{k}-\mathbf {y} _{l}\rVert ^{2})^{-1}}}}

and set 
 
 
 
 
 q
 
 i
 i
 
 
 =
 0
 
 
 {\displaystyle q_{ii}=0}
 
. 
Herein a heavy-tailed <a href="/facts/Student_t-distribution/DeT1SDqH">Student t-distribution</a> (with one-degree of freedom, which is the same as a <a href="/facts/Cauchy_distribution/3HS6D3Xk">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.
The locations of the points 
 
 
 
 
 
 y
 
 
 i
 
 
 
 
 {\displaystyle \mathbf {y} _{i}}
 
 in the map are determined by minimizing the (non-symmetric) <a href="/facts/Kullback%E2%80%93Leibler_divergence/nh7SjlPE">Kullback–Leibler divergence</a> of the distribution 
 
 
 
 P
 
 
 {\displaystyle P}
 
 from the distribution 
 
 
 
 Q
 
 
 {\displaystyle Q}
 
, that is:

K
          L
        
        
          (
          
            P
            ∥
            Q
          
          )
        
        =
        
          ∑
          
            i
            ≠
            j
          
        
        
          p
          
            i
            j
          
        
        log
        ⁡
        
          
            
              p
              
                i
                j
              
            
            
              q
              
                i
                j
              
            
          
        
      
    
    {\displaystyle \mathrm {KL} \left(P\parallel Q\right)=\sum _{i\neq j}p_{ij}\log {\frac {p_{ij}}{q_{ij}}}}

The minimization of the Kullback–Leibler divergence with respect to the points 
 
 
 
 
 
 y
 
 
 i
 
 
 
 
 {\displaystyle \mathbf {y} _{i}}
 
 is performed using <a href="/facts/Gradient_descent/pFFrek0F">gradient descent</a>. 
The result of this optimization is a map that reflects the similarities between the high-dimensional inputs.

<h2 id="output">Output</h2>
While t-SNE plots often seem to display <a href="/facts/Cluster_analysis/HUeDnWPt">clusters</a>, the visual clusters can be strongly influenced by the chosen parameterization (especially the perplexity) and so a good understanding of the parameters for t-SNE is needed. Such "clusters" can be shown to even appear in structured data with no clear clustering,<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a> and so may be false findings. Similarly, the size of clusters produced by t-SNE is not informative, and neither is the distance between clusters.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a> Thus, interactive exploration may be needed to choose parameters and validate results.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a><a class="footnote-ref" id="fnref:18" href="#fn:18">18</a> It has been shown that t-SNE can often recover well-separated clusters, and with special parameter choices, approximates a simple form of <a href="/facts/Spectral_clustering/GorfFMRT">spectral clustering</a>.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a>

<h2 id="software">Software</h2>
<ul><li>A C++ implementation of Barnes-Hut is available on the <a href="https://github.com/lvdmaaten/bhtsne">github account</a> of one of the original authors.</li>
<li>The R package <a href="https://CRAN.R-project.org/package=Rtsne">Rtsne</a> implements t-SNE in <a href="/facts/R_(programming_language)/LSrkr8K8">R</a>.</li>
<li><a href="/facts/ELKI/C2LWqW13">ELKI</a> contains tSNE, also with Barnes-Hut approximation</li>
<li><a href="/facts/Scikit-learn/HqxDHRMJ">scikit-learn</a>, a popular machine learning library in Python implements t-SNE with both exact solutions and the Barnes-Hut approximation.</li>
<li>Tensorboard, the visualization kit associated with <a href="/facts/TensorFlow/kDwfBeNd">TensorFlow</a>, also implements t-SNE (<a href="https://projector.tensorflow.org/">online version</a>)</li>
<li>The <a href="/facts/Julia_(programming_language)/AoB0PJ9C">Julia</a> package <a href="https://juliapackages.com/p/tsne">TSne</a> implements t-SNE</li></ul>

<h2 id="external-links">External links</h2>
<ul><li>Wattenberg, Martin; Viégas, Fernanda; Johnson, Ian (2016-10-13). <a href="https://distill.pub/2016/misread-tsne/">"How to Use t-SNE Effectively"</a>. Distill. 1 (10): e2. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.23915%2Fdistill.00002">10.23915/distill.00002</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/2476-0757">2476-0757</a>.. Interactive demonstration and tutorial.</li>
<li><a href="https://www.youtube.com/watch?v=RJVL80Gg3lA">Visualizing Data Using t-SNE</a>, Google Tech Talk about t-SNE</li>
<li><a href="https://lvdmaaten.github.io/tsne/">Implementations of t-SNE in various languages</a>, A link collection maintained by Laurens van der Maaten</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Hinton, Geoffrey; Roweis, Sam (January 2002). Stochastic neighbor embedding (PDF). Neural Information Processing Systems. <a href="https://cs.nyu.edu/~roweis/papers/sne_final.pdf" target="_blank">https://cs.nyu.edu/~roweis/papers/sne_final.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">van der Maaten, L.J.P.; Hinton, G.E. (Nov 2008). "Visualizing Data Using t-SNE" (PDF). Journal of Machine Learning Research. 9: 2579–2605. <a href="http://jmlr.org/papers/volume9/vandermaaten08a/vandermaaten08a.pdf" target="_blank">http://jmlr.org/papers/volume9/vandermaaten08a/vandermaaten08a.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Gashi, I.; Stankovic, V.; Leita, C.; Thonnard, O. (2009). "An Experimental Study of Diversity with Off-the-shelf AntiVirus Engines". Proceedings of the IEEE International Symposium on Network Computing and Applications: 4–11. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Hamel, P.; Eck, D. (2010). "Learning Features from Music Audio with Deep Belief Networks". Proceedings of the International Society for Music Information Retrieval Conference: 339–344. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Jamieson, A.R.; Giger, M.L.; Drukker, K.; Lui, H.; Yuan, Y.; Bhooshan, N. (2010). "Exploring Nonlinear Feature Space Dimension Reduction and Data Representation in Breast CADx with Laplacian Eigenmaps and t-SNE". Medical Physics. 37 (1): 339–351. doi:10.1118/1.3267037. PMC 2807447. PMID 20175497. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2807447" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2807447</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Wallach, I.; Liliean, R. (2009). "The Protein-Small-Molecule Database, A Non-Redundant Structural Resource for the Analysis of Protein-Ligand Binding". Bioinformatics. 25 (5): 615–620. doi:10.1093/bioinformatics/btp035. PMID 19153135. <a href="https://doi.org/10.1093%2Fbioinformatics%2Fbtp035" target="_blank">https://doi.org/10.1093%2Fbioinformatics%2Fbtp035</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Balamurali, Mehala; Silversides, Katherine L.; Melkumyan, Arman (2019-04-01). "A comparison of t-SNE, SOM and SPADE for identifying material type domains in geological data". Computers & Geosciences. 125: 78–89. Bibcode:2019CG....125...78B. doi:10.1016/j.cageo.2019.01.011. ISSN 0098-3004. S2CID 67926902. <a href="https://www.sciencedirect.com/science/article/pii/S0098300418306010" target="_blank">https://www.sciencedirect.com/science/article/pii/S0098300418306010</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Balamurali, Mehala; Melkumyan, Arman (2016). "t-SNE Based Visualisation and Clustering of Geological Domain". In Hirose, Akira; Ozawa, Seiichi; Doya, Kenji; Ikeda, Kazushi; Lee, Minho; Liu, Derong (eds.). Neural Information Processing. Lecture Notes in Computer Science. Vol. 9950. Cham: Springer International Publishing. pp. 565–572. doi:10.1007/978-3-319-46681-1_67. ISBN 978-3-319-46681-1. <a href="978-3-319-46681-1" target="_blank">978-3-319-46681-1</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Leung, Raymond; Balamurali, Mehala; Melkumyan, Arman (2021-01-01). "Sample Truncation Strategies for Outlier Removal in Geochemical Data: The MCD Robust Distance Approach Versus t-SNE Ensemble Clustering". Mathematical Geosciences. 53 (1): 105–130. Bibcode:2021MatGe..53..105L. doi:10.1007/s11004-019-09839-z. ISSN 1874-8953. S2CID 208329378. <a href="https://doi.org/10.1007/s11004-019-09839-z" target="_blank">https://doi.org/10.1007/s11004-019-09839-z</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Birjandtalab, J.; Pouyan, M. B.; Nourani, M. (2016-02-01). "Nonlinear dimension reduction for EEG-based epileptic seizure detection". 2016 IEEE-EMBS International Conference on Biomedical and Health Informatics (BHI). pp. 595–598. doi:10.1109/BHI.2016.7455968. ISBN 978-1-5090-2455-1. S2CID 8074617. <a href="978-1-5090-2455-1" target="_blank">978-1-5090-2455-1</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Pezzotti, Nicola (2015). "Approximated and User Steerable tSNE for Progressive Visual Analytics". arXiv:1512.01655 [cs.CV]. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">van der Maaten, L.J.P.; Hinton, G.E. (Nov 2008). "Visualizing Data Using t-SNE" (PDF). Journal of Machine Learning Research. 9: 2579–2605. <a href="http://jmlr.org/papers/volume9/vandermaaten08a/vandermaaten08a.pdf" target="_blank">http://jmlr.org/papers/volume9/vandermaaten08a/vandermaaten08a.pdf</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">van der Maaten, L.J.P.; Hinton, G.E. (Nov 2008). "Visualizing Data Using t-SNE" (PDF). Journal of Machine Learning Research. 9: 2579–2605. <a href="http://jmlr.org/papers/volume9/vandermaaten08a/vandermaaten08a.pdf" target="_blank">http://jmlr.org/papers/volume9/vandermaaten08a/vandermaaten08a.pdf</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Schubert, Erich; Gertz, Michael (2017-10-04). Intrinsic t-Stochastic Neighbor Embedding for Visualization and Outlier Detection. SISAP 2017 – 10th International Conference on Similarity Search and Applications. pp. 188–203. doi:10.1007/978-3-319-68474-1_13. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">"K-means clustering on the output of t-SNE". Cross Validated. Retrieved 2018-04-16. <a href="https://stats.stackexchange.com/a/264647" target="_blank">https://stats.stackexchange.com/a/264647</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Wattenberg, Martin; Viégas, Fernanda; Johnson, Ian (2016-10-13). "How to Use t-SNE Effectively". Distill. 1 (10): e2. doi:10.23915/distill.00002. ISSN 2476-0757. <a href="http://distill.pub/2016/misread-tsne" target="_blank">http://distill.pub/2016/misread-tsne</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Pezzotti, Nicola; Lelieveldt, Boudewijn P. F.; Maaten, Laurens van der; Hollt, Thomas; Eisemann, Elmar; Vilanova, Anna (2017-07-01). "Approximated and User Steerable tSNE for Progressive Visual Analytics". IEEE Transactions on Visualization and Computer Graphics. 23 (7): 1739–1752. arXiv:1512.01655. doi:10.1109/tvcg.2016.2570755. ISSN 1077-2626. PMID 28113434. S2CID 353336. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Wattenberg, Martin; Viégas, Fernanda; Johnson, Ian (2016-10-13). "How to Use t-SNE Effectively". Distill. 1 (10). doi:10.23915/distill.00002. Retrieved 4 December 2017. <a href="https://distill.pub/2016/misread-tsne/" target="_blank">https://distill.pub/2016/misread-tsne/</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Linderman, George C.; Steinerberger, Stefan (2017-06-08). "Clustering with t-SNE, provably". arXiv:1706.02582 [cs.LG]. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
</ol>

t-distributed stochastic neighbor embedding open-in-new

t-distributed stochastic neighbor embedding