Comments - A Different Breed of Mathematics: Topology - Data Science Central2021-04-12T15:47:14Zhttps://www.datasciencecentral.com/profiles/comment/feed?attachedTo=6448529%3ABlogPost%3A437752&xn_auth=noBased on very helpful comment…tag:www.datasciencecentral.com,2016-07-02:6448529:Comment:4440982016-07-02T22:14:16.283ZSyed Danish Alihttps://www.datasciencecentral.com/profile/SyedDanishAli
<p>Based on very helpful comments by Michel Baudin and Sione Palu I have briefly revised some comments to make this post more accurate. </p>
<p>Based on very helpful comments by Michel Baudin and Sione Palu I have briefly revised some comments to make this post more accurate. </p> I second to Michel Baudin' co…tag:www.datasciencecentral.com,2016-06-29:6448529:Comment:4418392016-06-29T23:56:45.101ZSione Paluhttps://www.datasciencecentral.com/profile/SioneKPalu
<p>I second to Michel Baudin' comment. The author of this article is misleading or simply its unbeknownst to him the history of topology. It is more than 100 years old, so the author of this article makes it out as if topology is new. Besides, machine learning has been adopting concept of topology in the last 10 years or so, from text-mining, image recognition and so forth.</p>
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<p>One example of topology is Riemannian Manifold topological space (or simply Riemannian Tensor) which was…</p>
<p>I second to Michel Baudin' comment. The author of this article is misleading or simply its unbeknownst to him the history of topology. It is more than 100 years old, so the author of this article makes it out as if topology is new. Besides, machine learning has been adopting concept of topology in the last 10 years or so, from text-mining, image recognition and so forth.</p>
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<p>One example of topology is Riemannian Manifold topological space (or simply Riemannian Tensor) which was popularized by Einstein in 1916 when he used Riemannian Tensor to develop his General Theory of Relativity.</p>
<p>Bernhard Riemann first published his work on what we know now as "Riemann Geometry" (or topology) in 1854.</p>
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<p>Here are some researches on Manifold (<a href="http://mathworld.wolfram.com/Manifold.html" target="_blank">http://mathworld.wolfram.com/Manifold.html</a>) in Machine Learning that has been available in the literature.</p>
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<p>1) Non-negative Matrix Factorization on Manifold</p>
<p><a href="http://www.cse.wustl.edu/~zhang/teaching/cs517/Spring12/CourseProjects/nmf-cai2008.pdf" target="_blank">http://www.cse.wustl.edu/~zhang/teaching/cs517/Spring12/CourseProjects/nmf-cai2008.pdf</a></p>
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<p>2) Non-Negative Tensor Factorization with Applications to Statistics and Computer Vision</p>
<p><a href="http://www.cs.huji.ac.il/~shashua/papers/NTF-icml.pdf" target="_blank">http://www.cs.huji.ac.il/~shashua/papers/NTF-icml.pdf</a></p>
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<p>3) Discriminant Analysis on Riemannian Manifold of Gaussian Distributions for Face Recognition with Image Sets</p>
<p><a href="http://www.cv-foundation.org/openaccess/content_cvpr_2015/papers/Wang_Discriminant_Analysis_on_2015_CVPR_paper.pdf" target="_blank">http://www.cv-foundation.org/openaccess/content_cvpr_2015/papers/Wang_Discriminant_Analysis_on_2015_CVPR_paper.pdf</a></p>
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<p>4) Large-Scale Manifold Learning</p>
<p><a href="http://web.cs.ucla.edu/~ameet/largeManifold.pdf" target="_blank">http://web.cs.ucla.edu/~ameet/largeManifold.pdf</a></p>
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<p>As I mentioned above, there are tons of machine learning models that have been developed in the manifold topology framework concepts, not limited to the 4 listed above.</p> While the idea of topological…tag:www.datasciencecentral.com,2016-06-29:6448529:Comment:4419072016-06-29T22:14:40.222ZMichel Baudinhttps://www.datasciencecentral.com/profile/MichelBaudin
<p>While the idea of topological data analysis is new to me and I find it intriguing, I have to take exception to the description of topology itself as "a new breed of math."</p>
<p>Topology, which I would describe as the study of continuity in its most general form, is not new. I first studied it in Bourbaki's 1971 textbook "Topologie Générale," which contained many results from the early 20th century, and later studied algebraic topology under Henri Cartan, whose own work was from the…</p>
<p>While the idea of topological data analysis is new to me and I find it intriguing, I have to take exception to the description of topology itself as "a new breed of math."</p>
<p>Topology, which I would describe as the study of continuity in its most general form, is not new. I first studied it in Bourbaki's 1971 textbook "Topologie Générale," which contained many results from the early 20th century, and later studied algebraic topology under Henri Cartan, whose own work was from the mid-20th century. </p>
<p>It was fascinating stuff, but hardly new.</p>