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UMAP: Uniform Manifold Approximation and Projec...

UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction

This talk will present a new approach to dimension reduction called UMAP. UMAP is grounded in manifold learning and topology, making an effort to preserve the topological structure of the data. The resulting algorithm can provide both 2D visualizations of data of comparable quality to t-SNE, and general purpose dimension reduction. UMAP has been implemented as a (scikit-learn compatible) python library that can perform efficient dimension reduction, scaling out to much larger datasets than t-SNE or other comparable algorithms (see http://github.com/lmcinnes/umap).

Leland McInnes

July 12, 2018
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  1. Who am I? I am a research mathematician at the

    Tutte Institute for Mathematics and Computing My Ph.D. was in Profinite Lie Rings (no, you don’t care) I now work on applying topological techniques to unsupervised learning problems
  2. Matrix Factorization Principal Component Analysis Non-negative Matrix Factorization Latent Dirichlet

    Allocation Word2Vec GloVe Generalised Low Rank Models Linear Autoencoder
  3. Theorem 1 (Nerve theorem). Let U = {Ui }i2I be

    a cover of a topological space X. If, for all ⇢ I T i2 Ui is either contractible or empty, then N(U) is homtopically equivalent to X. <latexit sha1_base64="8ITSuq3xcb28tfscSBtUyYXmYf8=">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</latexit> <latexit sha1_base64="8ITSuq3xcb28tfscSBtUyYXmYf8=">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</latexit> <latexit sha1_base64="8ITSuq3xcb28tfscSBtUyYXmYf8=">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</latexit> <latexit sha1_base64="8ITSuq3xcb28tfscSBtUyYXmYf8=">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</latexit>
  4. Theorem 2 (UMAP Adjunction). The functors FinReal : sFuzz !

    FinEPMet and FinSing : FinEPMet ! sFuzz form an adjunction FinReal a FinSing. <latexit sha1_base64="14RNK4O3bzzPAFIzAUCYYtOueE8=">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</latexit> <latexit sha1_base64="14RNK4O3bzzPAFIzAUCYYtOueE8=">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</latexit> <latexit sha1_base64="14RNK4O3bzzPAFIzAUCYYtOueE8=">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</latexit> <latexit sha1_base64="14RNK4O3bzzPAFIzAUCYYtOueE8=">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</latexit>
  5. Theorem 2 (UMAP Adjunction). The functors FinReal : sFuzz !

    FinEPMet and FinSing : FinEPMet ! sFuzz form an adjunction FinReal a FinSing. <latexit sha1_base64="14RNK4O3bzzPAFIzAUCYYtOueE8=">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</latexit> <latexit sha1_base64="14RNK4O3bzzPAFIzAUCYYtOueE8=">AAADhnicfVLbbtNAEF3XQIO5pfDIy4oEiafITihp+5QKqHiJCDRpK8VRtF6P46X22tpdt6SWH4Cv5CP4B9bOhYTbSJbGZ+bsXM54acSksu3vxo556/ad3dpd6979Bw8f1fcen8kkExRGNIkSceERCRHjMFJMRXCRCiCxF8G5d/m6jJ9fgZAs4UM1T2ESkxlnAaNEaWha/+FKUDTJuAKRqxASAXGRO4XlejBjfA2NR/3jAT72P2WclsyJhYch4KD8TYTETfeE8Y9AInyE3Zio0AtyeZLd3BTYVcka0klvB31QRRNbhPsL2injsw3aOmeLuXhM04JExJhwTNa9bBR3fSLDK7x6tdmyXOD+eoppvWG3nPbhS6eNtXNgdw5Lp+N0uu197LTsyhpoaYPpnvHF9ROaxcAVjYiUY8dO1SQnQjEagV5TJiEl9JLMYKxdTmKQk7zSpcDPNeJj3bD+uMIVusnISSzlPPZ0Zjml/D1Wgn+LjTMVHExyxtNMAaeLQkEWYb2vUmTsMwFURXPtECqY7hXTkAhCtcp/VFFhrOfgcL3c0q8zGC4dqzySCPhMhbmr4LO6Zr7uLH9Fy9gb0JsR0Nddvk9BEH0P+UqQIq/0k5WqFfAfQqnYFqECStFWyuB/O2ftlqMl/dBu9IZL+WroKXqGXiAHdVEPvUMDNELUODXmxlfjm1kzW+a+2V2k7hhLzhO0ZWbvJ8jSKCc=</latexit> <latexit sha1_base64="14RNK4O3bzzPAFIzAUCYYtOueE8=">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</latexit> <latexit sha1_base64="14RNK4O3bzzPAFIzAUCYYtOueE8=">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</latexit>
  6. f(↵, ) = ↵ + ↵ · <latexit sha1_base64="iZE9o2NLLCVem811EJWhe5r3D70=">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</latexit> <latexit

    sha1_base64="iZE9o2NLLCVem811EJWhe5r3D70=">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</latexit> <latexit sha1_base64="iZE9o2NLLCVem811EJWhe5r3D70=">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</latexit> <latexit sha1_base64="iZE9o2NLLCVem811EJWhe5r3D70=">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</latexit> Under a probabilistic fuzzy union the combination of weights on edges is given by
  7. X a2A µ(a) log ✓ µ(a) ⌫(a) ◆ + (1

    µ(a)) log ✓ 1 µ(a) 1 ⌫(a) ◆ <latexit sha1_base64="u7fUXwg3iBtccLqdNJCqNBZ5RiA=">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</latexit> <latexit sha1_base64="u7fUXwg3iBtccLqdNJCqNBZ5RiA=">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</latexit> <latexit sha1_base64="u7fUXwg3iBtccLqdNJCqNBZ5RiA=">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</latexit> <latexit sha1_base64="u7fUXwg3iBtccLqdNJCqNBZ5RiA=">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</latexit>
  8. X a2A µ(a) log ✓ µ(a) ⌫(a) ◆ + (1

    µ(a)) log ✓ 1 µ(a) 1 ⌫(a) ◆ <latexit sha1_base64="u7fUXwg3iBtccLqdNJCqNBZ5RiA=">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</latexit> <latexit sha1_base64="u7fUXwg3iBtccLqdNJCqNBZ5RiA=">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</latexit> <latexit sha1_base64="u7fUXwg3iBtccLqdNJCqNBZ5RiA=">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</latexit> <latexit sha1_base64="u7fUXwg3iBtccLqdNJCqNBZ5RiA=">AAADFXicfVHLjtMwFHXCY4byamHJxqJC6ghRJSloZnaDQIgNYoB2ZqS6qlz3JrHGcSLbYaisbPgJvoYdYsuaJX+Ck2YEHR5Xsnx0jo/u9T2LQnBtguC751+6fOXq1va1zvUbN2/d7vbuHOm8VAwmLBe5OllQDYJLmBhuBJwUCmi2EHC8OH1W68fvQWmey7FZFTDLaCJ5zBk1jpp3LZkSXWZzSwmX+GmFSVYO6A4ReUIExGZAYkWZXbOVJbK5ieJJanbwQzwI8aPW84fpl1St8YaZzObdfjAMo/3HYYQd2AtG+zUYhaPd6AkOh0FTfdTW4bznfSTLnJUZSMME1XoaBoWZWaoMZwKqDik1FJSd0gSmDkqagZ7ZZksVfuCYJY5z5Y40uGF/d1iaab3KFu5lRk2qL2o1+TdtWpp4b2a5LEoDkq0bxaXAJsf1yvGSK2BGrBygTHE3K2YpdRsyLpiLXUyauX9IODMp5Aoy296VHbegQzS4nGViUksMfDBnfOkms2HEavE5uNUoeOXGfF2AoiZXlrzg8i1Q4QJsxo/tOfEfwzsukw1DQ1QutfNo8L/BUTQMXaZvov7BuM1vG91D99EAhWgXHaCX6BBNEEM/vC2v6/X8T/5n/4v/df3U91rPXbRR/refSHP9Ew==</latexit> Get the clumps right Get the gaps right
  9. +

  10. Performance Comparison t-SNE UMAP COIL20 20 seconds 7 seconds MNIST

    22 minutes 98 seconds Fashion MNIST 15 minutes 78 seconds GoogleNews 4.5 hours 14 minutes UMAP speed up over t-SNE COIL20 3x MNIST 13x Fashion MNIST 11x GoogleNews 19x
  11. As long as you can provide a metric for the

    datatype UMAP can combine it with other datatypes!