Learning Topology: topological methods for unsupervised learning

4c76f001e0a3d59cc5a269df70940dfd?s=47 Leland McInnes
February 23, 2019

Learning Topology: topological methods for unsupervised learning

A whirlwind tour of how topological ideas and methods can provide powerful solutions to unsupervised learning problems.

4c76f001e0a3d59cc5a269df70940dfd?s=128

Leland McInnes

February 23, 2019
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  1. 6.
  2. 7.

    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>
  3. 8.
  4. 9.
  5. 10.
  6. 17.
  7. 19.
  8. 23.
  9. 25.
  10. 27.
  11. 29.

    Theorem 2 (UMAP Adjunction). The functors FinReal : sFuzz !

    FinEPMet and FinSing : FinEPMet ! sFuzz form an adjunction FinReal a FinSing. <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> <latexit sha1_base64="14RNK4O3bzzPAFIzAUCYYtOueE8=">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</latexit>
  12. 30.
  13. 33.

    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>
  14. 34.

    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> Get the clumps right Get the gaps right
  15. 39.
  16. 46.
  17. 47.
  18. 48.
  19. 54.

    A connected component of a level set of the probability

    density function of the underlying (and unknown) distribution from which our data samples are drawn.
  20. 55.
  21. 56.
  22. 57.
  23. 58.
  24. 59.
  25. 61.
  26. 66.

    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>
  27. 67.
  28. 71.
  29. 72.
  30. 73.
  31. 74.