Learning Topology: topological methods for unsupervised learning

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

Transcript

1. Learning Topology Topological Methods for Unsupervised Learning
2. A whirlwind tour of some topological data analysis techniques
3. Sound theory Practical application
4. A Topology Primer
5. Simplices
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>
11. Functor Adjuction Limit Colimit
12. Functor: A function between domains of discourse
13. Adjunction: A near equivalence between domains of discourse
14. Limit: A solution to a system of constraints
15. Colimit: Gluing together a system of objects
16. Dimension Reduction
18. If the data is uniformly distributed on the manifold then the cover will be “good”
20. When is data that nicely behaved?
21. Assumption: Data is uniformly distributed on the manifold
22. Define a Riemannian metric on the manifold to make this assumption true
24. Assumption: The manifold is locally connected
26. But our local metrics are all incompatible!
28. Glue things together with colimits?
29. 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>
31. Suppose we were given a low dimensional representation
32. We can apply the same process to get a probabilistic graph!
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=">AAADFXicfVHLjtMwFHXCY4byamHJxqJC6ghRJSloZnaDQIgNYoB2ZqS6qlz3JrHGcSLbYaisbPgJvoYdYsuaJX+Ck2YEHR5Xsnx0jo/u9T2LQnBtguC751+6fOXq1va1zvUbN2/d7vbuHOm8VAwmLBe5OllQDYJLmBhuBJwUCmi2EHC8OH1W68fvQWmey7FZFTDLaCJ5zBk1jpp3LZkSXWZzSwmX+GmFSVYO6A4ReUIExGZAYkWZXbOVJbK5ieJJanbwQzwI8aPW84fpl1St8YaZzObdfjAMo/3HYYQd2AtG+zUYhaPd6AkOh0FTfdTW4bznfSTLnJUZSMME1XoaBoWZWaoMZwKqDik1FJSd0gSmDkqagZ7ZZksVfuCYJY5z5Y40uGF/d1iaab3KFu5lRk2qL2o1+TdtWpp4b2a5LEoDkq0bxaXAJsf1yvGSK2BGrBygTHE3K2YpdRsyLpiLXUyauX9IODMp5Aoy296VHbegQzS4nGViUksMfDBnfOkms2HEavE5uNUoeOXGfF2AoiZXlrzg8i1Q4QJsxo/tOfEfwzsukw1DQ1QutfNo8L/BUTQMXaZvov7BuM1vG91D99EAhWgXHaCX6BBNEEM/vC2v6/X8T/5n/4v/df3U91rPXbRR/refSHP9Ew==</latexit> <latexit sha1_base64="u7fUXwg3iBtccLqdNJCqNBZ5RiA=">AAADFXicfVHLjtMwFHXCY4byamHJxqJC6ghRJSloZnaDQIgNYoB2ZqS6qlz3JrHGcSLbYaisbPgJvoYdYsuaJX+Ck2YEHR5Xsnx0jo/u9T2LQnBtguC751+6fOXq1va1zvUbN2/d7vbuHOm8VAwmLBe5OllQDYJLmBhuBJwUCmi2EHC8OH1W68fvQWmey7FZFTDLaCJ5zBk1jpp3LZkSXWZzSwmX+GmFSVYO6A4ReUIExGZAYkWZXbOVJbK5ieJJanbwQzwI8aPW84fpl1St8YaZzObdfjAMo/3HYYQd2AtG+zUYhaPd6AkOh0FTfdTW4bznfSTLnJUZSMME1XoaBoWZWaoMZwKqDik1FJSd0gSmDkqagZ7ZZksVfuCYJY5z5Y40uGF/d1iaab3KFu5lRk2qL2o1+TdtWpp4b2a5LEoDkq0bxaXAJsf1yvGSK2BGrBygTHE3K2YpdRsyLpiLXUyauX9IODMp5Aoy296VHbegQzS4nGViUksMfDBnfOkms2HEavE5uNUoeOXGfF2AoiZXlrzg8i1Q4QJsxo/tOfEfwzsukw1DQ1QutfNo8L/BUTQMXaZvov7BuM1vG91D99EAhWgXHaCX6BBNEEM/vC2v6/X8T/5n/4v/df3U91rPXbRR/refSHP9Ew==</latexit> <latexit sha1_base64="u7fUXwg3iBtccLqdNJCqNBZ5RiA=">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</latexit>
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=">AAADFXicfVHLjtMwFHXCY4byamHJxqJC6ghRJSloZnaDQIgNYoB2ZqS6qlz3JrHGcSLbYaisbPgJvoYdYsuaJX+Ck2YEHR5Xsnx0jo/u9T2LQnBtguC751+6fOXq1va1zvUbN2/d7vbuHOm8VAwmLBe5OllQDYJLmBhuBJwUCmi2EHC8OH1W68fvQWmey7FZFTDLaCJ5zBk1jpp3LZkSXWZzSwmX+GmFSVYO6A4ReUIExGZAYkWZXbOVJbK5ieJJanbwQzwI8aPW84fpl1St8YaZzObdfjAMo/3HYYQd2AtG+zUYhaPd6AkOh0FTfdTW4bznfSTLnJUZSMME1XoaBoWZWaoMZwKqDik1FJSd0gSmDkqagZ7ZZksVfuCYJY5z5Y40uGF/d1iaab3KFu5lRk2qL2o1+TdtWpp4b2a5LEoDkq0bxaXAJsf1yvGSK2BGrBygTHE3K2YpdRsyLpiLXUyauX9IODMp5Aoy296VHbegQzS4nGViUksMfDBnfOkms2HEavE5uNUoeOXGfF2AoiZXlrzg8i1Q4QJsxo/tOfEfwzsukw1DQ1QutfNo8L/BUTQMXaZvov7BuM1vG91D99EAhWgXHaCX6BBNEEM/vC2v6/X8T/5n/4v/df3U91rPXbRR/refSHP9Ew==</latexit> <latexit sha1_base64="u7fUXwg3iBtccLqdNJCqNBZ5RiA=">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</latexit> Get the clumps right Get the gaps right
35. UMAP on MNIST digits
36. UMAP on Fashion MNIST
37. UMAP on Kuzushiji-MNIST
38. Derived from Adam Bielski’s Siamese/Triplet repository: https://github.com/adambielski/siamese-triplet Metric Learning
40. Word Embeddings
41. You shall know a word by the company it keeps — John Rupert Firth
42. Represent a word as a multinomial distribution of words that co-occur with it
43. A very large and very sparse matrix
44. Find the manifold on which the words lie
45. Assumptions: Uniform distribution Locally connected
49. Use the correct metric for multinomial parameter space
50. Example embedding of Yelp reviews
51. Clustering
52. What do we mean by a cluster?
53. A cluster is …
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.
60. How do we compute that without knowing the PDF?
62. Assumption: Data is distributed on the manifold according to some PDF
63. Choose a Riemannian metric that preserves the distribution
64. But our local metrics may be incompatible…
65. Solve the system of constraints using limits?
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>
68. We have captured the topology of the PDF
69. The connected components functor p0 produces a fuzzy set of connected components π0
70. Exclude components below a threshold cluster size and sort points by component membership
75. Conclusions
76. Topology and category theory provide a different language to frame problems
77. Topological techniques can provide powerful solutions
78. Hopefully I have motivated you to learn more! leland.mcinnes@gmail.com @leland_mcinnes