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Learning Topology: topological methods for unsupervised learning

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.

Leland McInnes

February 23, 2019
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  1. Learning
    Topology
    Topological Methods for
    Unsupervised Learning

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  2. A whirlwind tour of some
    topological data analysis
    techniques

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  3. Sound theory
    Practical application

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  4. A Topology Primer

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  5. 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.
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  6. Functor
    Adjuction
    Limit
    Colimit

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  7. Functor: A function
    between domains of
    discourse

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  8. Adjunction: A near
    equivalence between
    domains of discourse

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  9. Limit: A solution to a
    system of constraints

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  10. Colimit: Gluing together
    a system of objects

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  11. Dimension
    Reduction

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  12. If the data is uniformly
    distributed on the manifold
    then the cover will be “good”

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  13. When is data that
    nicely behaved?

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  14. Assumption:
    Data is uniformly
    distributed on the manifold

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  15. Define a Riemannian
    metric on the manifold to
    make this assumption true

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  16. Assumption:
    The manifold is locally
    connected

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  17. But our local metrics
    are all incompatible!

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  18. Glue things together
    with colimits?

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  19. Theorem 2 (UMAP Adjunction). The
    functors FinReal : sFuzz ! FinEPMet
    and FinSing : FinEPMet ! sFuzz
    form an adjunction FinReal a FinSing.
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  20. Suppose we were given a
    low dimensional
    representation

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  21. We can apply the same
    process to get a
    probabilistic graph!

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  22. X
    a2A
    µ(a) log

    µ(a)
    ⌫(a)

    + (1 µ(a)) log

    1 µ(a)
    1 ⌫(a)

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  23. X
    a2A
    µ(a) log

    µ(a)
    ⌫(a)

    + (1 µ(a)) log

    1 µ(a)
    1 ⌫(a)

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    Get the clumps right
    Get the gaps right

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  24. UMAP on MNIST digits

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  25. UMAP on Fashion MNIST

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  26. UMAP on Kuzushiji-MNIST

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  27. Derived from Adam Bielski’s Siamese/Triplet repository:
    https://github.com/adambielski/siamese-triplet
    Metric Learning

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  28. Word Embeddings

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  29. You shall know a word
    by the company it keeps
    — John Rupert Firth

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  30. Represent a word as a
    multinomial
    distribution of words
    that co-occur with it

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  31. A very large and very
    sparse matrix

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  32. Find the manifold on
    which the words lie

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  33. Assumptions:
    Uniform distribution
    Locally connected

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  34. Use the correct metric
    for multinomial
    parameter space

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  35. Example embedding
    of Yelp reviews

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  36. What do we mean by a
    cluster?

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  37. A cluster is …

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  38. 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.

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  39. How do we compute
    that without knowing
    the PDF?

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  40. Assumption:
    Data is distributed on the
    manifold according to
    some PDF

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  41. Choose a Riemannian
    metric that preserves
    the distribution

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  42. But our local metrics
    may be incompatible…

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  43. Solve the system of
    constraints using limits?

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  44. Theorem 2 (UMAP Adjunction). The
    functors FinReal : sFuzz ! FinEPMet
    and FinSing : FinEPMet ! sFuzz
    form an adjunction FinReal a FinSing.
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  45. We have captured the
    topology of the PDF

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  46. The connected
    components functor p0
    produces a fuzzy set of
    connected components
    π0

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  47. Exclude components
    below a threshold cluster
    size and sort points by
    component membership

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  48. Topology and category
    theory provide a
    different language to
    frame problems

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  49. Topological techniques
    can provide powerful
    solutions

    View full-size slide

  50. Hopefully I have
    motivated you to learn
    more!
    [email protected] @leland_mcinnes

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