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nLab > Latest Changes: inner product of multisets

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  1.  
    • CommentRowNumber1.
    • CommentAuthorEric
    • CommentTimeOct 15th 2009

    I created inner product of multisets, which I hope will help make some sense of some speculations over at the discussion of magnitude of metric spaces.

    Thank you Toby for your help on my personal wikiweb.

    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeOct 15th 2009

    You're welcome!

    • CommentRowNumber3.
    • CommentAuthorEric
    • CommentTimeOct 15th 2009

    The solution turned out to be very pretty. What do you think? It is rare I can contribute something technical, so I'm happy at the moment :)

    • CommentRowNumber4.
    • CommentAuthorTobyBartels
    • CommentTimeOct 15th 2009

    I fixed a typo and added a note to the end.

    • CommentRowNumber5.
    • CommentAuthorEric
    • CommentTimeOct 15th 2009

    Cool. By the way, I defined a product of multisets, which turns them into a rig (at least for finite multisets).

    • CommentRowNumber6.
    • CommentAuthorTobyBartels
    • CommentTimeOct 16th 2009

    It's a rig regardless of cardinalities, since arithmetic of arbitrary cardinal numbers is riggish.

    • CommentRowNumber7.
    • CommentAuthorEric
    • CommentTimeOct 16th 2009

    Neat :)

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 27th 2009
    As I said at the cafe, this inner product arises from what machine learning people call the normalized bag-of-words kernel.
    • CommentRowNumber9.
    • CommentAuthorEric
    • CommentTimeOct 27th 2009
    • (edited Oct 27th 2009)

    Thanks. I added a section "Machine Learning" with a reference to the n-Cafe.

    By the way, the reason I'm looking into this is that I'm trying to relate it somehow to the cardinality of the colimit of a Set-valued functor as I outlined here. It would be neat if your discussion about kernel methods became relevant to weighted colimits (which are categorified integrals).

    • CommentRowNumber10.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 2nd 2021

    Added related concepts.

    diff, v7, current

1 to 10 of 10

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