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• 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 1st 2021