Monday, June 28, 2010

Metrics via pullbacks

One of the things I end up having to do a lot of is ponder distance metrics. Not the nicely behaved norm-induced ones, but more bizarre entities. metrics over shapes, metrics on lattices, metrics on distributions, and the like.

There are many constructions for building metric structures on spaces for which it's not clear how to do so. One of the neatest methods is via pullbacks, exploiting the algebraic and continuous duals for vector spaces.

The basic idea is as follows: You want to build a metric on some (usually ill-formed) vector space V. Fortunately for you, the space V* of linear functionals over V is better behaved. Even better, you can define a norm on V*. This allows you to do a little magic.

Define a function || || on V as ||x|| = sup f(v), over all f in V*, where ||f|| <= 1. This is of course the "dual" norm. It can be shown that it indeed satisfies the properties of a norm. Once you have a norm, then d(x,y) = ||x -y ||. Voila !

These types of constructions are particularly useful when dealing with distributions (the Schwarz kind) and their geometric generalizations, the currents (which are a measure-theoretic way of defining surfaces). Distributions can be nasty - you can only interact with them via their linear functionals (the space of smooth functions with compact support). But this construction allows you to put nice metric structures on them.

Some examples of metrics arising in this manner:

  • The l_1 distance between probability measures (or the total variation distance)
  • The earthmover distance between probability measures (this is quite nifty)
  • The current distance (between measures, or between currents). 

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