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Stream: learning: questions

Topic: Piecewise linear functions

Bruno Gavranović (Oct 30 2022 at 20:11):

Is there a category of piecewise linear functions? Piecewise linear functions are often used in math, but I've never seen a category theoretic account of them.

Jean-Baptiste Vienney (Oct 30 2022 at 20:31):

The identity function is piecewise linear and piecewise linear functions are stable by composition so yes there is for example a category with only one object $\mathbb{R}$ and and all the piecewise linear functions $\mathbb{R} \rightarrow \mathbb{R}$ . You can make linear combinations of your piecewise linear functions and the composition is bilinear, so it is a category enriched over $\mathbb{R}$ -vector spaces ie. a $\mathbb{R}$ -linear category.

A category with only one object is not very interesting, it would be more interesting to have more objects, for instance all the $\mathbb{R}^{n}$ but I'm not sure what could be the definition of a piecewise linear function $\mathbb{R}^{n} \rightarrow \mathbb{R}^{p}$ ?

Mike Shulman (Oct 31 2022 at 05:19):

Wikipedia suggests that in higher dimensions, the domain of each piece should be a polygon or polytope. There is a well-studied category of piecewise-linear manifolds.

Reid Barton (Oct 31 2022 at 05:42):

A piecewise linear function $f: \mathbb{R}^n \to \mathbb{R}^m$ is a continuous function satisfying either of the following equivalent properties:

$f$ is "locally conical" in the sense that any $x \in \mathbb{R}^n$ admits a conical neighborhood $U$ (e.g. a ball centered on $x$ ) on which $f$ preserves the conical structure: $f(ax + by) = af(x) + bf(y)$ for $y \in U$ , $0 \le a, b$ , $a + b = 1$ .
The domain $\mathbb{R}^n$ admits a locally finite decomposition into simplices such that $f$ is (affine) linear on each simplex.

From Roarke-Sanderson, chapters 1 and 2.

Robin Piedeleu (Oct 31 2022 at 08:18):

There is even a (symmetric monoidal) category of piecewise-linear relations! @Guillaume Boisseau and I have a paper giving a presentation of it. For us a piecewise linear relation is a finite union of convex polyhedra (which, in our definition, includes convex cones, with potentially unbounded faces).

Robin Piedeleu (Oct 31 2022 at 08:26):

I think in this setting the maps - the single-valued, total relations - are the piecewise-linear functions in the sense that Reid described above (and Reid helped us a lot in working out some of the details of the completeness proof, in this zulip thread btw).