Nested Barycentric Coordinate System as an Explicit Feature Map

We introduce a new embedding technique based on barycentric coordinate system. We show that our embedding can be used to transforms the problem of polytope approximation into that of finding a \emph{linear} classifier in a higher (but nevertheless quite sparse) dimensional representation. This embedding in effect maps a piecewise linear function into a single linear function, and allows us to invoke well-known algorithms for the latter problem to solve the former.

We demonstrate that our embedding has applications to the problems of approximating separating polytopes -- in fact, it can approximate any convex body and multiple convex bodies -- as well as to classification by separating polytopes and piecewise linear regression.