Marie-Charlotte Brandenburg: Separating points by piecewise linear functions: The real tropical geometry of neural networks

Abstract

Given a fixed set of points, it is well-known that the combinatorics of separating these points by a hyperplane (defined through a linear function) is described through

• the chambers of a hyperplane arrangement
• the vertices of a zonotope
• covectors of a realizable oriented matroid

But what if we replace the linear by a piecewise linear function, e.g. defined through a ReLU neural network?
In this talk, we consider the analogous objects when the linear function which defines a hyperplane is replaced by a piecewise linear function. Among others, this yields generalizations into an arrangement of polyhedral fans, whose combinatorial essence can be described by a collection of bipartite graphs.