Representing Piecewise-Linear Functions by Functions with Minimal Arity

Any continuous piecewise-linear function F Rⁿ→ R can be represented as a linear combination of max functions of at most $n+1$ affine-linear functions. In our previous paper [``Representing piecewise linear functions by functions with small arity'', AAECC, 2023], we showed that this upper bound of $n+1$ arguments is tight. In the present paper, we extend this result by establishing a correspondence between the function F and the minimal number of arguments that are needed in any such decomposition. We show that the tessellation of the input space Rⁿ induced by the function F has a direct connection to the number of arguments in the max functions.