Decomposition Polyhedra of Piecewise Linear Functions

Abstract

In this paper we contribute to the frequently studied question of how to decompose a continuous piecewise linear (CPWL) function into a difference of two convex CPWL functions. Every CPWL function has infinitely many such decompositions, but for applications in optimization and neural network theory, it is crucial to find decompositions with as few linear pieces as possible. This is a highly challenging problem, as we further demonstrate by disproving a recently proposed approach by Tran and Wang [Minimal representations of tropical rational functions. Algebraic Statistics, 15(1):27-59, 2024]. To make the problem more tractable, we propose to fix an underlying polyhedral complex determining the possible locus of nonlinearity. Under this assumption, we prove that the set of decompositions forms a polyhedron that arises as intersection of two translated cones. We prove that irreducible decompositions correspond to the bounded faces of this polyhedron and minimal solutions must be vertices. We then identify cases with a unique minimal decomposition, and illustrate how our insights have consequences in the theory of submodular functions. Finally, we improve upon previous constructions of neural networks for a given convex CPWL function and apply our framework to obtain results in the nonconvex case.

Figures (5)

• Figure 1: Two different parameterizations of the function that computes the median of $\{0,x_1,x_2\}$; see \ref{['ex:parameterizations']} for more details. In \ref{['fig:parameterization-weights']}, the convex breakpoints are colored in blue, and concave breakpoints are dashed and colored in red.
• Figure 2: Visualization of minimality, where a decomposition $(g,h)$ is described by the number of pieces of $g$ and $h$. A decomposition is minimal, if the rectangle spanned with $(0,0)$ does not contain another decomposition.
• Figure 3: The hyperplane extension of the median (second largest number) of $0,x_1,x_2$ (i.e., $n=3$) (\ref{['ex:hyperplane-median']}), which agrees with the local maxima decomposition (\ref{['ex:lattice-median']}) up to a factor $2$. These representations do not agree for the median when $n>3$.
• Figure 4: 4 polygons that cannot be placed in a coherent way in $\mathbb{R}^3$
• Figure 5: A $2$-dimensional representation of $\mathcal{P}$. The blue lines correspond to convex breakpoints of the function $f$, that is, a cone $\sigma \in \mathcal{P}^2$ such that $w(\sigma) > 0$. The concave breakpoints ($w(\sigma)<0)$ are dashed and colored in orange. $f$ has no breakpoints on the gray, dotted lines ($w(\sigma)=0$).

Theorems & Definitions (82)

• Lemma 3.1
• Lemma 3.2
• Proposition 3.3
• Definition 3.4
• Theorem 3.5
• Remark 3.6
• Definition 3.7
• Theorem 3.8
• Definition 3.9
• Lemma 3.10