[Paper Review] Tropical Geometry and Piecewise-Linear Approximation of Curves and Surfaces on Weighted Lattices

Petros Maragos, Emmanouil Theodosis|arXiv (Cornell University)|Dec 9, 2019

Polynomial and algebraic computation79 references4 citations

TL;DR

This paper introduces a generalized max-⋆ algebra on weighted lattices to unify tropical geometry and mathematical morphology, enabling optimal piecewise-linear approximation of curves and surfaces via morphological adjunctions. It derives efficient algorithms for max-affine regression and convex fitting, extending max-plus and min-plus frameworks to arbitrary distributive operations ⋆.

ABSTRACT

Tropical Geometry and Mathematical Morphology share the same max-plus and min-plus semiring arithmetic and matrix algebra. In this chapter we summarize some of their main ideas and common (geometric and algebraic) structure, generalize and extend both of them using weighted lattices and a max-$\star$ algebra with an arbitrary binary operation $\star$ that distributes over max, and outline applications to geometry, machine learning, and optimization. Further, we generalize tropical geometrical objects using weighted lattices. Finally, we provide the optimal solution of max-$\star$ equations using morphological adjunctions that are projections on weighted lattices, and apply it to optimal piecewise-linear regression for fitting max-$\star$ tropical curves and surfaces to arbitrary data that constitute polygonal or polyhedral shape approximations. This also includes an efficient algorithm for solving the convex regression problem of data fitting with max-affine functions.

Motivation & Objective

• Unify tropical geometry and mathematical morphology through shared max-plus/min-plus semiring arithmetic and lattice-based algebra.
• Extend both fields using weighted lattices and a generalized max-⋆ algebra with arbitrary binary operations ⋆ distributing over max.
• Develop optimal solutions for max-⋆ equations using morphological adjunctions as projections on weighted lattices.
• Enable optimal piecewise-linear regression for fitting max-⋆ tropical curves and surfaces to arbitrary data, including polygonal and polyhedral approximations.
• Provide an efficient algorithm for convex regression with max-affine functions using the generalized framework.

Proposed method

• Define a max-⋆ algebra on a complete lattice (K, ∨, ∧, ⋆, ⋆′) where ⋆ distributes over ∨, generalizing max-plus and min-plus arithmetic.
• Extend scalar operations to functions on a domain E via pointwise operations: (F ∨ G)(x) = F(x) ∨ G(x), (a ⋆ F)(x) = a ⋆ F(x).
• Introduce ⋆-translations τ_a(F)(x) = a ⋆ F(x) and dual ⋆′-translations τ′_a(F)(x) = a ⋆′ F(x), defining invariance under these operations.
• Define dilation δ and erosion ε operators as Δ⋆I and E⋆I, respectively, with kernel representations H(x,y) = δ(q_y)(x) and H′(x,y) = ε(q′_y)(x).
• Derive unified representations: δ(F)(x) = ⋁_y H(x,y) ⋆ F(y) and ε(F)(x) = ⋀_y H′(x,y) ⋆′ F(y), generalizing convolution-like operations.
• Use morphological adjunctions (residuation pairs) to solve max-⋆ equations and project data onto tropical curves/surfaces, enabling optimal fitting.

Experimental results

Research questions

• RQ1 How can tropical geometry and mathematical morphology be systematically unified through a generalized algebraic framework?
• RQ2 What is the role of weighted lattices in extending max-plus and min-plus algebras to arbitrary distributive operations ⋆?
• RQ3 How can optimal piecewise-linear approximation of curves and surfaces be achieved using morphological adjunctions on weighted lattices?
• RQ4 Can the framework efficiently solve convex regression problems using max-affine functions?
• RQ5 What is the structure of dilation and erosion operators in the generalized max-⋆ algebra, and how do they enable optimal data fitting?

Key findings

• The paper establishes a generalized max-⋆ algebra on weighted lattices that unifies tropical geometry and mathematical morphology via shared semiring and lattice structures.
• It proves that dilation δ and erosion ε operators are Δ⋆I and E⋆I invariant if and only if their kernel representations satisfy δ(F)(x) = ⋁_y H(x,y) ⋆ F(y) and ε(F)(x) = ⋀_y H′(x,y) ⋆′ F(y).
• Optimal solutions to max-⋆ equations are derived using morphological adjunctions, which act as projections on weighted lattices.
• The framework enables optimal piecewise-linear fitting of max-⋆ tropical curves and surfaces to arbitrary data, producing polygonal or polyhedral approximations.
• An efficient algorithm is provided for convex regression with max-affine functions, leveraging the generalized algebraic structure.
• The theory generalizes the Log-Sum-Exp approximation and Maslov dequantization, showing that the max-plus semiring arises as a limit θ↓0 of a family of isomorphic semirings S_θ.