Reciprocal Convex Costs for Ratio Matching: Functional-Equation Characterization and Decision Geometry

TL;DR

This paper characterizes the mathematical form of ratio-induced mismatch costs using functional equations and explores the geometric decision boundaries and properties of the associated argmin mapping.

Contribution

It provides a functional-equation characterization of reciprocal convex costs and analyzes the decision geometry and stability of the argmin mapping under various conditions.

Findings

Cost function J(x) has a hyperbolic cosine form.

Existence of argmin mapping under subspace-closedness.

Geometric-mean decision boundaries for finite dictionaries.

Abstract

We study ratio-induced mismatch costs of the form $c(s,o)=J(\iota_S(s)/\iota_O(o))$, built from positive scale maps $\iota_S:S\to(0,\infty)$ and $\iota_O:O\to(0,\infty)$ and a penalty $J:(0,\infty)\to[0,\infty)$. Assuming inversion symmetry, strict convexity, normalization $J(1)=0$, and a multiplicative d'Alembert identity, we show that $f(u):=1+J(e^u)$ satisfies the additive d'Alembert equation and hence $J(x)=\cosh(a\log x)-1=\tfrac12(x^a+x^{-a})-1$ for some $a>0$. We then analyze the associated argmin mapping over feasible scale sets: existence under explicit subspace-closedness hypotheses, geometric-mean decision boundaries for finite dictionaries with stability away from boundaries, exact compositionality for product models, and an optimal sequential mediation principle given by a geometric mean (or its log-space projection when infeasible). The paper is purely mathematical; any semantic interpretation is optional and external to the theorems proved here.