Reciprocal Convex Costs for Ratio Matching: Functional-Equation Characterization and Decision Geometry
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866918360789286912 |
|---|---|
| author | Washburn, Jonathan Barghi, Amir Rahnamai |
| author_facet | Washburn, Jonathan Barghi, Amir Rahnamai |
| contents | We study ratio-induced mismatch costs of the form $c(s,o)=J(ι_S(s)/ι_O(o))$, built from positive scale maps $ι_S:S\to(0,\infty)$ and $ι_O:O\to(0,\infty)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_00006 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Reciprocal Convex Costs for Ratio Matching: Functional-Equation Characterization and Decision Geometry Washburn, Jonathan Barghi, Amir Rahnamai Optimization and Control 39B22, 49J40, 26A51, 90C25, 90C31, 94A17 We study ratio-induced mismatch costs of the form $c(s,o)=J(ι_S(s)/ι_O(o))$, built from positive scale maps $ι_S:S\to(0,\infty)$ and $ι_O:O\to(0,\infty)$ 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. |
| title | Reciprocal Convex Costs for Ratio Matching: Functional-Equation Characterization and Decision Geometry |
| topic | Optimization and Control 39B22, 49J40, 26A51, 90C25, 90C31, 94A17 |
| url | https://arxiv.org/abs/2603.00006 |