识别符号式凸性是困难的

中文摘要 AI 辅助

符号式（Signomials）——具有实数指数的广义单项式的有限和——是一种自然的机器学习模型，结合了幂律、倒数关系和乘法交互的简约表示，以及通用逼近和可解释参数。事实上，AI Feynman基准中的100个方程中有45个承认符号式表示。此外，符号式优化广泛应用于工程设计、通信、经济学和机器学习，但通常计算上难以处理。凸性，作为高效优化的黄金标准，为符号式提供了三条可处理性的途径：符号式可能在其原始变量中是凸的，在变量对数变换后变为凸的，或者当其为正时，在对其值取对数后变为凸的——这是纪律性几何规划的基础结构。我们证明，识别每种形式在紧致域上和全局上都是强NP困难的。我们的证明从多项式凸性的间隙承诺变体出发，并开发了在相关对数变换下保持曲率间隙的多项式到符号式的归约。

英文摘要

Signomials---finite sums of generalized monomials with real exponents---are a natural machine learning model, combining parsimonious representations of power laws, inverse relationships, and multiplicative interactions with universal approximation and interpretable parameters. In fact, 45 of the 100 equations in the AI Feynman benchmark admit signomial representations. Moreover, signomial optimization is widely used in engineering design, communications, economics, and machine learning, but is computationally intractable in general. Convexity, the gold standard for efficient optimization, offers three routes to tractability for signomials: a signomial may be convex in its original variables, become convex after a logarithmic change of variables, or, when positive, become convex after additionally taking the logarithm of its value---the structure underlying disciplined geometric programming. We show that recognizing each form is strongly NP-hard, both on compact domains and globally. Our proofs start from gap-promise variants of polynomial convexity and develop polynomial-to-signomial reductions that preserve curvature gaps under the relevant logarithmic transformations.

发表机构

• Singapore University of Technology and Design（新加坡科技设计大学）
• Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）
• Archimedes/Athena Research Center（阿基米德/雅典研究中心）

机构由 AI 辅助整理，请以论文原文为准。