论文标题
非线性光谱二元性
Nonlinear Spectral Duality
论文作者
论文摘要
对成对均匀凸功能的非线性特征值问题是在各种环境中出现的特定非线性约束优化问题，包括图形挖掘，机器学习和网络科学。通过考虑二元性的不同概念从经典和最近的凸几何理论转变，在这项工作中，我们表明，人们可以从原始人转变为保持光谱，变量光谱以及相应多重性的双重非线性特征值公式。这些非线性频谱二元性能可用于将原始优化问题转换为各种替代方案，并且可能更可治疗的双重问题。我们说明了在各种示例设置中使用非线性光谱二重性，这些设置涉及图表，非线性拉普拉斯人以及凸体之间的距离。
Nonlinear eigenvalue problems for pairs of homogeneous convex functions are particular nonlinear constrained optimization problems that arise in a variety of settings, including graph mining, machine learning, and network science. By considering different notions of duality transforms from both classical and recent convex geometry theory, in this work we show that one can move from the primal to the dual nonlinear eigenvalue formulation maintaining the spectrum, the variational spectrum as well as the corresponding multiplicities unchanged. These nonlinear spectral duality properties can be used to transform the original optimization problem into various alternative and possibly more treatable dual problems. We illustrate the use of nonlinear spectral duality in a variety of example settings involving optimization problems on graphs, nonlinear Laplacians, and distances between convex bodies.
