摘要翻译：
本文讨论紧流形$(M,g)$和组合图$g=(V,E)$上的Laplacian本征函数$-\delta\phi=\lambda\phi$的几何。在$\mathbb{T}^d$($\mathbb{Z}^d$)和$\mathbb{R}^n$(self-dual-dual)上可以很好地理解拉普拉斯本征函数的“对偶”几何。对偶几何在纯数学和应用数学的各个领域都发挥着巨大的作用。本文的目的是指出一个特征函数之间的相似性概念，它允许重建几何学。本征函数$\phi_{\lambda},\phi_{\mu}$和$\phi_{\mu}$之间的“相似性”$\alpha(\phi_{\lambda},\phi_{\mu}）的度量是由局部相关性的全局平均值$$\alpha(\phi_{\lambda},\phi_{\mu}）^2=\\phi_{\lambda}\phi_{\mu}\_{l}^2}^{-2}\int_{M}{\left(\int_{M}{p（t,x,y）(\phi_{\lambda}(y)-mu}(y)-\phi_{\mu}(x))dy}\right)^2dx}，$$其中$p(t,x,y)$是经典的热核，$e^{-t\lambda}+e^{-t\mu}=1$。这个概念恢复了所有经典的对偶概念，但同样适用于其他（粗糙的）几何和图；在不同的连续和离散环境下的数值算例说明了这一结果。
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英文标题：
《On the Dual Geometry of Laplacian Eigenfunctions》
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作者：
Alexander Cloninger, Stefan Steinerberger
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最新提交年份：
2018
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分类信息：

一级分类：Electrical Engineering and Systems Science        电气工程与系统科学
二级分类：Signal Processing        信号处理
分类描述：Theory, algorithms, performance analysis and applications of signal and data analysis, including physical modeling, processing, detection and parameter estimation, learning, mining, retrieval, and information extraction. The term "signal" includes speech, audio, sonar, radar, geophysical, physiological, (bio-) medical, image, video, and multimodal natural and man-made signals, including communication signals and data. Topics of interest include: statistical signal processing, spectral estimation and system identification; filter design, adaptive filtering / stochastic learning; (compressive) sampling, sensing, and transform-domain methods including fast algorithms; signal processing for machine learning and machine learning for signal processing applications; in-network and graph signal processing; convex and nonconvex optimization methods for signal processing applications; radar, sonar, and sensor array beamforming and direction finding; communications signal processing; low power, multi-core and system-on-chip signal processing; sensing, communication, analysis and optimization for cyber-physical systems such as power grids and the Internet of Things.
信号和数据分析的理论、算法、性能分析和应用，包括物理建模、处理、检测和参数估计、学习、挖掘、检索和信息提取。“信号”一词包括语音、音频、声纳、雷达、地球物理、生理、（生物）医学、图像、视频和多模态自然和人为信号，包括通信信号和数据。感兴趣的主题包括：统计信号处理、谱估计和系统辨识；滤波器设计；自适应滤波/随机学习；（压缩）采样、传感和变换域方法，包括快速算法；用于机器学习的信号处理和用于信号处理应用的机器学习；网络与图形信号处理；信号处理中的凸和非凸优化方法；雷达、声纳和传感器阵列波束形成和测向；通信信号处理；低功耗、多核、片上系统信号处理；信息物理系统的传感、通信、分析和优化，如电网和物联网。
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一级分类：Computer Science        计算机科学
二级分类：Machine Learning        机器学习
分类描述：Papers on all aspects of machine learning research (supervised, unsupervised, reinforcement learning, bandit problems, and so on) including also robustness, explanation, fairness, and methodology. cs.LG is also an appropriate primary category for applications of machine learning methods.
关于机器学习研究的所有方面的论文（有监督的，无监督的，强化学习，强盗问题，等等），包括健壮性，解释性，公平性和方法论。对于机器学习方法的应用，CS.LG也是一个合适的主要类别。
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一级分类：Mathematics        数学
二级分类：Analysis of PDEs        偏微分方程分析
分类描述：Existence and uniqueness, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDE's, conservation laws, qualitative dynamics
存在唯一性，边界条件，线性和非线性算子，稳定性，孤子理论，可积偏微分方程，守恒律，定性动力学
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一级分类：Mathematics        数学
二级分类：Spectral Theory        光谱理论
分类描述：Schrodinger operators, operators on manifolds, general differential operators, numerical studies, integral operators, discrete models, resonances, non-self-adjoint operators, random operators/matrices
薛定谔算子，流形上的算子，一般微分算子，数值研究，积分算子，离散模型，共振，非自伴算子，随机算子/矩阵
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英文摘要：
  We discuss the geometry of Laplacian eigenfunctions $-\Delta \phi = \lambda \phi$ on compact manifolds $(M,g)$ and combinatorial graphs $G=(V,E)$. The 'dual' geometry of Laplacian eigenfunctions is well understood on $\mathbb{T}^d$ (identified with $\mathbb{Z}^d$) and $\mathbb{R}^n$ (which is self-dual). The dual geometry is of tremendous role in various fields of pure and applied mathematics. The purpose of our paper is to point out a notion of similarity between eigenfunctions that allows to reconstruct that geometry. Our measure of 'similarity' $ \alpha(\phi_{\lambda}, \phi_{\mu})$ between eigenfunctions $\phi_{\lambda}$ and $\phi_{\mu}$ is given by a global average of local correlations $$ \alpha(\phi_{\lambda}, \phi_{\mu})^2 = \| \phi_{\lambda} \phi_{\mu} \|_{L^2}^{-2}\int_{M}{ \left( \int_{M}{ p(t,x,y)( \phi_{\lambda}(y) - \phi_{\lambda}(x))( \phi_{\mu}(y) - \phi_{\mu}(x)) dy} \right)^2 dx},$$ where $p(t,x,y)$ is the classical heat kernel and $e^{-t \lambda} + e^{-t \mu} = 1$. This notion recovers all classical notions of duality but is equally applicable to other (rough) geometries and graphs; many numerical examples in different continuous and discrete settings illustrate the result.
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PDF链接：
https://arxiv.org/pdf/1804.09816