下组合图Laplacian估计的闭式解 无环拓扑约束

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摘要翻译：
如何从数据样本中获取图形是图形信号处理中的一个重要问题。在特定的拓扑约束下，一种基于高斯极大似然估计的图学习方法。为了解决这个问题，我们通常需要迭代凸优化求解器。在本文中，我们证明了当目标图拓扑不包含任何圈时，解就经验协方差矩阵而言具有闭合形式。这使得我们能够有效地从数据中构造树图，即使只有一个可用的数据样本。我们也给出了用相同解逼近循环图时目标函数的误差界。作为一个例子，我们考虑了一个图像去噪问题，其中对于每一个输入图像，我们根据理论结果构造一个图。然后我们在此图的基础上应用低通图滤波器。实验结果表明，在一定条件下，图学习算法给出的权值比双边权值具有更好的去噪效果。
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英文标题：
《Closed Form Solutions of Combinatorial Graph Laplacian Estimation under
  Acyclic Topology Constraints》
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作者：
Keng-Shih Lu and Antonio Ortega
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最新提交年份：
2017
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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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英文摘要：
  How to obtain a graph from data samples is an important problem in graph signal processing. One way to formulate this graph learning problem is based on Gaussian maximum likelihood estimation, possibly under particular topology constraints. To solve this problem, we typically require iterative convex optimization solvers. In this paper, we show that when the target graph topology does not contain any cycle, then the solution has a closed form in terms of the empirical covariance matrix. This enables us to efficiently construct a tree graph from data, even if there is only a single data sample available. We also provide an error bound of the objective function when we use the same solution to approximate a cyclic graph. As an example, we consider an image denoising problem, in which for each input image we construct a graph based on the theoretical result. We then apply low-pass graph filters based on this graph. Experimental results show that the weights given by the graph learning solution lead to better denoising results than the bilateral weights under some conditions.
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PDF链接：
https://arxiv.org/pdf/1711.00213

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关键词：ACI LAC CIA APL PLA data weights 权值 具有 problem