摘要翻译：
使用椭圆分布的混合模型比广泛使用的高斯混合模型(GMM)具有更强的鲁棒性、灵活性和稳定性。然而，现有的基于椭圆混合模型(EMM)的研究局限于几种特定类型的椭圆概率密度函数，没有得到一般的解决方案或系统分析框架的支持；这极大地限制了EMMs在应用中的严格性和能力。为此，我们提出了一个新的估计和分析EMMs的通用框架，通过黎曼流形优化实现。首先，我们研究了黎曼流形与椭圆分布之间的关系，原来的黎曼流形与重新构造的黎曼流形之间的这种联系表明了黎曼流形之间的不匹配，这是现有的求解一般EMMS的优化方法失败的主要原因。然后，我们提出了一个基于重新设计成本优化的通用求解器，并证明了与原问题相同的最优解的存在性；这是以一种简单、快速和稳定的方式实现的。我们进一步计算EMM的影响函数作为理论界，以量化对离群点的鲁棒性。综合数值结果表明，该框架能够稳定地适应具有不同性质的EMMs函数，并具有较快的收敛速度。最后，通过分析和综合仿真，证明了该框架比标准GMM增强了鲁棒性和灵活性。
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英文标题：
《A universal framework for learning the elliptical mixture model》
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作者：
Shengxi Li, Zeyang Yu, Danilo Mandic
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最新提交年份：
2020
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分类信息：

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：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：Statistics        统计学
二级分类：Machine Learning        机器学习
分类描述：Covers machine learning papers (supervised, unsupervised, semi-supervised learning, graphical models, reinforcement learning, bandits, high dimensional inference, etc.) with a statistical or theoretical grounding
覆盖机器学习论文（监督，无监督，半监督学习，图形模型，强化学习，强盗，高维推理等）与统计或理论基础
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英文摘要：
  Mixture modelling using elliptical distributions promises enhanced robustness, flexibility and stability over the widely employed Gaussian mixture model (GMM). However, existing studies based on the elliptical mixture model (EMM) are restricted to several specific types of elliptical probability density functions, which are not supported by general solutions or systematic analysis frameworks; this significantly limits the rigour and the power of EMMs in applications. To this end, we propose a novel general framework for estimating and analysing the EMMs, achieved through Riemannian manifold optimisation. First, we investigate the relationships between Riemannian manifolds and elliptical distributions, and the so established connection between the original manifold and a reformulated one indicates a mismatch between those manifolds, the major cause of failure of the existing optimisation for solving general EMMs. We next propose a universal solver which is based on the optimisation of a re-designed cost and prove the existence of the same optimum as in the original problem; this is achieved in a simple, fast and stable way. We further calculate the influence functions of the EMM as theoretical bounds to quantify robustness to outliers. Comprehensive numerical results demonstrate the ability of the proposed framework to accommodate EMMs with different properties of individual functions in a stable way and with fast convergence speed. Finally, the enhanced robustness and flexibility of the proposed framework over the standard GMM are demonstrated both analytically and through comprehensive simulations.
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PDF链接：
https://arxiv.org/pdf/1805.08045