摘要翻译：
估计一组随机变量的联合概率质量函数(PMF)是统计学习和信号处理的核心。如果没有结构假设，例如将变量建模为马尔可夫链、树或其他图形模型，联合PMF估计通常被认为是不可能的任务--未知量的数量随着变量的数量呈指数增长。但是谁给了我们结构模型？是否有一种通用的、“非参数”的方法来控制联合PMF复杂性，而不依赖于关于潜在概率模型的先验结构假设？是否有可能在不影响分析的前提下发现操作结构？如果我们只观察变量的随机子集，我们还能可靠地估计所有变量的联合PMF吗？本文令人惊讶地表明，如果任何三个变量的联合PMF都可以估计，那么所有变量的联合PMF都可以在相对温和的条件下被证明恢复。这个结果让人想起Kolmogorov的扩张定理--低维分布的一致规范为整个过程导出了一个唯一的概率测度。不同的是，对于有限复杂度（高维PMF的秩）的过程，仅从三维分布就可以获得完整的表征。事实上，并不是所有的三维PMF都是需要的；在更严格的条件下，即使是二维也可以。利用多重线性代数，证明了这样的高维PMF完备性是可以保证的，并得到了几个相关的可辨识性结果。并提供了一种实用高效的算法来执行恢复任务。通过对电影推荐和数据分类的仿真和实际数据实验，验证了该方法的有效性。
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英文标题：
《Tensors, Learning, and 'Kolmogorov Extension' for Finite-alphabet Random
  Vectors》
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作者：
Nikos Kargas, Nicholas D. Sidiropoulos, Xiao Fu
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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：Information Theory        信息论
分类描述：Covers theoretical and experimental aspects of information theory and coding. Includes material in ACM Subject Class E.4 and intersects with H.1.1.
涵盖信息论和编码的理论和实验方面。包括ACM学科类E.4中的材料，并与H.1.1有交集。
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一级分类：Mathematics        数学
二级分类：Information Theory        信息论
分类描述：math.IT is an alias for cs.IT. Covers theoretical and experimental aspects of information theory and coding.
它是cs.it的别名。涵盖信息论和编码的理论和实验方面。
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一级分类：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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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：
  Estimating the joint probability mass function (PMF) of a set of random variables lies at the heart of statistical learning and signal processing. Without structural assumptions, such as modeling the variables as a Markov chain, tree, or other graphical model, joint PMF estimation is often considered mission impossible - the number of unknowns grows exponentially with the number of variables. But who gives us the structural model? Is there a generic, `non-parametric' way to control joint PMF complexity without relying on a priori structural assumptions regarding the underlying probability model? Is it possible to discover the operational structure without biasing the analysis up front? What if we only observe random subsets of the variables, can we still reliably estimate the joint PMF of all? This paper shows, perhaps surprisingly, that if the joint PMF of any three variables can be estimated, then the joint PMF of all the variables can be provably recovered under relatively mild conditions. The result is reminiscent of Kolmogorov's extension theorem - consistent specification of lower-dimensional distributions induces a unique probability measure for the entire process. The difference is that for processes of limited complexity (rank of the high-dimensional PMF) it is possible to obtain complete characterization from only three-dimensional distributions. In fact not all three-dimensional PMFs are needed; and under more stringent conditions even two-dimensional will do. Exploiting multilinear algebra, this paper proves that such higher-dimensional PMF completion can be guaranteed - several pertinent identifiability results are derived. It also provides a practical and efficient algorithm to carry out the recovery task. Judiciously designed simulations and real-data experiments on movie recommendation and data classification are presented to showcase the effectiveness of the approach.
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PDF链接：
https://arxiv.org/pdf/1712.00205