Modeling Higher-Order Dependence With Cumulant Tensors

多元分布的第二个累积张量是它的协方差矩阵，它提供了它的依赖结构的部分描述（在高斯情况下是完整的）。无数成功的统计方法都是基于协方差矩阵的分析，例如在主成分分析中施加秩限制，或者在高斯图形模型中施加逆零。此外，协方差矩阵在金融和其他涉及风险收益优化的领域的优化中起着关键作用，因为它是产生变量线性组合方差的双线性形式。对于多元非高斯数据，协方差矩阵是依赖结构的不完全描述。累积张量是单变量偏度和峰度的多元推广，是协方差矩阵的高阶推广，可以更完整地描述相关性。本研究探讨了围绕用累积张量建模高阶非高斯依赖的理论、估计、算法和应用中的一些问题。从计算机视觉、金融和计算生物学等现代应用中产生的数据很少能用正态分布很好地描述，尽管分析通常是这样进行的。例如，金融危机及其对许多投资者造成的损害的一个原因是过度依赖主要适用于正态分布的基于方差的风险度量。这使得风险在某种意义上隐藏在高阶结构中，在那里它可以被忽略，甚至通过应用传统的风险度量而变得更糟。累积张量为高阶相关性的建模提供了一条很有前途的途径。开发这些模型的成功将对具有复杂依赖性的现实世界数据的分析产生广泛的影响，特别是在建模和管理金融风险以及降维方面，并有助于提高金融系统部分的稳健性。

The second cumulant tensor of a multivariate distribution is itscovariance matrix, which provides a partial description of itsdependence structure (complete in the Gaussian case). Innumerablesuccessful statistical methods are based on analyzing the covariancematrix, e.g. imposing rank restrictions as in principal componentanalysis or zeros in its inverse as in Gaussian graphical models.Moreover, the covariance matrix plays a critical role in optimizationin finance and other areas involving optimization of risky payoffs,since it is the bilinear form yielding the variance of a linearcombination of variables. For multivariate, non-Gaussian data, thecovariance matrix is an incomplete description of the dependencestructure. Cumulant tensors are the multivariate generalization ofunivariate skewness and kurtosis and the higher-order generalizationof covariance matrices, and allow a more complete description ofdependence. The research investigates a number of problems in theory,estimation, algorithms, and applications around modeling higher-ordernon-Gaussian dependence with cumulant tensors.Data arising from modern applications like computer vision, finance,and computational biology are rarely well described by a normaldistribution, though analysis often proceeds as if they were. Forexample, one cause of the financial crisis and the damage it did tomany investors was an over-reliance on the variance-based riskmeasures appropriate primarily for normal distributions. This canallow risk to be in a sense hidden in the higher-order structure,where it can be ignored or even made worse by application oftraditional risk metrics. Cumulant tensors provide a promising avenuefor modeling higher-order dependence. Success in developing thesemodels will have broad impacts in the analysis of real-world data withcomplex dependence, particularly in modeling and managing financialrisk and in dimension reduction, and could help improve the robustnessof parts of the financial system.