Gaussian determinantal processes: A new model for directionality in data
Gaussian determinantal processes: A new model for directionality in data
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DOI：
10.1073/pnas.1917151117
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发表时间：
2020-06
期刊：
影响因子：
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通讯作者：
Subhro Ghosh;P. Rigollet
中科院分区：
文献类型：
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作者：
Subhro Ghosh;P. Rigollet
数据日益复杂的本质导致统计学家重新思考即使是最基本的建模假设。在这种情况下，决定性点过程（DPP）建模范式促进了手头样本的多样性。在这项工作中，我们引入了一个简单而灵活的高斯DPP模型来捕获数据中的方向性。使用高斯DPP作为ansatz，我们获得了一种降维方法，该方法比标准主成分分析（PCA）产生更好、更可读的原始数据表示。这些发现得到了我们的估计器性能的有限样本分析的支持，特别是在一个类似于用于分析PCA的尖峰模型中。确定性点过程（DPPs）最近成为流行的工具，用于模拟数据中的负依赖或排斥现象。然而，我们对经典参数统计理论的模拟的理解对于这类模型是相当有限的。在这项工作中，我们研究了高斯dpp的参数族，参数调制对观测点有明显的可解释的影响。我们证明了参数调制通过在它们的斥力结构中引入方向性来影响观测点，并且主方向对应于最大（即最长距离）依赖的方向。该模型很容易产生主成分分析（PCA）的可行替代方案，作为一种降维工具，它倾向于数据最分散的方向。这种方法上的贡献是由一个类似于用于协方差矩阵作为研究PCA框架的尖刺模型的统计分析来补充的。这些理论研究为随机矩阵理论、随机几何和相关主题的进一步研究揭示了有趣的问题。
Significance The increasingly complex nature of data has led statisticians to rethinking even the most basic of modeling assumptions. In this context, a determinantal point process (DPP) modeling paradigm promotes diversity in the sample at hand. In this work, we introduce a simple and flexible Gaussian DPP model to capture directionality in the data. Using the Gaussian DPP as an ansatz, we obtain an approach for dimensionality reduction that produces a better and more readable representation of the original data than standard principal component analysis (PCA). These findings are supported by a finite sample analysis of the performance of our estimator, in particular in a spiked model similar to the one employed to analyze PCA. Determinantal point processes (DPPs) have recently become popular tools for modeling the phenomenon of negative dependence, or repulsion, in data. However, our understanding of an analogue of a classical parametric statistical theory is rather limited for this class of models. In this work, we investigate a parametric family of Gaussian DPPs with a clearly interpretable effect of parametric modulation on the observed points. We show that parameter modulation impacts the observed points by introducing directionality in their repulsion structure, and the principal directions correspond to the directions of maximal (i.e., the most long-ranged) dependency. This model readily yields a viable alternative to principal component analysis (PCA) as a dimension reduction tool that favors directions along which the data are most spread out. This methodological contribution is complemented by a statistical analysis of a spiked model similar to that employed for covariance matrices as a framework to study PCA. These theoretical investigations unveil intriguing questions for further examination in random matrix theory, stochastic geometry, and related topics.