arXiv 2105.024872024-01-26stat.MLcs.LGstat.ME

基于邻域选择方法的高维函数图模型结构学习

High-dimensional Functional Graphical Model Structure Learning via Neighborhood Selection Approach

Booth School of Business, The University of Chicago（芝加哥大学布斯商学院）；Department of Statistics and Data Science, Cornell University（康奈尔大学统计与数据科学系）；Department of Data Sciences and Operations, Marshall School of Business, University of Southern California（南加州大学马歇尔商学院数据科学与运营系）

Boxin Zhao, Percy S. Zhai, Y. Samuel Wang, Mladen Kolar

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AI总结：

针对高维函数型数据的条件独立结构建模需求，提出基于函数对函数回归的邻域选择法估计高斯函数图模型结构，兼具计算效率与统计一致性，还给出了降维用函数基的选择准则与实用方案。

摘要：

无向图模型被广泛用于建模向量值数据的条件独立结构。然而在诸多现代应用场景中，例如涉及脑电图（EEG）和功能性磁共振成像（fMRI）数据的场景，将观测值建模为多元随机函数而非向量会更为恰当。目前已有研究者提出函数图模型来建模这类函数型数据的条件独立结构。\n我们提出一种邻域选择方法来估计高斯函数图模型的结构：首先通过函数对函数回归估计每个节点的邻域，随后通过合并估计得到的邻域恢复整个图的结构。该方法仅需对随机函数的条件分布作出假设，可直接估计条件独立结构，从而规避了对定义良好的精度算子的需求——而当函数为无限维时，这类精度算子可能不存在。此外，该邻域选择方法计算效率高，且易于并行化。\n理论分析与实验结果均验证了所提方法在高维场景下的统计一致性。我们还研究了中间降维步骤中所用函数基的选择对结果的影响，给出了选择函数基的启发式准则，并提出了两种具有实用价值的基函数选择方案，二者均得到了理论与实验的支撑。

英文摘要：

Undirected graphical models are widely used to model the conditional independence structure of vector-valued data. However, in many modern applications, for example those involving EEG and fMRI data, observations are more appropriately modeled as multivariate random functions rather than vectors. Functional graphical models have been proposed to model the conditional independence structure of such functional data. We propose a neighborhood selection approach to estimate the structure of Gaussian functional graphical models, where we first estimate the neighborhood of each node via a function-on-function regression and subsequently recover the entire graph structure by combining the estimated neighborhoods. Our approach only requires assumptions on the conditional distributions of random functions, and we estimate the conditional independence structure directly. We thus circumvent the need for a well-defined precision operator that may not exist when the functions are infinite dimensional. Additionally, the neighborhood selection approach is computationally efficient and can be easily parallelized. The statistical consistency of the proposed method in the high-dimensional setting is supported by both theory and experimental results. In addition, we study the effect of the choice of the function basis used for dimensionality reduction in an intermediate step. We give a heuristic criterion for choosing a function basis and motivate two practically useful choices, which we justify by both theory and experiments.

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