Abstract

We propose a one-step procedure to estimate the latent positions in random dot product graphs efficiently. Unlike the classical spectral-based methods, the proposed one-step procedure takes advantage of both the low-rank structure of the expected adjacency matrix and the Bernoulli likelihood information of the sampling model simultaneously. We show that for each vertex, the corresponding row of the one-step estimator (OSE) converges to a multivariate normal distribution after proper scaling and centering up to an orthogonal transformation, with an efficient covariance matrix. The initial estimator for the one-step procedure needs to satisfy the so-called approximate linearization property. The OSE improves the commonly adopted spectral embedding methods in the following sense: Globally for all vertices, it yields an asymptotic sum of squares error no greater than those of the spectral methods, and locally for each vertex, the asymptotic covariance matrix of the corresponding row of the OSE dominates those of the spectral embeddings in spectra. The usefulness of the proposed one-step procedure is demonstrated via numerical examples and the analysis of a real-world Wikipedia graph dataset.

摘要

我们提出了一种单步程序来有效地估计随机点积图中的潜在位置。与经典的基于谱的方法不同，所提出的一步过程同时利用了预期邻接矩阵的低秩结构和采样模型的伯努利似然信息。我们表明，对于每个顶点，一步估计器 (OSE) 的相应行在适当缩放和居中到正交变换后收敛到多元正态分布，具有有效的协方差矩阵。一步过程的初始估计量需要满足所谓的近似线性化特性。OSE 在以下意义上改进了常用的谱嵌入方法：全局地针对所有顶点，它产生的渐近平方和误差不大于光谱方法的误差，并且对于每个顶点局部，OSE 相应行的渐近协方差矩阵支配光谱中光谱嵌入的渐近协方差矩阵。通过数值示例和对现实世界的维基百科图形数据集的分析，证明了所提出的一步过程的有用性。