Asymptotically efficient estimators for stochastic blockmodels: The naive MLE, the rank-constrained MLE, and the spectral estimator
Asymptotically efficient estimators for stochastic blockmodels: The naive MLE, the rank-constrained MLE, and the spectral estimator
复制标题
随机块模型的渐近有效估计器：朴素 MLE、秩约束 MLE 和谱估计器
作者：
M. Tang;Joshua Cape;C. Priebe
我们建立了渐近正态性的结果估计的块概率矩阵$\mathbf{B}$在随机块模型图使用谱嵌入时的平均度增长率为$\omega（\sqrt{n}）$在$n$，顶点数。作为推论，我们证明了当$\mathbf{B}$满秩时，由谱嵌入得到的$\mathbf{B}$的估计是渐近有效的.当$\mathbf{B}$奇异时，谱嵌入估计的均方误差比无秩假设下的对数似然最大化估计的均方误差小，而且与假设$\mathrm{rk}（\mathbf{B}）$已知的真实极大似然估计几乎一样有效.我们的研究结果表明，在随机块模型图的背景下，谱嵌入不仅是计算上容易处理的，但由此产生的估计也是可接受的，即使相比，据称是最佳的，但计算上难以处理的最大似然估计无秩假设。
We establish asymptotic normality results for estimation of the block probability matrix $\mathbf{B}$ in stochastic blockmodel graphs using spectral embedding when the average degrees grows at the rate of $\omega(\sqrt{n})$ in $n$, the number of vertices. As a corollary, we show that when $\mathbf{B}$ is of full-rank, estimates of $\mathbf{B}$ obtained from spectral embedding are asymptotically efficient. When $\mathbf{B}$ is singular the estimates obtained from spectral embedding can have smaller mean square error than those obtained from maximizing the log-likelihood under no rank assumption, and furthermore, can be almost as efficient as the true MLE that assume known $\mathrm{rk}(\mathbf{B})$. Our results indicate, in the context of stochastic blockmodel graphs, that spectral embedding is not just computationally tractable, but that the resulting estimates are also admissible, even when compared to the purportedly optimal but computationally intractable maximum likelihood estimation under no rank assumption.
DOI：
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发表时间：
2017-09
期刊：
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影响因子：
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作者：
Jiaming Xu
通讯作者：
Jiaming Xu