In statistical connectomics, the quantitative study of brain networks,
estimating the mean of a population of graphs based on a sample is a core
problem. Often, this problem is especially difficult because the sample or
cohort size is relatively small, sometimes even a single subject. While using
the element-wise sample mean of the adjacency matrices is a common approach,
this method does not exploit any underlying structural properties of the
graphs. We propose using a low-rank method which incorporates tools for
dimension selection and diagonal augmentation to smooth the estimates and
improve performance over the naive methodology for small sample sizes.
Theoretical results for the stochastic blockmodel show that this method offers
major improvements when there are many vertices. Similarly, we demonstrate that
the low-rank methods outperform the standard sample mean for a variety of
independent edge distributions as well as human connectome data derived from
magnetic resonance imaging, especially when sample sizes are small. Moreover,
the low-rank methods yield "eigen-connectomes", which correlate with the
lobe-structure of the human brain and superstructures of the mouse brain. These
results indicate that low-rank methods are an important part of the tool box
for researchers studying populations of graphs in general, and statistical
connectomics in particular.

本文提出了低秩方法来解决小样本量情况下人脑网络连接分布的估计问题，并通过实验证明这一方法在特定研究领域中的重要性。