Measuring geometric similarity between high-dimensional network
representations is a topic of longstanding interest to neuroscience and deep
learning. Although many methods have been proposed, only a few works have
rigorously analyzed their statistical efficiency or quantified estimator
uncertainty in data-limited regimes. Here, we derive upper and lower bounds on
the worst-case convergence of standard estimators of shape
distance$\unicode{x2014}$a measure of representational dissimilarity proposed
by Williams et al. (2021). These bounds reveal the challenging nature of the
problem in high-dimensional feature spaces. To overcome these challenges, we
introduce a new method-of-moments estimator with a tunable bias-variance
tradeoff. We show that this estimator achieves superior performance to standard
estimators in simulation and on neural data, particularly in high-dimensional
settings. Thus, we lay the foundation for a rigorous statistical theory for
high-dimensional shape analysis, and we contribute a new estimation method that
is well-suited to practical scientific settings.

通过推导上、下界，分析了高维网络表示的几何相似性测量方法的统计效率和估计不确定性，提出了一种新的矩法估计器，在高维特征空间中优于标准估计器，为高维形状分析奠定了统计理论基础。