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。

利用有限样本在神经表示上估计形状距离

Estimating Shape Distances on Neural Representations with Limited  Samples

Articulated object manipulation is a fundamental yet challenging task in
robotics. Due to significant geometric and semantic variations across object
categories, previous manipulation models struggle to generalize to novel
categories. Few-shot learning is a promising solution for alleviating this
issue by allowing robots to perform a few interactions with unseen objects.
However, extant approaches often necessitate costly and inefficient test-time
interactions with each unseen instance. Recognizing this limitation, we observe
that despite their distinct shapes, different categories often share similar
local geometries essential for manipulation, such as pullable handles and
graspable edges - a factor typically underutilized in previous few-shot
learning works. To harness this commonality, we introduce 'Where2Explore', an
affordance learning framework that effectively explores novel categories with
minimal interactions on a limited number of instances. Our framework explicitly
estimates the geometric similarity across different categories, identifying
local areas that differ from shapes in the training categories for efficient
exploration while concurrently transferring affordance knowledge to similar
parts of the objects. Extensive experiments in simulated and real-world
environments demonstrate our framework's capacity for efficient few-shot
exploration and generalization.

通过引入‘Where2Explore’，一个能够有效地在有限数量的实例上探索新类别的可行性学习框架，本文解决了在机器人学中基础而又具有挑战性的关节物体操作问题，实现了有效的少样本学习与推广。