标题： 关于巴洛双胞胎最优表示效率的论文：信息几何解释 显示英文标题

标题： On the Optimal Representation Efficiency of Barlow Twins: An Information-Geometric Interpretation

摘要： 自监督学习（SSL）通过在没有标签数据的情况下学习有意义的表示取得了显著的成功。 然而，对于理解并比较不同SSL范式的效率，一个统一的理论框架仍然难以捉摸。 在本文中，我们引入了一个新颖的信息几何框架来量化表示效率。 我们将表示效率$\eta$定义为所学表示空间的有效内在维度与其环境维度之间的比率，其中有效维度是从由编码器引起的统计流形上的Fisher信息矩阵（FIM）的谱特性推导出来的。 在这个框架内，我们对Barlow Twins方法进行了理论分析。 在特定但自然的假设下，我们证明Barlow Twins通过将表示的交叉相关矩阵推向单位矩阵，从而实现了最优表示效率（$\eta = 1$），这反过来会导致各向同性的FIM。 这项工作为理解Barlow Twins的有效性提供了严格的理论基础，并为分析SSL算法提供了一个新的几何视角。 显示英文摘要

摘要： Self-supervised learning (SSL) has achieved remarkable success by learning meaningful representations without labeled data. However, a unified theoretical framework for understanding and comparing the efficiency of different SSL paradigms remains elusive. In this paper, we introduce a novel information-geometric framework to quantify representation efficiency. We define representation efficiency $\eta$ as the ratio between the effective intrinsic dimension of the learned representation space and its ambient dimension, where the effective dimension is derived from the spectral properties of the Fisher Information Matrix (FIM) on the statistical manifold induced by the encoder. Within this framework, we present a theoretical analysis of the Barlow Twins method. Under specific but natural assumptions, we prove that Barlow Twins achieves optimal representation efficiency ($\eta = 1$) by driving the cross-correlation matrix of representations towards the identity matrix, which in turn induces an isotropic FIM. This work provides a rigorous theoretical foundation for understanding the effectiveness of Barlow Twins and offers a new geometric perspective for analyzing SSL algorithms.