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

Abstract: 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.