On the Optimal Representation Efficiency of Barlow Twins: An Information-Geometric Interpretation
Di Zhang
TL;DR
This work introduces an information-geometric framework to quantify representation efficiency in self-supervised learning, defining $η = \frac{d_{eff}}{d}$ where $d_{eff}$ is derived from the eigen-spectrum of the average Fisher Information Matrix $\bar{G}$. By modeling representations with a Gaussian conditional $p(t|z)$ and linking the cross-correlation $C$ of augmented views to $\bar{G}$, the authors show that, under idealized assumptions, driving $C$ toward the identity yields an isotropic $\bar{G}$ and maximal efficiency $η = 1$. The principal result is that Barlow Twins achieves optimal representation efficiency under these conditions by enforcing $C = I$, which corresponds to uniform, non-redundant use of all representation dimensions. This provides a geometric justification for Barlow Twins and offers a diagnostic lens for comparing SSL methods, with potential extensions to other paradigms like VICReg.
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 $η$ 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 ($η= 1$) 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.