IdEst: Assessing Self-Supervised Learning Representations via Intrinsic Dimension

The analysis of neural representation has become an integral part of research aiming to better understand the inner workings of neural networks. While there are many different approaches to investigate neural representations, an important line of research has focused on doing so through the lens of intrinsic dimensions (IDs). Although this perspective has provided valuable insights and stimulated substantial follow-up research, important limitations of this approach have remained largely unaddressed. In this paper, we highlight a crucial discrepancy between theory and practice of IDs in neural representations, theoretically and empirically showing that common ID estimators are, in fact, not tracking the true underlying ID of the representation. We contrast this negative result with an investigation of the underlying factors that may drive commonly reported ID-related results on neural representation in the literature. Building on these insights, we offer a new perspective on ID estimation in neural representations.

Self-supervised learning (SSL) has achieved remarkable representation learning performance, but many existing methods rely on large batch sizes, memory banks, momentum encoders, or global synchronization mechanisms that substantially increase computational cost and training complexity. In this work, we propose Semantic Mutual Information (SMI), a lightweight self-supervised objective derived from a mutual-information-inspired dependency formulation under Gaussian assumptions. Unlike conventional correlation matching objectives that operate on high-dimensional feature correlation matrices, SMI performs optimization on a sample-level dependency matrix through a nonlinear transformation of pairwise correlations. This formulation induces distinct optimization dynamics that emphasize strongly dependent semantic pairs while maintaining representation diversity. Experimental results on ImageNet using a ResNet-50 backbone demonstrate that SMI achieves competitive linear evaluation performance relative to state-of-the-art SSL approaches while substantially reducing computational complexity. Across multiple low-resource benchmarks, SMI consistently improves transfer performance over Barlow Twins, particularly on fine-grained datasets. Furthermore, analyses of optimization dynamics and representation geometry suggest improved alignment--redundancy balance, greater feature diversity, and more spatially localized semantic representations. These results indicate that nonlinear dependency optimization provides an effective and computationally efficient alternative to conventional correlation-based self-supervised learning objectives.

Self-supervised learning (SSL) excels at finding general-purpose latent representations from complex data, yet lacks a unifying theoretical framework that explains the diverse existing methods and guides the design of new ones. We cast SSL as latent distribution matching (LDM): learning representations that maximize their log-probability under an assumed latent model (alignment), while maximizing latent entropy to prevent collapse (uniformity). This view unifies independent component analysis with contrastive, non-contrastive, and predictive SSL methods, including stop gradient approaches. Leveraging LDM, we derive a nonlinear, sampling-free Bayesian filtering model with a Kalman-based predictor for high-dimensional timeseries. We further prove that predictive LDM yields identifiable latent representations under mild assumptions, even with nonlinear predictors. Overall, LDM clarifies the assumptions behind established SSL methods and provides principled guidance for developing new approaches.