Radial Basis Function Networks as Projection Heads in Self-Supervised Learning

There has been growing interest in studying the complexity of Rectified Linear Unit (ReLU) based activation networks. Recent work investigates the evolution of the number of piecewise-linear partitions (linear regions) that are formed during training. However, current research is limited to examining the complexity of models trained in a supervised way. Self-Supervised Learning (SSL) differs in that it directly optimises the representation space using a loss function to enhance the model's performance across multiple downstream tasks. This study investigates the local distribution of linear regions produced by SSL models. We demonstrate that the evolution of linear regions correlates with the representation quality by utilising SplineCam to extract two-dimensional polytopes near the data distribution. We track the number, area, eccentricity, and boundaries of regions throughout training. The study compares supervised, contrastive, and self-distillation methods over two standard benchmark datasets, MNIST and FashionMNIST. The analysis of the experimental results shows that self-supervised methods create substantially fewer regions to achieve comparable accuracy to supervised models. Contrastive methods rapidly expand regions over time, whereas self-distillation methods tend to consolidate by merging neighbouring regions. Lastly, we can detect representation collapse early within the geometric space of linear regions. Our analysis suggests that polytopal metrics can serve as reliable indicators of representation quality and model performance.

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) has emerged as a powerful paradigm for learning meaningful representations from unlabeled data. However, the standard protocol for evaluating these representations, linear probing, is computationally expensive, sensitive to hyperparameters, and provides limited insight into the geometric structure of the representation space. In this work, motivated by connections between neural network generalization and intrinsic dimension (ID) we propose IdEst, a method for estimating the ID of SSL representations via the Minimum Spanning Tree dimension estimator ($\mathrm{dim}_\mathrm{MST}$). Across diverse datasets, architectures, and SSL pretraining objectives, we show that IdEst strongly correlates with downstream linear probe performances. Furthermore, we demonstrate that IdEst enables efficient hyperparameter selection, significantly reducing the computational cost compared to supervised alternatives. Our results highlight intrinsic dimensionality as a principled geometric proxy for assessing SSL representations, complementing standard supervised probing protocols.