Self-supervised learning (SSL) has revolutionized learning from large-scale unlabeled datasets, yet the intrinsic relationship between pretraining data and the learned representations remains poorly understood. Traditional supervised learning benefits from gradient-based data attribution tools like influence functions that measure the contribution of an individual data point to model predictions. However, existing definitions of influence rely on labels, making them unsuitable for SSL settings. We address this gap by introducing Influence-SSL, a novel and label-free approach for defining influence functions tailored to SSL. Our method harnesses the stability of learned representations against data augmentations to identify training examples that help explain model predictions. We provide both theoretical foundations and empirical evidence to show the utility of Influence-SSL in analyzing pre-trained SSL models. Our analysis reveals notable differences in how SSL models respond to influential data compared to supervised models. Finally, we validate the effectiveness of Influence-SSL through applications in duplicate detection, outlier identification and fairness analysis. Code is available at: \url{https://github.com/cryptonymous9/Influence-SSL}.

Advances in neural architecture search, as well as explainability and interpretability of connectionist architectures, have been reported in the recent literature. However, our understanding of how to design Bayesian Deep Learning (BDL) hyperparameters, specifically, the depth, width and ensemble size, for robust function mapping with uncertainty quantification, is still emerging. This paper attempts to further our understanding by mapping Bayesian connectionist representations to polynomials of different orders with varying noise types and ratios. We examine the noise-contaminated polynomials to search for the combination of hyperparameters that can extract the underlying polynomial signals while quantifying uncertainties based on the noise attributes. Specifically, we attempt to study the question that an appropriate neural architecture and ensemble configuration can be found to detect a signal of any n-th order polynomial contaminated with noise having different distributions and signal-to-noise (SNR) ratios and varying noise attributes. Our results suggest the possible existence of an optimal network depth as well as an optimal number of ensembles for prediction skills and uncertainty quantification, respectively. However, optimality is not discernible for width, even though the performance gain reduces with increasing width at high values of width. Our experiments and insights can be directional to understand theoretical properties of BDL representations and to design practical solutions.