Self-supervised learning (SSL) has recently advanced through non-contrastive methods that couple an invariance term with variance, covariance, or redundancy-reduction penalties. While such objectives shape first- and second-order statistics of the representation, they largely ignore the local geometry of the underlying data manifold. In this paper, we introduce CurvSSL, a curvature-regularized self-supervised learning framework, and its RKHS extension, kernel CurvSSL. Our approach retains a standard two-view encoder-projector architecture with a Barlow Twins-style redundancy-reduction loss on projected features, but augments it with a curvature-based regularizer. Each embedding is treated as a vertex whose $k$ nearest neighbors define a discrete curvature score via cosine interactions on the unit hypersphere; in the kernel variant, curvature is computed from a normalized local Gram matrix in an RKHS. These scores are aligned and decorrelated across augmentations by a Barlow-style loss on a curvature-derived matrix, encouraging both view invariance and consistency of local manifold bending. Experiments on MNIST and CIFAR-10 datasets with a ResNet-18 backbone show that curvature-regularized SSL yields competitive or improved linear evaluation performance compared to Barlow Twins and VICReg. Our results indicate that explicitly shaping local geometry is a simple and effective complement to purely statistical SSL regularizers.

In this paper, we explore the properties of loss curvature with respect to input data in deep neural networks. Curvature of loss with respect to input (termed input loss curvature) is the trace of the Hessian of the loss with respect to the input. We investigate how input loss curvature varies between train and test sets, and its implications for train-test distinguishability. We develop a theoretical framework that derives an upper bound on the train-test distinguishability based on privacy and the size of the training set. This novel insight fuels the development of a new black box membership inference attack utilizing input loss curvature. We validate our theoretical findings through experiments in computer vision classification tasks, demonstrating that input loss curvature surpasses existing methods in membership inference effectiveness. Our analysis highlights how the performance of membership inference attack (MIA) methods varies with the size of the training set, showing that curvature-based MIA outperforms other methods on sufficiently large datasets. This condition is often met by real datasets, as demonstrated by our results on CIFAR10, CIFAR100, and ImageNet. These findings not only advance our understanding of deep neural network behavior but also improve the ability to test privacy-preserving techniques in machine learning.

Deep neural networks are over-parameterized and easily overfit the datasets they train on. In the extreme case, it has been shown that these networks can memorize a training set with fully randomized labels. We propose using the curvature of loss function around each training sample, averaged over training epochs, as a measure of memorization of the sample. We use this metric to study the generalization versus memorization properties of different samples in popular image datasets and show that it captures memorization statistics well, both qualitatively and quantitatively. We first show that the high curvature samples visually correspond to long-tailed, mislabeled, or conflicting samples, those that are most likely to be memorized. This analysis helps us find, to the best of our knowledge, a novel failure mode on the CIFAR100 and ImageNet datasets: that of duplicated images with differing labels. Quantitatively, we corroborate the validity of our scores via two methods. First, we validate our scores against an independent and comprehensively calculated baseline, by showing high cosine similarity with the memorization scores released by Feldman & Zhang (2020). Second, we inject corrupted samples which are memorized by the network, and show that these are learned with high curvature. An added advantage of our method is that it is scalable, as it requires training only a single network as opposed to the thousands trained by the baseline, while capturing the aforementioned failure mode that the baseline fails to identify. Deep learning has been hugely successful in many fields.

Publicly available data sets of structural MRIs might not contain specific measurements of brain Regions of Interests (ROIs) that are important for training machine learning models. For example, the curvature scores computed by Freesurfer are not released by the Adolescent Brain Cognitive Development (ABCD) Study. One can address this issue by simply reapplying Freesurfer to the data set. However, this approach is generally computationally and labor intensive (e.g., requiring quality control). An alternative is to impute the missing measurements via a deep learning approach. However, the state-of-the-art is designed to estimate randomly missing values rather than entire measurements. We therefore propose to re-frame the imputation problem as a prediction task on another (public) data set that contains the missing measurements and shares some ROI measurements with the data sets of interest. A deep learning model is then trained to predict the missing measurements from the shared ones and afterwards is applied to the other data sets. Our proposed algorithm models the dependencies between ROI measurements via a graph neural network (GNN) and accounts for demographic differences in brain measurements (e.g. sex) by feeding the graph encoding into a parallel architecture. The architecture simultaneously optimizes a graph decoder to impute values and a classifier in predicting demographic factors. We test the approach, called Demographic Aware Graph-based Imputation (DAGI), on imputing those missing Freesurfer measurements of ABCD (N=3760) by training the predictor on those publicly released by the National Consortium on Alcohol and Neurodevelopment in Adolescence (NCANDA, N=540)...