AtlasD: Automatic Local Symmetry Discovery

Existing symmetry discovery methods predominantly focus on global transformations across the entire system or space, but they fail to consider the symmetries in local neighborhoods. This may result in the reported symmetry group being a misrepresentation of the true symmetry. In this paper, we formalize the notion of local symmetry as atlas equivariance. Our proposed pipeline, automatic local symmetry discovery (AtlasD), recovers the local symmetries of a function by training local predictor networks and then learning a Lie group basis to which the predictors are equivariant. We demonstrate AtlasD is capable of discovering local symmetry groups with multiple connected components in top-quark tagging and partial differential equation experiments. The discovered local symmetry is shown to be a useful inductive bias that improves the performance of downstream tasks in climate segmentation and vision tasks.

Impact of Regularization on Calibration and Robustness: from the Representation Space Perspective

Recent studies have shown that regularization techniques using soft labels, e.g., label smoothing, Mixup, and CutMix, not only enhance image classification accuracy but also improve model calibration and robustness against adversarial attacks. However, the underlying mechanisms of such improvements remain underexplored. In this paper, we offer a novel explanation from the perspective of the representation space (i.e., the space of the features obtained at the penultimate layer). Our investigation first reveals that the decision regions in the representation space form cone-like shapes around the origin after training regardless of the presence of regularization. However, applying regularization causes changes in the distribution of features (or representation vectors). The magnitudes of the representation vectors are reduced and subsequently the cosine similarities between the representation vectors and the class centers (minimal loss points for each class) become higher, which acts as a central mechanism inducing improved calibration and robustness. Our findings provide new insights into the characteristics of the high-dimensional representation space in relation to training and regularization using soft labels.