Neural Information Processing Systems http://nips.cc/

Modern deep neural networks areoverparameterizedwith respect to the amount of training data and achieve zero training error, yet generalize well on test data.

We show that this laziness directly relates to volume comparison results from Riemannian geometry. We then extend this scalar curvature notion to an entire quadratic form using neural network estimations based on the diffusion map of point-clouddata.

Inspired by graph-based methodologies, we introduce a novel graph-spanning algorithm designed to identify changes in both offline and online data across low to high dimensions. This versatile approach is applicable to Euclidean and graph-structured data with unknown distributions, while maintaining control over error probabilities. Theoretically, we demonstrate that the algorithm achieves high detection power when the magnitude of the change surpasses the lower bound of the minimax separation rate, which scales on the order of $\sqrt{nd}$. Our method outperforms other techniques in terms of accuracy for both Gaussian and non-Gaussian data. Notably, it maintains strong detection power even with small observation windows, making it particularly effective for online environments where timely and precise change detection is critical.

In this study, we introduce a method for learning group (known or unknown) equivariant functions by learning the associated quadratic form $x^T A x$ corresponding to the group from the data. Certain groups, known as orthogonal groups, preserve a specific quadratic form, and we leverage this property to uncover the underlying symmetry group under the assumption that it is orthogonal. By utilizing the corresponding unique symmetric matrix and its inherent diagonal form, we incorporate suitable inductive biases into the neural network architecture, leading to models that are both simplified and efficient. Our approach results in an invariant model that preserves norms, while the equivariant model is represented as a product of a norm-invariant model and a scale-invariant model, where the ``product'' refers to the group action. Moreover, we extend our framework to a more general setting where the function acts on tuples of input vectors via a diagonal (or product) group action. In this extension, the equivariant function is decomposed into an angular component extracted solely from the normalized first vector and a scale-invariant component that depends on the full Gram matrix of the tuple. This decomposition captures the inter-dependencies between multiple inputs while preserving the underlying group symmetry. We assess the effectiveness of our framework across multiple tasks, including polynomial regression, top quark tagging, and moment of inertia matrix prediction. Comparative analysis with baseline methods demonstrates that our model consistently excels in both discovering the underlying symmetry and efficiently learning the corresponding equivariant function.

The (adjusted) twisted conjugation, used as the action of the Clifford group.