Graph Hierarchical Recurrence for Long-Range Generalization

Graph Neural Networks (GNNs) and Graph Transformers (GTs) are now a fundamental paradigm for graph learning, combining the representation-learning capabilities of deep models with the sample efficiency induced by their inductive biases. Despite their effectiveness, a large body of work has shown that these models still face fundamental limitations in tasks that require capturing correlations between distant regions of a graph. To address this issue, we introduce Graph Hierarchical Recurrence (GHR), a novel framework that operates jointly on the input graph and on a hierarchical abstraction obtained through pooling. We also show that the limitations of existing models are even more pronounced in out-of-range generalization, where test instances involve interactions over distances longer than those observed during training. By contrast, despite its simple design, GHR provides three key advantages: strong performance on long-range dependencies, improved out-of-range generalization, and high parameter efficiency. To corroborate these claims, we show that across a broad set of long-range benchmarks, GHR consistently outperforms existing graph models while using as little as 1% of the parameters of current state-of-the-art models. These results suggest a complementary direction to the current trend of scaling architectures to obtain graph foundation models, indicating that increased model capacity alone may not be sufficient for generalization.

On Uncertainty Calibration for Equivariant Functions

Data-sparse settings such as robotic manipulation, molecular physics, and galaxy morphology classification are some of the hardest domains for deep learning. For these problems, equivariant networks can help improve modeling across undersampled parts of the input space, and uncertainty estimation can guard against overconfidence. However, until now, the relationships between equivariance and model confidence, and more generally equivariance and model calibration, has yet to be studied. Since traditional classification and regression error terms show up in the definitions of calibration error, it is natural to suspect that previous work can be used to help understand the relationship between equivariance and calibration error. In this work, we present a theory relating equivariance to uncertainty estimation. By proving lower and upper bounds on uncertainty calibration errors (ECE and ENCE) under various equivariance conditions, we elucidate the generalization limits of equivariant models and illustrate how symmetry mismatch can result in miscalibration in both classification and regression. We complement our theoretical framework with numerical experiments that clarify the relationship between equivariance and uncertainty using a variety of real and simulated datasets, and we comment on trends with symmetry mismatch, group size, and aleatoric and epistemic uncertainties.

Separation Power of Equivariant Neural Networks

The separation power of a machine learning model refers to its capacity to distinguish distinct inputs, and it is often employed as a proxy for its expressivity. In this paper, we propose a theoretical framework to investigate the separation power of equivariant neural networks with point-wise activations. Using the proposed framework, we can derive an explicit description of inputs indistinguishable by a family of neural networks with given architecture, demonstrating that it remains unaffected by the choice of non-polynomial activation function employed. We are able to understand the role played by activation functions in separability. Indeed, we show that all non-polynomial activations, such as ReLU and sigmoid, are equivalent in terms of expressivity, and that they reach maximum discrimination capacity. We demonstrate how assessing the separation power of an equivariant neural network can be simplified to evaluating the separation power of minimal representations. We conclude by illustrating how these minimal components form a hierarchy in separation power.

A Characterization Theorem for Equivariant Networks with Point-wise Activations

Equivariant neural networks have shown improved performance, expressiveness and sample complexity on symmetrical domains. But for some specific symmetries, representations, and choice of coordinates, the most common point-wise activations, such as ReLU, are not equivariant, hence they cannot be employed in the design of equivariant neural networks. The theorem we present in this paper describes all possible combinations of finite-dimensional representations, choice of coordinates and point-wise activations to obtain an exactly equivariant layer, generalizing and strengthening existing characterizations. Notable cases of practical relevance are discussed as corollaries. Indeed, we prove that rotation-equivariant networks can only be invariant, as it happens for any network which is equivariant with respect to connected compact groups. Then, we discuss implications of our findings when applied to important instances of exactly equivariant networks. First, we completely characterize permutation equivariant networks such as Invariant Graph Networks with point-wise nonlinearities and their geometric counterparts, highlighting a plethora of models whose expressive power and performance are still unknown. Second, we show that feature spaces of disentangled steerable convolutional neural networks are trivial representations.