geometric deep learning

Existing graph neural network (GNN) pooling methods often compromise graph structural fidelity and interpretability, leading to unstable performance across datasets, tasks, and pooling ratios. To address this, we propose a geometry-aware edge-collapsing pooling method. Our approach is the first to adopt “magnitude”—a concept from diffusion geometry—as a measure of structural diversity, and introduces “metric spread” to enhance numerical stability and computational efficiency. By integrating spectral-preserving optimization with edge-collapsing operations, the method significantly retains the original graph’s metric structure and spectral properties while reducing its scale. Extensive experiments demonstrate that our method consistently outperforms state-of-the-art pooling baselines across diverse graph classification benchmarks. Moreover, it maintains high accuracy and strong robustness under varying pooling ratios, offering improved generalizability and reliability for downstream GNN applications.

Neural models trained on identical tasks exhibit geometrically heterogeneous latent representations due to stochasticity and architectural differences, rendering cross-model representations incomparable. To address this, we propose relative geodesic representations grounded in pullback metrics—a novel application of differential-geometric pullback metrics to latent space alignment—explicitly modeling intrinsic geometric transformations between latent manifolds of distinct models. Unlike conventional linear alignment methods, our approach operates without supervision and generalizes across diverse architectures and pretraining paradigms. Experiments on autoencoders and vision foundation discriminative models demonstrate substantial improvements in cross-model retrieval accuracy and model stitching performance. Moreover, the method scales effectively to large-scale settings.

Existing graph pooling methods fail to preserve higher-order combinatorial and topological consistency when applied to simplicial complexes—topological data structures capable of encoding high-order relational information. Method: We propose NervePool, the first learnable downsampling layer specifically designed for simplicial complexes. It introduces a vertex-clustering-driven hierarchical coarsening framework that deterministically, differentiably, and topologically awarely compresses from vertices to higher-dimensional simplices via star unions and nerve complex construction. To ensure differentiability and computational efficiency, we integrate GNN-Sinkhorn joint optimization with simplicial adjacency algebra. Contribution/Results: On multiple benchmark tasks, NervePool achieves an average accuracy improvement of 2.3%, significantly enhancing generalization and computational efficiency. It represents the first systematic extension of neural pooling to higher-order topological data, establishing a foundation for deep learning on simplicial complexes.

This work investigates how training dynamically shapes the Riemannian geometric structure induced by neural network representations. **Problem**: While deep networks operate in high-dimensional feature spaces, the geometric evolution of their induced Riemannian metric during training remains poorly understood. **Method**: We integrate Riemannian geometric analysis, infinite-width network theory, and feature-space metric modeling, conducting systematic experiments across supervised and self-supervised learning paradigms. **Contribution/Results**: We theoretically prove that infinitely wide random networks initially possess an isotropic (highly symmetric) Riemannian metric; training actively breaks this symmetry by locally amplifying the metric tensor near decision boundaries—a phenomenon we term *boundary-sensitive metric amplification*. Empirical validation across deep image classification and self-supervised learning confirms its robustness, revealing a self-emergent geometric inductive bias. This provides a novel geometric paradigm for understanding the intrinsic geometry of nonlinear feature learning.

Controllable geometric generation remains challenging in scenarios lacking large-scale 3D shape datasets. Method: This paper proposes a data-free neural implicit field generation framework that encodes user-specified design objectives—such as smoothness, genus (number of holes), and connectivity—as partial differential equation (PDE) constraints, geometric differential operator regularizers, and a multi-objective Lagrangian optimization objective, all directly embedded into neural field training. Contribution/Results: It establishes the first data-free paradigm for implicit shape generation; introduces explicit diversity constraints to mitigate mode collapse; and enables joint yet disentangled control over geometric and topological attributes. Experiments on multiple benchmarks and real-world engineering design tasks demonstrate precise, stable control over surface smoothness, connectivity, and genus, while consistently producing high-quality, diverse, and feasible shape ensembles.

This work addresses the lack of theoretical foundations for substructure transferability in graph data by bridging transferable substructures with the intrinsic geometry of graph representation spaces from a functional behavior perspective. It proposes the first Riemannian geometry–based framework for learning intrinsic graph geometry, innovatively introducing neural vector bundles and local coordinate charts to construct the GAUGE pretraining architecture. A Dirichlet loss function is designed to enable explicit modeling of intrinsic graph geometry and quantification of transfer difficulty. The method demonstrates significant performance gains over existing models on zero-shot link prediction and graph isomorphism tasks, validating its expressive power and cross-task transferability.

Existing sheaf neural networks lack an effective hierarchical pooling mechanism for multiscale graph modeling. This work proposes HiSP, a framework that introduces, for the first time, a learnable sheaf-aware hierarchical pooling scheme. Built upon local spectral coarsening, HiSP projects fine-grained stalk features onto low-frequency modes of the cluster-wise sheaf Laplacian and preserves sheaf energy consistency via Galerkin operators. The method integrates cochain-level prolongation maps with a lifting-based sheaf Laplacian, enabling efficient batch processing within PyTorch Geometric. HiSP not only explicitly distinguishes and quantifies truncation and realization losses incurred during coarsening but also significantly enhances multiscale representation capacity while preserving the underlying sheaf structure.

This work investigates whether the predictions of geometric neural networks—such as sheaf neural networks—genuinely rely on intrinsic geometric mechanisms like SO(2) rotations and parallel transport, rather than merely reflecting task performance indirectly. To this end, the authors propose the first basis-invariant triangular loop product metric and employ intervention experiments to disentangle the effects of rotation, stalk-space area, and orientation. Evaluation on a custom high-homophily GraphUniverse dataset demonstrates that neural sheaf propagation (NSP) substantially enhances loop rotation (up to 0.388 radians), while substituting identity transport drastically increases error, confirming the critical role of geometric mechanisms in connection sensitivity. Furthermore, a graph-summary ridge regression predictor outperforms others, underscoring the importance of structural inductive bias.

This work addresses a critical gap in topological deep learning: the absence of native benchmark datasets that inherently embody higher-order topological structures. Current research often relies on elevating graph-based data to construct such structures, which limits rigorous model evaluation and hinders progress. The study systematically identifies this benchmark deficiency and advocates for the development of genuinely native datasets rooted in intrinsic higher-order topology. Drawing upon established higher-order modeling paradigms—such as message passing frameworks and sheaf theory—the paper proposes concrete directions and standardization guidelines for dataset construction. These contributions aim to establish a solid foundation for topological deep learning and catalyze the future development of benchmarks in higher-order machine learning.

This study addresses the challenge of evaluating cross-layer and cross-network similarity of internal neural representations across multiple scales. The authors propose a novel framework that integrates diffusion geometry with multi-view learning: by modeling the data manifold via a Markov transition matrix, they leverage its powers to construct multi-scale variants of Centered Kernel Alignment (CKA) and distance correlation. Furthermore, alternating diffusion is introduced to fuse information across layers, enabling a paradigm shift from local inter-layer comparisons to global inter-network assessments. Evaluated on the ReSi benchmark—spanning 14 architectures, 7 datasets, and 3 domains—the method achieves state-of-the-art performance in representation similarity and out-of-distribution generalization across both language and vision tasks.