Topological deep learning (TDL) is a rapidly growing field that seeks to leverage topological structure in data and facilitate learning from data supported on topological objects, ranging from molecules to 3D shapes. Most TDL architectures can be unified under the framework of higher-order message-passing (HOMP), which generalizes graph message-passing to higher-order domains. In the first part of the paper, we explore HOMP's expressive power from a topological perspective, demonstrating the framework's inability to capture fundamental topological and metric invariants such as diameter, orientability, planarity, and homology. In addition, we demonstrate HOMP's limitations in fully leveraging lifting and pooling methods on graphs. To the best of our knowledge, this is the first work to study the expressivity of TDL from a topological perspective. In the second part of the paper, we develop two new classes of architectures -- multi-cellular networks (MCN) and scalable MCN (SMCN) -- which draw inspiration from expressive GNNs. MCN can reach full expressivity, but scaling it to large data objects can be computationally expansive. Designed as a more scalable alternative, SMCN still mitigates many of HOMP's expressivity limitations. Finally, we design new benchmarks for evaluating models based on their ability to learn topological properties of complexes. We then evaluate SMCN on these benchmarks as well as on real-world graph datasets, demonstrating improvements over both HOMP baselines and expressive graph methods, highlighting the value of expressively leveraging topological information.

Subgraph GNNs have emerged as promising architectures that overcome the expressiveness limitations of Graph Neural Networks (GNNs) by processing bags of subgraphs. Despite their compelling empirical performance, these methods are afflicted by a high computational complexity: they process bags whose size grows linearly in the number of nodes, hindering their applicability to larger graphs. In this work, we propose an effective and easy-to-implement approach to dramatically alleviate the computational cost of Subgraph GNNs and unleash broader applications thereof. Our method, dubbed HyMN, leverages walk-based centrality measures to sample a small number of relevant subgraphs and drastically reduce the bag size. By drawing a connection to perturbation analysis, we highlight the strength of the proposed centrality-based subgraph sampling, and further prove that these walk-based centralities can be additionally used as Structural Encodings for improved discriminative power. A comprehensive set of experimental results demonstrates that HyMN provides an effective synthesis of expressiveness, efficiency, and downstream performance, unlocking the application of Subgraph GNNs to dramatically larger graphs. Not only does our method outperform more sophisticated subgraph sampling approaches, it is also competitive, and sometimes better, than other state-of-the-art approaches for a fraction of their runtime.

Gradients of neural networks encode valuable information for optimization, editing, and analysis of models. Therefore, practitioners often treat gradients as inputs to task-specific algorithms, e.g., using gradient statistics for pruning or optimization. Recent works explore *learning* algorithms that operate directly on gradients but use architectures that are not specifically designed for gradient processing, hindering their applicability. In this paper, we present a principled approach for designing architectures that process gradients. Our approach is guided by three principles: (1) equivariant design that preserves neuron permutation symmetries, (2) processing sets of gradients across multiple data points to capture curvature information, and (3) efficient gradient representation through rank-1 decomposition. Based on these principles, we introduce GradMetaNet, a novel architecture for learning on gradients, constructed from simple equivariant blocks. We prove universality results for GradMetaNet, and show that previous approaches cannot approximate natural gradient-based functions that GradMetaNet can. We then demonstrate GradMetaNet's effectiveness on a diverse set of gradient-based tasks for *MLPs* and *transformers*, such as learned optimization, INR editing, and loss landscape curvature estimation.

Subgraph GNNs enhance message-passing GNNs expressivity by representing graphs as sets of subgraphs, demonstrating impressive performance across various tasks. However, their scalability is hindered by the need to process large numbers of subgraphs. While previous approaches attempted to generate smaller subsets of subgraphs through random or learnable sampling, these methods often yielded suboptimal selections or were limited to small subset sizes, ultimately compromising their effectiveness. This paper introduces a new Subgraph GNN framework to address these issues. Our approach diverges from most previous methods by associating subgraphs with node clusters rather than with individual nodes. We show that the resulting collection of subgraphs can be viewed as the product of coarsened and original graphs, unveiling a new connectivity structure on which we perform generalized message passing. Crucially, controlling the coarsening function enables meaningful selection of any number of subgraphs. In addition, we reveal novel permutation symmetries in the resulting node feature tensor, characterize associated linear equivariant layers, and integrate them into our Subgraph GNN. We also introduce novel node marking strategies and provide a theoretical analysis of their expressive power and other key aspects of our approach. Extensive experiments on multiple graph learning benchmarks demonstrate that our method is significantly more flexible than previous approaches, as it can seamlessly handle any number of subgraphs, while consistently outperforming baseline approaches. Our code is available at https://github.com/BarSGuy/Efficient-Subgraph-GNNs.