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版本更新 · v2arXiv:2610.21820

A Closed Form Depth-Width Law for Neural Collapse in Deep Linear Networks

Pankhil Gawade, Kirsten Fischer, Guy Wolf, Ewa Szczurek

原文摘要

How the depth and width of neural networks shape learned representations is difficult to predict from the data and architecture. We use neural collapse (NC) as a probe of the representation geometry and express its first three properties, within-class variability collapse (NC1), simplex class means (NC2) and classifier alignment (NC3), through the learned feature kernel, extending a known identity for NC1 to NC2 and NC3. For deep linear networks with an exchangeable input kernel, the kernel equations yield a geometric law, exact within this theory, for the layerwise decrease of NC1. At large width $N$ the accumulated decrease becomes a function of the depth-to-width ratio $L/N$, with its rate determined self-consistently by the data and architecture, yielding a scaling law for representation geometry.

cs.LG

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2605.06466

Diversity Curves for Graph Representation Learning

Katharina Limbeck, Nadja Häusermann, Martin Carrasco, Guy Wolf, Bastian Rieck

原文摘要

Graph-level representations are crucial tools for characterising structural differences between graphs. However, comparing graphs with different cardinalities, even when sampled from the same underlying distribution, remains challenging. Unsupervised tasks in particular require interpretable, scalable, and reliable size-aware graph representations. Our work addresses these issues by tracking the structural diversity of a graph across coarsening levels. The resulting graph embeddings, which we denote diversity curves, are interpretable by construction, efficient, and directly comparable across coarsening hierarchies. Specifically, we track the spread of graphs, a novel isometry invariant that is inherently well-suited for encoding the metric diversity and geometry of graphs. We utilise edge contraction coarsening and prove that this improves expressivity, thus leading to more powerful graph-level representations than structural descriptors alone. Demonstrating their utility over a range of baseline methods in practice, we use diversity curves to (i) cluster and visualise simulated graphs across varying sizes, (ii) distinguish the geometry of single-cell graphs, (iii) compare the structure of molecular graph datasets, and (iv) characterise geometric shapes.

cs.LG

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arXiv ↗PDF ↗
版本更新 · v3arXiv:2605.06467

No Triangulation Without Representation: Generalization in Topological Deep Learning

Johannes S. Schmidt, Martin Carrasco, Ernst Röell, Guy Wolf, Nello Blaser, Bastian Rieck

原文摘要

Despite an ever-increasing interest in topological deep learning models that target higher-order datasets, there is no consensus on how to evaluate such models. This is exacerbated by the fact that topological objects permit operations, such as structural refinements, that are not appropriate for graph data. In this work, we extend MANTRA, a benchmark dataset containing manifold triangulations, to a larger class of manifolds with more diverse homeomorphism types. We show that, contrary to prior claims, graph neural networks can saturate the benchmark, while higher-order message passing methods lag behind. However, we find that this is contingent on the right representation and feature assignment, emphasizing their importance in baseline models. Hence, we present a novel evaluation protocol based on representational diversity and triangulation refinement. Even when including refinement-like changes in the training data, existing models fail to generalize beyond the combinatorial structure of the data. This points towards a research gap in developing models that understand topological structure independent of scale. Our work thus provides the necessary scaffolding to evaluate future models and enable the development of topology-aware inductive biases.

cs.LG · math.AT

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arXiv ↗PDF ↗