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Melanie Weber

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7 papers

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JMLR Journal 2025 Journal Article

Curvature-based Clustering on Graphs

Yu Tian
Zachary Lubberts
Melanie Weber

Unsupervised node clustering (or community detection) is a classical graph learning task. In this paper, we study algorithms that exploit the geometry of the graph to identify densely connected substructures, which form clusters or communities. Our method implements discrete Ricci curvatures and their associated geometric flows, under which the edge weights of the graph evolve to reveal its community structure. We consider several discrete curvature notions and analyze the utility of the resulting algorithms. In contrast to prior literature, we study not only single-membership community detection, where each node belongs to exactly one community, but also mixed-membership community detection, where communities may overlap. For the latter, we argue that it is beneficial to perform community detection on the line graph, i.e., the graph's dual. We provide both theoretical and empirical evidence for the utility of our curvature-based clustering algorithms. In addition, we give several results on the relationship between the curvature of a graph and that of its dual, which enable the efficient implementation of our proposed mixed-membership community detection approach and which may be of independent interest for curvature-based network analysis. [abs] [ pdf ][ bib ] &copy JMLR 2025. ( edit, beta )

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NeurIPS Conference 2025 Conference Paper

Higher-Order Learning with Graph Neural Networks via Hypergraph Encodings

Raphaël Pellegrin
Lukas Fesser
Melanie Weber

Higher-order information is crucial for relational learning in many domains where relationships extend beyond pairwise interactions. Hypergraphs provide a natural framework for modeling such relationships, which has motivated recent extensions of graph neural network (GNN) architectures to hypergraphs. Most of these architectures rely on message-passing to encode higher-order information. In this paper, we propose to instead use hypergraph-level encodings based on characteristics such as hypergraph Laplacians and discrete curvature notions. These encodings can be used on datasets that are naturally parametrized as hypergraphs and on graph-level datasets, which we reparametrize as hypergraphs to compute encodings. In both settings, performance increases significantly, on social networks by more than 10 percent. Our theoretical analysis shows that hypergraph-level encodings provably increase the representational power of message-passing graph neural networks beyond that of their graph-level counterparts. For complete reproducibility, we release our codebase: https: //github. com/Weber-GeoML/Hypergraph_Encodings.

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AAAI Conference 2024 Conference Paper

Exploiting Data Geometry in Machine Learning

Melanie Weber

A key challenge in Machine Learning (ML) is the identification of geometric structure in high-dimensional data. Most algorithms assume that data lives in a high-dimensional vector space; however, many applications involve non-Euclidean data, such as graphs, strings and matrices, or data whose structure is determined by symmetries in the underlying system. Here, we discuss methods for identifying geometric structure in data and how leveraging data geometry can give rise to efficient ML algorithms with provable guarantees.

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TMLR Journal 2024 Journal Article

Graph Pooling via Ricci Flow

Amy Feng
Melanie Weber

Graph Machine Learning often involves the clustering of nodes based on similarity structure encoded in the graph’s topology and the nodes’ attributes. On homophilous graphs, the integration of pooling layers has been shown to enhance the performance of Graph Neural Networks by accounting for inherent multi-scale structure. Here, similar nodes are grouped together to coarsen the graph and reduce the input size in subsequent layers in deeper architectures. In both settings, the underlying clustering approach can be implemented via graph pooling operators, which often rely on classical tools from Graph Theory. In this work, we introduce a graph pooling operator (ORC-Pool), which utilizes a characterization of the graph’s geometry via Ollivier’s discrete Ricci curvature and an associated geometric flow. Previous Ricci flow based clustering approaches have shown great promise across several domains, but are by construction unable to account for similarity structure encoded in the node attributes. However, in many ML applications, such information is vital for downstream tasks. ORC-Pool extends such clustering approaches to attributed graphs, allowing for the integration of geometric coarsening into Graph Neural Networks as a pooling layer.

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NeurIPS Conference 2024 Conference Paper

Hardness of Learning Neural Networks under the Manifold Hypothesis

Bobak T. Kiani
Jason Wang
Melanie Weber

The manifold hypothesis presumes that high-dimensional data lies on or near a low-dimensional manifold. While the utility of encoding geometric structure has been demonstrated empirically, rigorous analysis of its impact on the learnability of neural networks is largely missing. Several recent results have established hardness results for learning feedforward and equivariant neural networks under i. i. d. Gaussian or uniform Boolean data distributions. In this paper, we investigate the hardness of learning under the manifold hypothesis. We ask, which minimal assumptions on the curvature and regularity of the manifold, if any, render the learning problem efficiently learnable. We prove that learning is hard under input manifolds of bounded curvature by extending proofs of hardness in the SQ and cryptographic settings for boolean data inputs to the geometric setting. On the other hand, we show that additional assumptions on the volume of the data manifold alleviate these fundamental limitations and guarantee learnability via a simple interpolation argument. Notable instances of this regime are manifolds which can be reliably reconstructed via manifold learning. Looking forward, we comment on and empirically explore intermediate regimes of manifolds, which have heterogeneous features commonly found in real world data.

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NeurIPS Conference 2024 Conference Paper

Unitary Convolutions for Learning on Graphs and Groups

Bobak T. Kiani
Lukas Fesser
Melanie Weber

Data with geometric structure is ubiquitous in machine learning often arising from fundamental symmetries in a domain, such as permutation-invariance in graphs and translation-invariance in images. Group-convolutional architectures, which encode symmetries as inductive bias, have shown great success in applications, but can suffer from instabilities as their depth increases and often struggle to learn long range dependencies in data. For instance, graph neural networks experience instability due to the convergence of node representations (over-smoothing), which can occur after only a few iterations of message-passing, reducing their effectiveness in downstream tasks. Here, we propose and study unitary group convolutions, which allow for deeper networks that are more stable during training. The main focus of the paper are graph neural networks, where we show that unitary graph convolutions provably avoid over-smoothing. Our experimental results confirm that unitary graph convolutional networks achieve competitive performance on benchmark datasets compared to state-of-the-art graph neural networks. We complement our analysis of the graph domain with the study of general unitary convolutions and analyze their role in enhancing stability in general group convolutional architectures.

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NeurIPS Conference 2020 Conference Paper

Robust large-margin learning in hyperbolic space

Melanie Weber
Manzil Zaheer
Ankit Singh Rawat
Aditya K. Menon
Sanjiv Kumar

Recently, there has been a surge of interest in representation learning in hyperbolic spaces, driven by their ability to represent hierarchical data with significantly fewer dimensions than standard Euclidean spaces. However, the viability and benefits of hyperbolic spaces for downstream machine learning tasks have received less attention. In this paper, we present, to our knowledge, the first theoretical guarantees for learning a classifier in hyperbolic rather than Euclidean space. Specifically, we consider the problem of learning a large-margin classifier for data possessing a hierarchical structure. Our first contribution is a hyperbolic perceptron algorithm, which provably converges to a separating hyperplane. We then provide an algorithm to efficiently learn a large-margin hyperplane, relying on the careful injection of adversarial examples. Finally, we prove that for hierarchical data that embeds well into hyperbolic space, the low embedding dimension ensures superior guarantees when learning the classifier directly in hyperbolic space.

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