Gabriela Moisescu-Pareja

Toward a general understanding of neural representations learned by deep neural networks on group multiplications

Arthur Ayestas Hilgert

We study the neural representations learned by deep neural networks trained on alternating group multiplication, discovering that they are S… (see more)chreier coset graphs, a generalization of Cayley graphs. Previous works inspected neural representations learned from cyclic group multiplication and identified them as Cayley graphs. Since cyclic group multiplication is Abelian, all subgroups are normal, and Schreier coset graphs reduce to Cayley graphs in this case. This contribution makes a step toward finding a general theory of what neural representations are in networks learning group multiplications.

2026-07-15

Topology, Algebra, and Geometry in Data Science (oral)

openreview.net

On the geometry and topology of representations: the manifolds of modular addition

Gabriela Moisescu-Pareja

Colin Daniels

Jonathan Love

The Clock and Pizza interpretations, associated with architectures differing in either uniform or learnable attention, were introduced to ar… (see more)gue that different architectural designs can yield distinct circuits for modular addition. In this work, we show that this is not the case, and that both uniform attention and trainable attention architectures implement the same algorithm via topologically and geometrically equivalent representations. Our methodology goes beyond the interpretation of individual neurons and weights. Instead, we identify all of the neurons corresponding to each learned representation and then study the collective group of neurons as one entity. This method reveals that each learned representation is a manifold that we can study utilizing tools from topology. Based on this insight, we can statistically analyze the learned representations across hundreds of circuits to demonstrate the similarity between learned modular addition circuits that arise naturally from common deep learning paradigms.

2026-01-25

International Conference on Learning Representations (poster)

doi.org

openreview.net

Deep neural networks divide and conquer dihedral multiplication

Sihui Wei

Gavin McCracken

Gabriela Moisescu-Pareja

Harley Wiltzer

Doina Precup

Irina Rish

Jonathan Love

We find multilayer perceptrons and transformers both universally learn an instantiation of the same divide-and-conquer algorithm that requir… (see more)es only a logarithmic number of neural representations to solve dihedral multiplication. Clustering neurons based on similar activation behaviour reveals remarkably clear structure: each neural representation corresponds to a Cayley graph. To our knowledge, this is the first work that fully characterizes and describes all neural representations that are learnable on a dataset, while prior work on group multiplications studied neuron-level behavior, or preliminarily investigated cluster behavior. Thus, we can understand the algorithm networks universally learn at three levels of abstraction: 1) Neurons activate on coset or approximate coset structure of the dihedral group. 2) Groups of neurons together form neural representations that act to divide the dataset into different subproblems, being Cayley graphs, where the equivalence class of the answer is computed. 3) The global algorithm then linearly combines each neural representation (subproblem) together at the logits. This work provides a deep case study and provides the community with a very well understood toy model for interpretability, as well as makes steps toward proving the conjecture that DNNs will divide and conquer all group multiplication tasks.

2025-12-31

International Conference on Machine Learning (Accept (regular))

openreview.net

The Geometry and Topology of Circuits: the Manifolds of Modular Addition

Gabriela Moisescu-Pareja

Colin Daniels

Jonathan Love

The Clock and Pizza interpretations, associated with architectures differing in either uniform or learnable attention, were introduced to ar… (see more)gue that different architectural designs can yield distinct circuits for modular addition. In this work, we show that this is not the case, and that both the uniform and trainable attention architectures implement the same algorithm via topologically and geometrically equivalent representations. Our methodology goes beyond the interpretation of individual neurons and weights. Instead, we identify all of the neurons corresponding to each learned representation and then study the collective group of neurons as one entity. This method reveals that each learned representation is a manifold that we can study utilizing tools from topology. Based on this insight, we can statistically analyze the learned representations across hundreds of circuits to demonstrate the similarity between learned modular addition circuits that arise naturally from common deep learning paradigms.

2025-12-31

International Conference on Learning Representations (Accept (Poster))

openreview.net

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks

Gavin McCracken

Gabriela Moisescu-Pareja

Vincent Létourneau

Doina Precup

Jonathan Love

We propose a testable universality hypothesis, asserting that seemingly disparate neural network solutions observed in the simple task of mo… (see more)dular addition are unified under a common abstract algorithm. While prior work interpreted variations in neuron-level representations as evidence for distinct algorithms, we demonstrate - through multi-level analyses spanning neurons, neuron clusters, and entire networks - that multilayer perceptrons and transformers universally implement the abstract algorithm we call the approximate Chinese Remainder Theorem. Crucially, we introduce approximate cosets and show that neurons activate exclusively on them. Furthermore, our theory works for deep neural networks (DNNs). It predicts that universally learned solutions in DNNs with trainable embeddings or more than one hidden layer require only O(log n) features, a result we empirically confirm. This work thus provides the first theory-backed interpretation of multilayer networks solving modular addition. It advances generalizable interpretability and opens a testable universality hypothesis for group multiplication beyond modular addition.

2025-12-02

Neural Information Processing Systems (Accept (poster))

doi.org

openreview.net

The Geometry and Topology of Modular Addition Representations

Gabriela Moisescu-Pareja

Colin Daniels

Jonathan Love

The Clock and Pizza interpretations, associated with neural architectures differing in either uniform or learnable attention, were introduce… (see more)d to argue that different architectural designs can yield distinct circuits for modular addition. Applying geometric and topological analyses to learned representations, we show that this is not the case: Clock and Pizza circuits are topologically and geometrically equivalent and are thus equivalent representations.

2025-11-12

TAG-DS/2025/Conference (poster)

openreview.net

Unifying Mechanistic Interpretations of Neural Networks Trained on Modular Addition

Gabriela Moisescu-Pareja

Gavin McCracken

Vincent Létourneau

Doina Precup

Jonathan Love

2025-09-21

NeurIPS.cc/2025/Workshop/WiML (published)

openreview.net

Towards Foundational Models for Molecular Learning on Large-Scale Multi-Task Datasets

Dominique Beaini

Shenyang Huang

Joao Alex Cunha

Zhiyi Li

Gabriela Moisescu-Pareja

Oleksandr Dymov

Samuel Maddrell-Mander

Callum McLean

Jama Hussein Mohamud

Michael Craig

Cristian Gabellini

Kerstin Klaser

Josef Dean

Cas Wognum … (see 14 more)

Maciej Sypetkowski

Hadrien Mary

Therence Bois

Andrew Fitzgibbon

Błażej Banaszewski

Chad Martin

Dominic Masters

Recently, pre-trained foundation models have shown significant advancements in multiple fields. However, the lack of datasets with labeled f… (see more)eatures and codebases has hindered the development of a supervised foundation model for molecular tasks. Here, we have carefully curated seven datasets specifically tailored for node- and graph-level prediction tasks to facilitate supervised learning on molecules. Moreover, to support the development of multi-task learning on our proposed datasets, we created the Graphium graph machine learning library. Our dataset collection encompasses two distinct categories. Firstly, the TOYMIX category modifies three small existing datasets with additional data for multi-task learning. Secondly, the LARGEMIX category includes four large-scale datasets with 344M graph-level data points and 409M node-level data points from ∼5M unique molecules. Finally, the ultra-large dataset contains 2,210M graph-level data points and 2,031M node-level data points coming from 86M molecules. Hence our datasets represent an order of magnitude increase in data volume compared to other 2D-GNN datasets. In addition, recognizing that molecule-related tasks often span multiple levels, we have designed our library to explicitly support multi-tasking, offering a diverse range of multi-level representations, i.e., representations at the graph, node, edge, and node-pair level. We equipped the library with an extensive collection of models and features to cover different levels of molecule analysis. By combining our curated datasets with this versatile library, we aim to accelerate the development of molecule foundation models. Datasets and code are available at https://github.com/datamol-io/graphium.

2024-05-06

International Conference on Learning Representations (Accept (poster))

doi.org

openreview.net

Towards Foundational Models for Molecular Learning on Large-Scale Multi-Task Datasets (Ultra Large Dataset)

Dominique Beaini

Shenyang Huang

Joao Alex Cunha

Zhiyi Li

Gabriela Moisescu-Pareja

Oleksandr Dymov

Samuel Maddrell-Mander