J. Elisenda Grigsby, Yiyao Huang, Arushi Katyal, Ryan Lin, Pranav Ramakrishnan, Weiya Wang
原文摘要
The field of mechanistic interpretability is well-suited to collaborative projects between mathematicians and AI researchers. This work describes one such project. We trained a small transformer to learn the language of the Coxeter group affine A2, best known as the symmetries of the tiling of the Euclidean plane by equilateral triangles. We then formulated and proved a word-theoretic conjecture about affine A2 after observing the attention patterns of the trained model.
Arif Dönmez, Axel Mosig, Ellen Fritsche, Katharina Koch
原文摘要
Neural Sheaf Diffusion (NSD) learns a task-adapted sheaf geometry for node classification. We connect the study of these learned geometries with Geometric Invariant Theory (GIT) to analyse their representation-theoretic structure and develop mechanisms for geometric control during training. Identifying cellular sheaves with representations of the incidence quiver connects direct-sum decompositions to the harmonic space and motivates the study of trivial or low-complexity summands as potential degeneration modes. We make this perspective concrete by focusing on the trivial representation: on an embedded copy, the sheaf Dirichlet energy reduces to weighted scalar graph energy. The GIT stability framework provides a criterion for excluding such copies within the semistable class. Using its moment-map characterisation of polystability, we construct differentiable surrogates that encourage the corresponding balance conditions. We identify an equal-stalk obstruction to assigning a destabilising weight to the trivial representation and show how nonuniform stalk dimensions remove it. Controlled energy-minimisation experiments reproduce trivial-component formation, while native feature-conditioned NSD experiments on heterophilic graphs show how moment-map regularisation changes the balance and Dirichlet-energy trajectories of learned geometries during training, alongside their classification behaviour.
Francesco Conti, Patrizio Frosini, Nicola Quercioli
原文摘要
Geometric and Topological Deep Learning enhance machine learning by exploiting geometric and topological structures in data. In this context, Group Equivariant Non-Expansive Operators (GENEOs) provide a principled framework for encoding symmetries and designing efficient, interpretable neural architectures. Originally introduced in Topological Data Analysis and later adopted in Deep Learning, GENEOs unify geometric and topological approaches and include, as a special case, the operator computing persistence diagrams. Their theoretical foundations rely on group actions, equivariance, and compactness properties of operator spaces. GENEOs are defined between data spaces endowed with transformation groups, referred to as perception pairs. While previous representation results characterized linear GENEOs acting on identical perception pairs, many applications require operators between heterogeneous ones. In this work, we introduce and prove a new algebraic representation theorem for linear GENEOs acting between different perception pairs. Under mild assumptions on data domains and group actions, this result provides a complete characterization of such operators and implicitly yields a computational method to generate all linear GENEOs between two perception pairs. The proof combines tools from the theory of stochastic matrices, group actions, and invariant measures. In our paper, we also show that the space of linear GENEOs is compact and convex, making easier their use for learning and optimization. Our new theorem enlarges the applicability of GENEOs, and allows for their composition with a broader class of current architectures. We show this with an application in image reconstruction.
Mahmut Levent Doğan, Vladimir Lysikov, Keiya Sakabe, Michael Walter
原文摘要
We give an exposition of OpenAI's recent result that the exponent of matrix multiplication is at most 9/4 valid over every field. Our main tool is a fractional symmetrized variant of the tensor value introduced by Coppersmith and Winograd, used in place of spectral points (tensor characters) in the original proof. We prove a superadditivity inequality for a power of this value on partially direct sums, whose summands share one tensor factor and occupy independent subspaces in the other two. Together with the degenerations of polynomial multiplication tensors from the OpenAI proof, this inequality yields the bound.