Machine Learning meets Algebraic Combinatorics: A Suite of Datasets Capturing Research-level Conjecturing Ability in Pure Mathematics

With recent dramatic increases in AI system capabilities, there has been growing interest in utilizing machine learning for reasoning-heavy, quantitative tasks, particularly mathematics. While there are many resources capturing mathematics at the high-school, undergraduate, and graduate level, there are far fewer resources available that align with the level of difficulty and open endedness encountered by professional mathematicians working on open problems. To address this, we introduce a new collection of datasets, the Algebraic Combinatorics Dataset Repository (ACD Repo), representing either foundational results or open problems in algebraic combinatorics, a subfield of mathematics that studies discrete structures arising from abstract algebra. Further differentiating our dataset collection is the fact that it aims at the conjecturing process. Each dataset includes an open-ended research-level question and a large collection of examples (up to 10M in some cases) from which conjectures should be generated. We describe all nine datasets, the different ways machine learning models can be applied to them (e.g., training with narrow models followed by interpretability analysis or program synthesis with LLMs), and discuss some of the challenges involved in designing datasets like these.

Adaptively evaluating models with task elicitation

Manual curation of evaluation datasets is struggling to keep up with the rapidly expanding capabilities and deployment scenarios of language models. Towards scalable model profiling, we introduce and validate a framework for evaluating LLMs, called Adaptive Evaluations. Adaptive evaluations use scaffolded language models (evaluator agents) to search through a target model's behavior on a domain dataset and create difficult questions (tasks) that can discover and probe the model's failure modes. We find that frontier models lack consistency when adaptively probed with our framework on a diverse suite of datasets and tasks, including but not limited to legal reasoning, forecasting, and online harassment. Generated questions pass human validity checks and often transfer to other models with different capability profiles, demonstrating that adaptive evaluations can also be used to create difficult domain-specific datasets.

Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes

Machine learning is becoming an increasingly valuable tool in mathematics, enabling one to identify subtle patterns across collections of examples so vast that they would be impossible for a single researcher to feasibly review and analyze. In this work, we use graph neural networks to investigate quiver mutation -- an operation that transforms one quiver (or directed multigraph) into another -- which is central to the theory of cluster algebras with deep connections to geometry, topology, and physics. In the study of cluster algebras, the question of mutation equivalence is of fundamental concern: given two quivers, can one efficiently determine if one quiver can be transformed into the other through a sequence of mutations? Currently, this question has only been resolved in specific cases. In this paper, we use graph neural networks and AI explainability techniques to discover mutation equivalence criteria for the previously unknown case of quivers of type $\tilde{D}_n$. Along the way, we also show that even without explicit training to do so, our model captures structure within its hidden representation that allows us to reconstruct known criteria from type $D_n$, adding to the growing evidence that modern machine learning models are capable of learning abstract and general rules from mathematical data.

Model editing for distribution shifts in uranium oxide morphological analysis

Deep learning still struggles with certain kinds of scientific data. Notably, pretraining data may not provide coverage of relevant distribution shifts (e.g., shifts induced via the use of different measurement instruments). We consider deep learning models trained to classify the synthesis conditions of uranium ore concentrates (UOCs) and show that model editing is particularly effective for improving generalization to distribution shifts common in this domain. In particular, model editing outperforms finetuning on two curated datasets comprising of micrographs taken of U$_{3}$O$_{8}$ aged in humidity chambers and micrographs acquired with different scanning electron microscopes, respectively.

Haldane Bundles: A Dataset for Learning to Predict the Chern Number of Line Bundles on the Torus

Characteristic classes, which are abstract topological invariants associated with vector bundles, have become an important notion in modern physics with surprising real-world consequences. As a representative example, the incredible properties of topological insulators, which are insulators in their bulk but conductors on their surface, can be completely characterized by a specific characteristic class associated with their electronic band structure, the first Chern class. Given their importance to next generation computing and the computational challenge of calculating them using first-principles approaches, there is a need to develop machine learning approaches to predict the characteristic classes associated with a material system. To aid in this program we introduce the {\emph{Haldane bundle dataset}}, which consists of synthetically generated complex line bundles on the $2$-torus. We envision this dataset, which is not as challenging as noisy and sparsely measured real-world datasets but (as we show) still difficult for off-the-shelf architectures, to be a testing ground for architectures that incorporate the rich topological and geometric priors underlying characteristic classes.

Understanding the Inner Workings of Language Models Through Representation Dissimilarity

As language models are applied to an increasing number of real-world applications, understanding their inner workings has become an important issue in model trust, interpretability, and transparency. In this work we show that representation dissimilarity measures, which are functions that measure the extent to which two model's internal representations differ, can be a valuable tool for gaining insight into the mechanics of language models. Among our insights are: (i) an apparent asymmetry in the internal representations of model using SoLU and GeLU activation functions, (ii) evidence that dissimilarity measures can identify and locate generalization properties of models that are invisible via in-distribution test set performance, and (iii) new evaluations of how language model features vary as width and depth are increased. Our results suggest that dissimilarity measures are a promising set of tools for shedding light on the inner workings of language models.

Attributing Learned Concepts in Neural Networks to Training Data

By now there is substantial evidence that deep learning models learn certain human-interpretable features as part of their internal representations of data. As having the right (or wrong) concepts is critical to trustworthy machine learning systems, it is natural to ask which inputs from the model's original training set were most important for learning a concept at a given layer. To answer this, we combine data attribution methods with methods for probing the concepts learned by a model. Training network and probe ensembles for two concept datasets on a range of network layers, we use the recently developed TRAK method for large-scale data attribution. We find some evidence for convergence, where removing the 10,000 top attributing images for a concept and retraining the model does not change the location of the concept in the network nor the probing sparsity of the concept. This suggests that rather than being highly dependent on a few specific examples, the features that inform the development of a concept are spread in a more diffuse manner across its exemplars, implying robustness in concept formation.

On Privileged and Convergent Bases in Neural Network Representations

In this study, we investigate whether the representations learned by neural networks possess a privileged and convergent basis. Specifically, we examine the significance of feature directions represented by individual neurons. First, we establish that arbitrary rotations of neural representations cannot be inverted (unlike linear networks), indicating that they do not exhibit complete rotational invariance. Subsequently, we explore the possibility of multiple bases achieving identical performance. To do this, we compare the bases of networks trained with the same parameters but with varying random initializations. Our study reveals two findings: (1) Even in wide networks such as WideResNets, neural networks do not converge to a unique basis; (2) Basis correlation increases significantly when a few early layers of the network are frozen identically. Furthermore, we analyze Linear Mode Connectivity, which has been studied as a measure of basis correlation. Our findings give evidence that while Linear Mode Connectivity improves with increased network width, this improvement is not due to an increase in basis correlation.

How many dimensions are required to find an adversarial example?

Past work exploring adversarial vulnerability have focused on situations where an adversary can perturb all dimensions of model input. On the other hand, a range of recent works consider the case where either (i) an adversary can perturb a limited number of input parameters or (ii) a subset of modalities in a multimodal problem. In both of these cases, adversarial examples are effectively constrained to a subspace $V$ in the ambient input space $\mathcal{X}$. Motivated by this, in this work we investigate how adversarial vulnerability depends on $\dim(V)$. In particular, we show that the adversarial success of standard PGD attacks with $\ell^p$ norm constraints behaves like a monotonically increasing function of $\epsilon (\frac{\dim(V)}{\dim \mathcal{X}})^{\frac{1}{q}}$ where $\epsilon$ is the perturbation budget and $\frac{1}{p} + \frac{1}{q} =1$, provided $p > 1$ (the case $p=1$ presents additional subtleties which we analyze in some detail). This functional form can be easily derived from a simple toy linear model, and as such our results land further credence to arguments that adversarial examples are endemic to locally linear models on high dimensional spaces.

Fast computation of permutation equivariant layers with the partition algebra

Linear neural network layers that are either equivariant or invariant to permutations of their inputs form core building blocks of modern deep learning architectures. Examples include the layers of DeepSets, as well as linear layers occurring in attention blocks of transformers and some graph neural networks. The space of permutation equivariant linear layers can be identified as the invariant subspace of a certain symmetric group representation, and recent work parameterized this space by exhibiting a basis whose vectors are sums over orbits of standard basis elements with respect to the symmetric group action. A parameterization opens up the possibility of learning the weights of permutation equivariant linear layers via gradient descent. The space of permutation equivariant linear layers is a generalization of the partition algebra, an object first discovered in statistical physics with deep connections to the representation theory of the symmetric group, and the basis described above generalizes the so-called orbit basis of the partition algebra. We exhibit an alternative basis, generalizing the diagram basis of the partition algebra, with computational benefits stemming from the fact that the tensors making up the basis are low rank in the sense that they naturally factorize into Kronecker products. Just as multiplication by a rank one matrix is far less expensive than multiplication by an arbitrary matrix, multiplication with these low rank tensors is far less expensive than multiplication with elements of the orbit basis. Finally, we describe an algorithm implementing multiplication with these basis elements.