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

• URL: http://arxiv.org/abs/2411.07467v1
• Date: Tue, 12 Nov 2024 01:09:41 GMT
• Title: Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes
• Authors: Jesse He, Helen Jenne, Herman Chau, Davis Brown, Mark Raugas, Sara Billey, Henry Kvinge,
• Abstract summary: We use graph neural networks and AI explainability techniques to discover mutation equivalence criteria for quivers of type $tildeD_n$. We also show that our model captures structure within its hidden representation that allows us to reconstruct known criteria from type $D_n$.
• Score: 4.229995708813431
• License:
• Abstract: 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.

Related papers

• Learning Linear Attention in Polynomial Time [115.68795790532289]
  We provide the first results on learnability of single-layer Transformers with linear attention. We show that linear attention may be viewed as a linear predictor in a suitably defined RKHS. We show how to efficiently identify training datasets for which every empirical riskr is equivalent to the linear Transformer.
  arXiv  Detail & Related papers  (2024-10-14T02:41:01Z)
• Stacked ensemble\-based mutagenicity prediction model using multiple modalities with graph attention network [0.9736758288065405]
  Mutagenicity is a concern due to its association with genetic mutations which can result in a variety of negative consequences. In this work, we introduce a novel stacked ensemble based mutagenicity prediction model.
  arXiv  Detail & Related papers  (2024-09-03T09:14:21Z)
• Machine Learning Clifford invariants of ADE Coxeter elements [2.0269884338680866]
  We perform exhaustive calculations of all Coxeter transformations for $A_8$, $D_8$ and $E_8$ for a choice of basis of simple roots. This computational algebra paradigm generates a dataset that can then be mined using techniques from data science. This paper is a pump-priming study in experimental mathematics using Clifford algebras.
  arXiv  Detail & Related papers  (2023-09-29T18:00:01Z)
• Unsupervised Learning of Invariance Transformations [105.54048699217668]
  We develop an algorithmic framework for finding approximate graph automorphisms. We discuss how this framework can be used to find approximate automorphisms in weighted graphs in general.
  arXiv  Detail & Related papers  (2023-07-24T17:03:28Z)
• Group-invariant tensor train networks for supervised learning [0.0]
  We introduce a new numerical algorithm to construct a basis of tensors that are invariant under the action of normal matrix representations. The group-invariant tensors are then combined into a group-invariant tensor train network, which can be used as a supervised machine learning model.
  arXiv  Detail & Related papers  (2022-06-30T06:33:08Z)
• Cluster Algebras: Network Science and Machine Learning [0.0]
  Cluster algebras have recently become an important player in mathematics and physics. We investigate them through the lens of modern data science, specifically with techniques from network science and machine learning.
  arXiv  Detail & Related papers  (2022-03-25T18:25:37Z)
• The Separation Capacity of Random Neural Networks [78.25060223808936]
  We show that a sufficiently large two-layer ReLU-network with standard Gaussian weights and uniformly distributed biases can solve this problem with high probability. We quantify the relevant structure of the data in terms of a novel notion of mutual complexity.
  arXiv  Detail & Related papers  (2021-07-31T10:25:26Z)
• Category-Learning with Context-Augmented Autoencoder [63.05016513788047]
  Finding an interpretable non-redundant representation of real-world data is one of the key problems in Machine Learning. We propose a novel method of using data augmentations when training autoencoders. We train a Variational Autoencoder in such a way, that it makes transformation outcome predictable by auxiliary network.
  arXiv  Detail & Related papers  (2020-10-10T14:04:44Z)
• Deducing neighborhoods of classes from a fitted model [68.8204255655161]
  In this article a new kind of interpretable machine learning method is presented. It can help to understand the partitioning of the feature space into predicted classes in a classification model using quantile shifts. Basically, real data points (or specific points of interest) are used and the changes of the prediction after slightly raising or decreasing specific features are observed.
  arXiv  Detail & Related papers  (2020-09-11T16:35:53Z)
• The data-driven physical-based equations discovery using evolutionary approach [77.34726150561087]
  We describe the algorithm for the mathematical equations discovery from the given observations data. The algorithm combines genetic programming with the sparse regression. It could be used for governing analytical equation discovery as well as for partial differential equations (PDE) discovery.
  arXiv  Detail & Related papers  (2020-04-03T17:21:57Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.