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1. Algebraic Statistics in Practice: Applications to Networks

   https://doi.org/10.1146/annurev-statistics-031017-100053  

   Casanellas, Marta; Petrović, Sonja; Uhler, Caroline (March 2020, Annual Review of Statistics and Its Application)

   Algebraic statistics uses tools from algebra (especially from multilinear algebra, commutative algebra, and computational algebra), geometry, and combinatorics to provide insight into knotty problems in mathematical statistics. In this review, we illustrate this on three problems related to networks: network models for relational data, causal structure discovery, and phylogenetics. For each problem, we give an overview of recent results in algebraic statistics, with emphasis on the statistical achievements made possible by these tools and their practical relevance for applications to other scientific disciplines. 

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2. Thinking With Algebra (TWA): A Project and Perspective

   Feikes, D.; Kafle, B.; McGathey, N.; & Walker, W. S. (January 2021, Proceedings of the Sixth Annual Indiana STEM Education Conference)

   W. S. Walker, III; null; null; null (Ed.)

   Thinking with Algebra (TWA) is a project to develop algebra curriculum for college classes. The overall goal is to develop curricular materials that will help prepare students conceptually for college algebra. The project is described and the importance of algebraic structure as a theoretical consideration is explained. 

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3. Learning algebraic representation for systematic generalization in abstract reasoning

   Zhang, C.; Xie, S.; Jia B.; Wu, Y. N.; Zhu, S.-C.; Zhu, Y. (January 2022, European Conference on Computer Vision (ECCV 2022))

   Is intelligence realized by connectionist or classicist? While connectionist approaches have achieved superhuman performance, there has been growing evidence that such task-specific superiority is particularly fragile in systematic generalization. This observation lies in the central debate between connectionist and classicist, wherein the latter continually advocates an algebraic treatment in cognitive architectures. In this work, we follow the classicist’s call and propose a hybrid approach to improve systematic generalization in reasoning. Specifically, we showcase a prototype with algebraic representation for the abstract spatial-temporal reasoning task of Raven’s Progressive Matrices (RPM) and present the ALgebra-Aware Neuro-Semi-Symbolic (ALANS) learner. The ALANS learner is motivated by abstract algebra and the representation theory. It consists of a neural visual perception frontend and an algebraic abstract reasoning backend: the frontend summarizes the visual information from object-based representation, while the backend transforms it into an algebraic structure and induces the hidden operator on the fly. The induced operator is later executed to predict the answer’s representation, and the choice most similar to the prediction is selected as the solution. Extensive experiments show that by incorporating an algebraic treatment, the ALANS learner outperforms various pure connectionist models in domains requiring systematic generalization. We further show the generative nature of the learned algebraic representation; it can be decoded by isomorphism to generate an answer. 

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4. Leveraging Early Algebraic Experiences

   https://doi.org/10.5951/MTLT.2023.0347  

   Walton, Margaret; Walkoe, Janet (May 2025, Mathematics Teacher: Learning and Teaching PK-12)

   Understand how to consider your students’ informal algebraic experiences from early childhood to support their formal learning of algebra. 

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5. Knowledgebra: An Algebraic Learning Framework for Knowledge Graph

   https://doi.org/10.3390/make4020019  

   Yang, Tong; Wang, Yifei; Sha, Long; Engelbrecht, Jan; Hong, Pengyu (June 2022, Machine Learning and Knowledge Extraction)

   Knowledge graph (KG) representation learning aims to encode entities and relations into dense continuous vector spaces such that knowledge contained in a dataset could be consistently represented. Dense embeddings trained from KG datasets benefit a variety of downstream tasks such as KG completion and link prediction. However, existing KG embedding methods fell short to provide a systematic solution for the global consistency of knowledge representation. We developed a mathematical language for KG based on an observation of their inherent algebraic structure, which we termed as Knowledgebra. By analyzing five distinct algebraic properties, we proved that the semigroup is the most reasonable algebraic structure for the relation embedding of a general knowledge graph. We implemented an instantiation model, SemE, using simple matrix semigroups, which exhibits state-of-the-art performance on standard datasets. Moreover, we proposed a regularization-based method to integrate chain-like logic rules derived from human knowledge into embedding training, which further demonstrates the power of the developed language. As far as we know, by applying abstract algebra in statistical learning, this work develops the first formal language for general knowledge graphs, and also sheds light on the problem of neural-symbolic integration from an algebraic perspective. 

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