# Learning Algebraic Representation for Systematic Generalization in Abstract Reasoning

arXiv: https://arxiv.org/abs/2111.12990 · HTML: https://mlsanity.com/paper/2111.12990/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 80 references that are in the map.
- First posted on arXiv 25 Nov 2021.

## Builds on (most cited first, 30 of 30)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [CLEVRER: CoLlision Events for Video REpresentation and Reasoning](https://mlsanity.com/paper/1910.01442/) (2019)
- [Neural Probabilistic Logic Programming in DeepProbLog](https://mlsanity.com/paper/1907.08194/) (2019)
- [The Neuro-Symbolic Concept Learner: Interpreting Scenes, Words, and Sentences From Natural Supervision](https://mlsanity.com/paper/1904.12584/) (2019)
- [A simple neural network module for relational reasoning](https://mlsanity.com/paper/1706.01427/) (2017)
- [Neural-Symbolic VQA: Disentangling Reasoning from Vision and Language Understanding](https://mlsanity.com/paper/1810.02338/) (2018)
- [RAVEN: A Dataset for Relational and Analogical Visual rEasoNing](https://mlsanity.com/paper/1903.02741/) (2019)
- [Learning Explanatory Rules from Noisy Data](https://mlsanity.com/paper/1711.04574/) (2017)
- [The Next Decade in AI: Four Steps Towards Robust Artificial Intelligence](https://mlsanity.com/paper/2002.06177/) (2020)
- [Measuring abstract reasoning in neural networks](https://mlsanity.com/paper/1807.04225/) (2018)
- [End-to-End Differentiable Proving](https://mlsanity.com/paper/1705.11040/) (2017)
- [Neural Logic Machines](https://mlsanity.com/paper/1904.11694/) (2019)
- [Learning by Abstraction: The Neural State Machine](https://mlsanity.com/paper/1907.03950/) (2019)
- [Compositional Generalization via Neural-Symbolic Stack Machines](https://mlsanity.com/paper/2008.06662/) (2020)
- [Grounding Physical Concepts of Objects and Events Through Dynamic Visual Reasoning](https://mlsanity.com/paper/2103.16564/) (2021)
- [Dark, Beyond Deep: A Paradigm Shift to Cognitive AI with Humanlike Common Sense](https://mlsanity.com/paper/2004.09044/) (2020)
- [Schema Networks: Zero-shot Transfer with a Generative Causal Model of Intuitive Physics](https://mlsanity.com/paper/1706.04317/) (2017)
- [Learning Perceptual Inference by Contrasting](https://mlsanity.com/paper/1912.00086/) (2019)
- [Learning to Make Analogies by Contrasting Abstract Relational Structure](https://mlsanity.com/paper/1902.00120/) (2019)
- [The Scattering Compositional Learner: Discovering Objects, Attributes, Relationships in Analogical Reasoning](https://mlsanity.com/paper/2007.04212/) (2020)
- [Abstract Spatial-Temporal Reasoning via Probabilistic Abduction and Execution](https://mlsanity.com/paper/2103.14230/) (2021)
- [Visual Concept-Metaconcept Learning](https://mlsanity.com/paper/2002.01464/) (2020)
- [Abstract Diagrammatic Reasoning with Multiplex Graph Networks](https://mlsanity.com/paper/2006.11197/) (2020)
- [Abstract Reasoning with Distracting Features](https://mlsanity.com/paper/1912.00569/) (2019)
- [ACRE: Abstract Causal REasoning Beyond Covariation](https://mlsanity.com/paper/2103.14232/) (2021)
- [Improving Generalization for Abstract Reasoning Tasks Using Disentangled Feature Representations](https://mlsanity.com/paper/1811.04784/) (2018)
- [Dynamic Inference with Neural Interpreters](https://mlsanity.com/paper/2110.06399/) (2021)
- [Machine Number Sense: A Dataset of Visual Arithmetic Problems for Abstract and Relational Reasoning](https://mlsanity.com/paper/2004.12193/) (2020)
- [MetaStyle: Three-Way Trade-Off Among Speed, Flexibility, and Quality in Neural Style Transfer](https://mlsanity.com/paper/1812.05233/) (2018)

## Built on by (most cited first, 27 of 27)

- [EgoTaskQA: Understanding Human Tasks in Egocentric Videos](https://mlsanity.com/paper/2210.03929/) (2022)
- [Deep Learning Methods for Abstract Visual Reasoning: A Survey on Raven's Progressive Matrices](https://mlsanity.com/paper/2201.12382/) (2022)
- [Learning to reason over visual objects](https://mlsanity.com/paper/2303.02260/) (2023)
- [GENOME: GenerativE Neuro-symbOlic visual reasoning by growing and reusing ModulEs](https://mlsanity.com/paper/2311.04901/) (2023)
- [MEWL: Few-shot multimodal word learning with referential uncertainty](https://mlsanity.com/paper/2306.00503/) (2023)
- [Look, Remember and Reason: Grounded reasoning in videos with language models](https://mlsanity.com/paper/2306.17778/) (2023)
- [Neural-Symbolic Recursive Machine for Systematic Generalization](https://mlsanity.com/paper/2210.01603/) (2022)
- [Abstract Visual Reasoning: An Algebraic Approach for Solving Raven's Progressive Matrices](https://mlsanity.com/paper/2303.11730/) (2023)
- [Multi-Viewpoint and Multi-Evaluation with Felicitous Inductive Bias Boost Machine Abstract Reasoning Ability](https://mlsanity.com/paper/2210.14914/) (2022)
- [Towards Learning to Reason: Comparing LLMs with Neuro-Symbolic on Arithmetic Relations in Abstract Reasoning](https://mlsanity.com/paper/2412.05586/) (2024)
- [Abstracting Concept-Changing Rules for Solving Raven's Progressive Matrix Problems](https://mlsanity.com/paper/2307.07734/) (2023)
- [Can Large Reasoning Models do Analogical Reasoning under Perceptual Uncertainty?](https://mlsanity.com/paper/2503.11207/) (2025)
- [DETR++: Taming Your Multi-Scale Detection Transformer](https://mlsanity.com/paper/2206.02977/) (2022)
- [A Cognitively-Inspired Neural Architecture for Visual Abstract Reasoning Using Contrastive Perceptual and Conceptual Processing](https://mlsanity.com/paper/2309.10532/) (2023)
- [A-I-RAVEN and I-RAVEN-Mesh: Two New Benchmarks for Abstract Visual Reasoning](https://mlsanity.com/paper/2406.11061/) (2024)
- [Learning Visual Abstract Reasoning through Dual-Stream Networks](https://mlsanity.com/paper/2411.19451/) (2024)
- [On the Complexity of Bayesian Generalization](https://mlsanity.com/paper/2211.11033/) (2022)
- [Unraveling the geometry of visual relational reasoning](https://mlsanity.com/paper/2502.17382/) (2025)
- [Towards Generative Abstract Reasoning: Completing Raven's Progressive Matrix via Rule Abstraction and Selection](https://mlsanity.com/paper/2401.09966/) (2024)
- [Triple-CFN: Separating Concepts and Features Enhances Machine Abstract Reasoning Ability](https://mlsanity.com/paper/2403.03190/) (2024)
- [Funny-Valen-Tine: Planning Solution Distribution Enhances Machine Abstract Reasoning Ability](https://mlsanity.com/paper/2407.02688/) (2024)
- [Systematic Abductive Reasoning via Diverse Relation Representations in Vector-symbolic Architecture](https://mlsanity.com/paper/2501.11896/) (2025)
- [D4C: Improving Negative Example Quality to Enhance Machine Abstract Reasoning Ability](https://mlsanity.com/paper/2403.03452/) (2024)
- [GenVP: Generating Visual Puzzles with Contrastive Hierarchical VAEs](https://mlsanity.com/paper/2503.23598/) (2025)
- [Advancing Generalization Across a Variety of Abstract Visual Reasoning Tasks](https://mlsanity.com/paper/2505.13391/) (2025)
- [Beyond Task-Specific Reasoning: A Unified Conditional Generative Framework for Abstract Visual Reasoning](https://mlsanity.com/paper/2507.11761/) (2025)
- [DIO: Refining Mutual Information and Causal Chain to Enhance Machine Abstract Reasoning Ability](https://mlsanity.com/paper/2508.15387/) (2025)

## Abstract (arXiv)

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.