# Open Problem: Separating Geometric and Algorithmic Compression via Cayley-Table Completion

arXiv: https://arxiv.org/abs/2605.29885 · HTML: https://mlsanity.com/paper/2605.29885/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 25 references that are in the map.
- First posted on arXiv 28 May 2026.

## Builds on (most cited first, 13 of 13)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Intrinsic Dimensionality Explains the Effectiveness of Language Model Fine-Tuning](https://mlsanity.com/paper/2012.13255/) (2020)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [Neural Probabilistic Logic Programming in DeepProbLog](https://mlsanity.com/paper/1907.08194/) (2019)
- [Implicit Regularization in Matrix Factorization](https://mlsanity.com/paper/1705.09280/) (2017)
- [Intrinsic dimension of data representations in deep neural networks](https://mlsanity.com/paper/1905.12784/) (2019)
- [Implicit Bias of Gradient Descent on Linear Convolutional Networks](https://mlsanity.com/paper/1806.00468/) (2018)
- [Neural Networks and the Chomsky Hierarchy](https://mlsanity.com/paper/2207.02098/) (2022)
- [The Low-Rank Simplicity Bias in Deep Networks](https://mlsanity.com/paper/2103.10427/) (2021)
- [Can You Learn an Algorithm? Generalizing from Easy to Hard Problems with Recurrent Networks](https://mlsanity.com/paper/2106.04537/) (2021)
- [Simplicity Bias in 1-Hidden Layer Neural Networks](https://mlsanity.com/paper/2302.00457/) (2023)
- [A Differentiable Measure of Algebraic Complexity: Provably Exact Discovery of Group Structures](https://mlsanity.com/paper/2511.23152/) (2025)

## Abstract (arXiv)

Modern statistical learning theory and deep learning characterize generalization primarily in terms of continuous capacity control (e.g., norm-based regularization, margin maximization, low-rank bias). While highly successful in continuous domains, deep learning consistently fails to extrapolate exact algorithmic or discrete algebraic rules, reflecting a missing inductive bias toward algorithmic complexity minimization. We propose the Cayley-table completion as the canonical testbed for this missing bias, serving as the discrete algebraic counterpart to matrix completion. Just as matrix factorization combined with weight decay yields an implicit geometric bias toward low linear rank, recent results demonstrate that operator-valued tensor factorizations paired with a flatness prior yield an implicit algorithmic bias toward exact discrete associativity. We pose the open problem of establishing formal exact recovery bounds for Cayley-table completion, and challenge the community to generalize continuous flatness priors to autonomously discover broader discrete algorithmic axioms without combinatorial search.