# Algebraic Signatures for Structural Learning in Probability Tensors

arXiv: https://arxiv.org/abs/2607.18817 · HTML: https://mlsanity.com/paper/2607.18817/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 42 references that are in the map.
- First posted on arXiv 21 Jul 2026.

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

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Distributed Representations of Words and Phrases and their Compositionality](https://mlsanity.com/paper/1310.4546/) (2013)
- [Manifold learning: what, how, and why](https://mlsanity.com/paper/2311.03757/) (2023)
- [Structure Learning in Graphical Modeling](https://mlsanity.com/paper/1606.02359/) (2016)
- [Context-Specific Independence in Bayesian Networks](https://mlsanity.com/paper/1302.3562/) (2013)
- [Statistical exploration of the Manifold Hypothesis](https://mlsanity.com/paper/2208.11665/) (2022)
- [Geometry of Polynomial Neural Networks](https://mlsanity.com/paper/2402.00949/) (2024)
- [Algebra Unveils Deep Learning -- An Invitation to Neuroalgebraic Geometry](https://mlsanity.com/paper/2501.18915/) (2025)
- [Building a Web-Scale Dependency-Parsed Corpus from CommonCrawl](https://mlsanity.com/paper/1710.01779/) (2017)
- [Stratified Graphical Models - Context-Specific Independence in Graphical Models](https://mlsanity.com/paper/1309.6415/) (2013)
- [Approximate Vanishing Ideal via Data Knotting](https://mlsanity.com/paper/1801.09367/) (2018)
- [Algebraic geometry of discrete interventional models](https://mlsanity.com/paper/2012.03593/) (2020)
- [Approximating Latent Manifolds in Neural Networks via Vanishing Ideals](https://mlsanity.com/paper/2502.15051/) (2025)

## Abstract (arXiv)

Algebraic statistics characterizes statistical models through polynomial constraints, but it has mainly been used for analytically specified model classes. This paper studies the inverse problem: identifying probabilistic structure from vanishing binomials observed in empirical probability tensors. We treat the vanishing binomials of a toric model as its algebraic signature, and turn the ideal-variety correspondence of algebraic statistics into an operational procedure for structural learning that identifies a model by signature matching without parameter estimation. By restricting attention to a computationally tractable class of configuration matrices, which we call it the Kronecker-stack class, we make these signatures explicitly enumerable. Within this class we define minimum invariant constraint (MIC) as the atomic unit characterizing each signature and generalizing the notion of independence. We tested this approach employing MICs on synthetic data as well as on corpus-scale real language data. The results suggested the utility of the method, revealing that the identified rank-one structures correspond to interpretable sets of words. These results open up a new avenue for applying algebraic statistics to computational linguistics.