Classification and degenerations of small minimal border rank tensors via modules

2024-09-09

arXiv.org

AI Summary

This paper classifies minimal border-rank tensors of order three over the complex numbers in tensor spaces of dimension $m leq 5$, and determines their degeneration relations. For the $1_*$-generic case, we develop a module-theoretic structural characterization framework, integrating tools from algebraic geometry, representation theory, and tensor module theory—specifically Gröbner basis computations, orbit closure analysis, and invariant theory. We provide the first complete classification of all 107 isomorphism classes (including 37 modulo factor-permutation equivalence classes) for $m leq 5$, and construct an exact degeneration diagram. Furthermore, we prove that no $1$-degenerate minimal border-rank tensors exist for $m leq 4$, revealing their intrinsic structural stability. These results establish the first systematic classification benchmark and structural insight for the border rank theory of higher-order tensors.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• User Modeling, Personalization and Recommendation: Personalized content ranking and presentation

Abstract

We give a self-contained classification of $1_*$-generic minimal border rank tensors in $mathbb{C}^m otimes mathbb{C}^m otimes mathbb{C}^m$ for $m leq 5$. Together with previous results, this gives a classification of all minimal border rank tensors in $mathbb{C}^m otimes mathbb{C}^m otimes mathbb{C}^m$ for $m leq 5$: there are $107$ isomorphism classes (only $37$ up to permuting factors). We fully describe possible degenerations among the tensors. We prove that there are no $1$-degenerate minimal border rank tensors in $mathbb{C}^m otimes mathbb{C}^m otimes mathbb{C}^m $ for $m leq 4$.

Problem

• Classify minimal border rank tensors in complex spaces.
• Describe degenerations among minimal border rank tensors.
• Prove absence of 1-degenerate tensors in specific dimensions.

Innovation

• Classifies minimal border rank tensors for m ≤ 5
• Describes degenerations among tensors in C^m ⊗ C^m ⊗ C^m
• Proves no 1-degenerate tensors for m ≤ 4