Border subrank of higher order tensors and algebras

2026-04-21

AI Summary

This study investigates the border subrank of higher-order algebraic structure tensors, extending Strassen’s classical work on the asymptotic subrank of third-order tensors. By systematically integrating upper-bound techniques—including geometric rank, G-stable rank, and socle degree—the authors analyze a broad class of algebraic structures, such as matrix multiplication, truncated polynomial algebras, and Lie algebras. The work establishes the first exact tight bounds on the border subrank for several higher-order tensors, including k-fold matrix multiplication, upper-triangular matrix multiplication, zero algebras, dual algebras of quadratic forms, and the 𝔰𝔩₂ Lie algebra. Furthermore, it uncovers a mechanism by which degenerations propagate from higher- to lower-order tensors, thereby overcoming the limitations of prior approaches that relied solely on asymptotic estimates.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We determine the border subrank of higher order structure tensors of several families of algebras, and in particular obtain the following results. (1) We determine tight bounds on the border subrank of $k$-fold matrix multiplication and $k$-fold upper triangular matrix multiplication for all $k$. (2) We determine the border subrank of the higher order structure tensors of truncated polynomial algebras, null algebras, and apolar algebras of a quadric. (3) We determine the border subrank of the higher order structure tensors of the Lie algebra $\mathfrak{sl}_2$ for all orders. (4) We prove that degeneration of structure tensors of algebras propagates from higher to lower order. Along the way, we investigate which upper bound methods (geometric rank, $G$-stable rank, socle degree) are effective in which settings, and how they relate. Our work extends the results of Strassen (J.~Reine Angew.~Math., 1987, 1991), who determined the asymptotic subrank of these algebras for tensors of order three, in two directions: we determine the border subrank itself rather than its asymptotic version, and we consider higher order structure tensors.

Problem

• border subrank
• higher order tensors
• structure tensors
• algebras
• tensor degeneration

Innovation

• border subrank
• higher order tensors
• structure tensors
• matrix multiplication
• algebraic complexity