Border subrank via a generalised Hilbert-Mumford criterion

2024-02-16

Advances in Mathematics

AI Summary

This paper investigates the asymptotic growth rate of the border subrank of generic tensors in the tensor product space $(mathbb{C}^n)^{otimes d}$, specifically addressing whether it matches the subrank’s known rate $Theta(n^{1/(d-1)})$ as $n o infty$. To resolve this, the authors introduce, for the first time, the generalized Hilbert–Mumford stability criterion into border subrank analysis, thereby establishing a deep connection between geometric invariant theory and tensor rank theory. They derive a novel algebro-geometric characterization of border subrank: a necessary and sufficient condition expressed in terms of orbit closures and semistability—unifying and generalizing prior border rank criteria. This framework overcomes a key obstacle in proving the asymptotic growth conjecture and provides refined tools for asymptotic complexity analysis of fundamental computational problems, including matrix multiplication.

Technology Category

Machine Learning: Matrix & Tensor MethodsKnowledge Representation and Reasoning: Computational Complexity of ReasoningSearch and Optimization: Non-convex Optimization

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingSecurity and Privacy: Large-scale security measurements

Problem

• Determine border subrank growth in general tensors
• Compare border and non-border subrank asymptotic rates
• Generalize Hilbert-Mumford criterion for tensor analysis

Innovation

• Generalized Hilbert-Mumford criterion application
• Border subrank growth rate analysis
• Tensor complexity asymptotic comparison