The next gap in the subrank of 3-tensors

2023-07-12

Linear Algebra and its Applications

AI Summary

This paper investigates the gap problem for asymptotic subrank and asymptotic slice rank of third-order tensors, aiming to determine the smallest attainable value beyond 1, 1.88, and 2—specifically, whether it is 2.68 or larger. Method: Leveraging combinatorial tensor theory, the asymptotic spectrum framework, and algebraic complexity analysis, the authors conduct a systematic study of structural discontinuities in subrank—both exact and asymptotic. Contribution/Results: The work establishes, for the first time, that subrank exhibits intrinsic structural non-continuity in the asymptotic regime. It proves rigorously that, for every integer (k geq 3), there exists a 3-tensor whose (exact or asymptotic) subrank cannot equal (k). Moreover, an explicit counterexample is constructed showing that (k = 4) is unattainable. Consequently, the subrank spectrum admits a strict lower bound gap strictly greater than 2—a fundamental boundary characterization for tensor rank structure theory.

Problem

• Determine gaps in asymptotic subrank of 3-tensors
• Identify discrete values for asymptotic tensor ranks
• Establish next threshold for asymptotic slice rank

Innovation

• Determining gaps in asymptotic subrank values
• Classifying 3-tensor asymptotic subrank cases
• Extending discreteness results for tensor ranks