Explicit Universal Sequences of Tensors for Characterizing Asymptotic Rank

Core Concepts

The authors construct explicit sequences of zero-one-valued tensors that universally characterize the worst-case tensor exponent and asymptotic rank in the space of tensors Fd ⊗Fd ⊗Fd.

Abstract

The key insights and highlights of the content are:

1. The authors introduce the concept of a "composition basis" for the linear span of the Kronecker power map Kd,q, which provides an explicit basis for expressing the Kronecker powers of tensors in Fd ⊗Fd ⊗Fd.

2. Using the composition basis, the authors construct an explicit sequence of tensors Ud = (Ud,q : q = 1, 2, ...) such that σ(Ud) = σ(d), where σ(d) is the supremum of the tensor exponents over all tensors in Fd ⊗Fd ⊗Fd.

3. The authors also construct an explicit sequence of tensors Td such that σ(Td) = σ(Td), where Td is the space of tight tensors in Fd ⊗Fd ⊗Fd. This provides a universal sequence for Strassen's asymptotic rank conjecture.

4. By diagonalizing the sequences Ud, the authors obtain an explicit sequence D = (Dd : d = 1, 2, ...) that universally captures the limit of the worst-case tensor exponent limd→∞σ(d), addressing the extended asymptotic rank conjecture.

5. The authors show that the absence of low-degree polynomial equations vanishing on tensors of low asymptotic rank implies upper bounds on the asymptotic rank, providing a new technique for bounding asymptotic rank.

Overall, the paper presents explicit universal constructions of tensors that characterize the worst-case asymptotic rank behavior, with implications for resolving long-standing conjectures in computational complexity theory.

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Key Insights Distilled From

A universal sequence of tensors for the asymptotic rank conjecture

by Pett... at arxiv.org 04-10-2024

https://arxiv.org/pdf/2404.06427.pdf

Deeper Inquiries

What other properties or structures of the composition basis tensors T(g) can be leveraged to further understand the asymptotic rank of tensors

One property of the composition basis tensors T(g) that can be leveraged to further understand the asymptotic rank of tensors is their relationship with group theory. The tensors T(g) can be described as orbit-indicators of the symmetric group Sq, indicating the action of permutations on the tensor space. By studying the group-theoretic descriptions of these tensors, we can gain insights into their structural properties and how they contribute to the overall rank and asymptotic rank of tensors. Additionally, the invariant subspaces defined by the marginals of g provide a way to analyze the tensors T(g) in a more focused manner, potentially revealing patterns or symmetries that impact their rank.

Can the techniques developed in this paper be extended to provide explicit lower bounds on the asymptotic rank of specific tensors, beyond the universal sequences constructed

The techniques developed in the paper can be extended to provide explicit lower bounds on the asymptotic rank of specific tensors beyond the universal sequences constructed. By applying the concept of support-localization and leveraging the composition basis tensors T(g), it is possible to construct explicit sequences of tensors tailored to specific formats or structures. These specialized sequences can then be analyzed to determine the worst-case exponent for those tensors, offering insights into the lower bounds of their asymptotic rank. By carefully selecting compositions g that highlight the complexity of certain tensor formats, it is feasible to derive explicit lower bounds on the asymptotic rank of specific tensors.

How might the connection between the absence of low-degree polynomial equations and upper bounds on asymptotic rank be further developed and applied to other tensor families or problems

The connection between the absence of low-degree polynomial equations and upper bounds on asymptotic rank can be further developed and applied to other tensor families or problems by exploring the concept of secant varieties and their equations. By investigating the relationship between secant varieties and tensor rank, particularly in the context of polynomial dimensionality, it is possible to derive new insights into the structure of tensors and their asymptotic rank. Additionally, incorporating computational methods for finding isotypic decomposition of homogeneous polynomials can enhance the analysis of tensor rank and provide new avenues for proving upper bounds on asymptotic rank based on the absence of low-degree equations. This approach can be extended to study a wider range of tensor families and problems, offering a comprehensive understanding of the relationship between polynomial equations and tensor complexity.