Efficient Tensor Isomorphism Reductions with Linear-Length Blow-Up and Applications to Algebraic Isomorphism Problems

Core Concepts

Tensor Isomorphism can be reduced to other algebraic isomorphism problems, such as Group Isomorphism and Cubic Form Equivalence, with only linear-length blow-up in the input sizes, leading to faster algorithms for these problems.

Abstract

The paper presents new techniques for reducing Tensor Isomorphism (TI) to other algebraic isomorphism problems, such as Group Isomorphism (GpI) and Cubic Form Equivalence (CFE), with only linear-length blow-up in the input sizes. This is a significant improvement over previous reductions that incurred quadratic blow-ups. The key contributions are: A new gadget construction for reducing Partitioned Tensor Isomorphism (part-TI) to plain TI, with only linear-length blow-up in the input sizes. This improves upon the previous quadratic blow-up. Using the new part-TI gadget, the authors show that if Graph Isomorphism (GI) is in P, then CFE over finite fields can be solved in quasilinear time, improving upon the previous brute-force upper bound. The authors devise a reduction from GpI for p-groups of class c and exponent p, where c < p, to GpI for the class c = 2 and exponent p. This, combined with a recent breakthrough algorithm for the class c = 2 case, yields faster algorithms for the more general class c < p. Polynomial-time search- and counting-to-decision reductions are presented for GpI of p-groups of class 2 and exponent p, answering a question of Arvind and Tóran. The techniques developed for TI and its connections to other algebraic isomorphism problems provide a unifying framework for tackling long-standing complexity-theoretic questions about these problems.

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Key Insights Distilled From

On the complexity of isomorphism problems for tensors, groups, and polynomials IV: linear-length reductions and their applications

by Joshua A. Gr... at arxiv.org 04-15-2024

https://arxiv.org/pdf/2306.16317.pdf

Deeper Inquiries

Can the techniques developed for tensor isomorphism reductions be extended to handle more general tensor structures, such as higher-order tensors or tensors with additional symmetries

The techniques developed for tensor isomorphism reductions can potentially be extended to handle more general tensor structures, such as higher-order tensors or tensors with additional symmetries. The key lies in adapting the reduction frameworks to accommodate the specific characteristics of these tensor structures. For higher-order tensors, the linear-length reductions may need to consider the interactions between multiple dimensions and the corresponding group actions. By carefully designing gadgets and transformations that preserve the essential properties of these tensors, it may be possible to achieve linear-length reductions for more complex tensor structures. Additionally, tensors with additional symmetries, such as symmetric tensors or tensors with specific patterns, can be addressed by incorporating these symmetries into the reduction techniques. This would involve leveraging the symmetries to simplify the isomorphism testing process and potentially reduce the blow-up in dimensions during the reduction process.

Are there other algebraic isomorphism problems, beyond the ones considered in this paper, that can benefit from the linear-length tensor isomorphism reductions

Beyond the algebraic isomorphism problems discussed in the paper, there are several other problems that could benefit from the linear-length tensor isomorphism reductions. One such problem is the isomorphism testing of structured matrices, where the matrices have specific patterns or constraints. By formulating the matrix isomorphism problem as a tensor isomorphism problem and applying the linear-length reductions, it may be possible to achieve faster algorithms for structured matrix isomorphism. Additionally, problems related to symmetry groups, permutation groups, or other algebraic structures that can be represented as tensors could also benefit from these reductions. The linear-length reductions provide a powerful tool for simplifying complex isomorphism problems across various algebraic domains, opening up possibilities for more efficient algorithms in a wide range of applications.

What are the implications of the faster algorithms for cubic form equivalence and algebra isomorphism on applications in areas such as cryptography, quantum information, and machine learning

The faster algorithms for cubic form equivalence and algebra isomorphism have significant implications for various applications in cryptography, quantum information, and machine learning. In cryptography, these algorithms can enhance the efficiency of cryptographic protocols that involve polynomial equivalence testing. By reducing the computational complexity of verifying the equivalence of cubic forms, cryptographic schemes that rely on algebraic properties can be implemented more efficiently and securely. In quantum information, the algorithms for algebra isomorphism can be utilized in quantum computing tasks that involve algebraic structures. The ability to test the isomorphism of algebras efficiently can contribute to the development of quantum algorithms for solving algebraic problems, which are fundamental in quantum information processing. In machine learning, the advancements in cubic form equivalence algorithms can benefit tasks that involve polynomial representations, such as polynomial regression or polynomial kernel methods in machine learning models. Faster algorithms for testing the equivalence of cubic forms enable more efficient computations and optimizations in polynomial-based machine learning algorithms, leading to improved performance and scalability in various applications.