Isomorphism problems for tensors, groups, and cubic forms: completeness and reductions

Abstract

In this paper we consider the problems of testing isomorphism of tensors, p-groups, cubic forms, algebras, and more, which arise from a variety of areas, including machine learning, group theory, and cryptography. These problems can all be cast as orbit problems on multi-way arrays under different group actions. Our first two main results are:1. All the aforementioned isomorphism problems are equivalent under polynomial-time reductions, in conjunction with the recent results of Futorny-Grochow-Sergeichuk (Lin. Alg. Appl., 2019).2. Isomorphism of d-tensors reduces to isomorphism of 3-tensors, for any d ≥ 3.All but one of the reductions for the preceding contributions work over arbitrary fields. Together they suggest that the aforementioned isomorphism problems form a rich and robust equivalence class, which we call Tensor Isomorphism-complete, or TI-complete for short. Furthermore, this provides a unified viewpoint on these hard isomorphism testing problems arising from a variety of areas.We then leverage the techniques used in the above results to prove two first-of-their-kind results for Group Isomorphism (GpI):3. We give a reduction from testing isomorphism of p-groups of exponent p and small class (c < p) to isomorphism of p-groups of exponent p and class 2. The latter are widely believed to be the hardest cases of GpI, but as far as we know, this is the first reduction from any more general class of groups to this class.4. We give a search-to-decision reduction for isomorphism of p-groups of exponent p and class 2 in time |G| O(log log |G|) . While search-to-decision reductions for Graph Isomorphism (GI) have been known for more than 40 years, as far as we know this is the first non-trivial search-to-decision reduction in the context of GpI.Our main technique for (1), (3), and ( 4) is a linear-algebraic analogue of the classical graph coloring gadget, which was used to obtain the search-to-decision reduction for GI. This gadget construction may be of independent interest and utility. The technique for (2) gives a method for encoding an arbitrary tensor into an algebra.

On the Baer-Lovász-Tutte construction of groups from graphs: isomorphism types and homomorphism notions

“…On one hand, when groups are given by Cayley tables, group isomorphism reduces to graph isomorphism [KST93]. On the other hand, graph isomorphism reduces to group isomorphism, when groups are given by generators as permutations [Luk93] or matrices over finite fields [GQ19]. The reduction in [GQ19] actually constructs p-groups of class 2 and exponent p from graphs.…”

“…The reduction in [GQ19] relies a reduction from graph isomorphism to alternating bilinear map isomorphism. However, there some gadget is needed to restrict from invertible matrices to monomial matrices.…”

Derandomization and absolute reconstruction for sums of powers of linear forms

“…More generally, one could ask whether two forms of degree 3 given as input are equivalent (by an invertible change of variables as above). This problem is known to be at least as hard as graph isomorphism [2,36] and is "tensor isomorphism complete" [24]. By contrast, equivalence of quadratic forms is "easy": it is classically known from linear algebra that two real quadratic forms are equivalent iff they have the same rank and signature (this is "Sylvester's law of inertia") and two quadratic forms over C n are equivalent iff they have the same rank.…”

“…Equivalence to P d for arbitrary d was studied by Kayal [36]. This paper also begins a study of equivalence to other specific polynomials such as the elementary symmetric polynomials; this study is continued in [20,23,24,37,38], in particular for the permanent and determinant polynomials. The contributions of the present paper are twofold:…”

On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials I: Tensor Isomorphism-Completeness

“…They produce a local algebra on n+1 2 + n generators, whose dimension is cubic in the number of generators. 6 These works were independent of one another, and provide two different reductions; we note that the former result seems to have first appeared in the 2019 preprint[GQ19].…”