ALGEBRA AND NUMBER THEORY

Algebraic proof of the equivalence of two variants of the cut-norm for multidimensional symmetric matrices

P. N. Shvedkov^a, K. V. Lykov<sup>b

a</sup> Belarusian State University, Minsk, Belarus
^b Institute of Mathematics of the National Academy of Sciences of Belarus, Minsk, Belarus

Abstract: The paper proves the equivalence of two special matrix norms. Both norms arise in models formulated in terms of interactions between binary variables. One norm is associated with the interaction of these variables within a single group, while the other is related to the interaction of variables from different groups. The statement allows for an easy transfer of meaningful results from the second (simpler) case to the first.

Keywords: cut-norm, matrix norm, multilinear forms, equivalence of norms, graph theory, combinatorial optimization, quantum computing, restricted Boltzmann machine, multidimensional array, tensor.

UDC: 519.16, 512.64:(004.85+538.9)

Received: 09.04.2025
Revised: 23.05.2025
Accepted: 23.05.2025