Thanks to everyone who helped pitch in on this. I was happy to finally track down a pdf of Van Loan & Pitsianis [1993]. As stated in this other article on "Automated Kronecker Product Approximation"[KoPA]:
"This was introduced in the matrix computation literature as the nearest Kronecker product problem in Van Loan and Pitsianis, who demonstrated its equivalence to the best rank-one approximation and therefore also to the SVD, after a proper rearrangement of the matrix entries." As well as: "Finding a low-rank approximation of a given matrix is closely related to the singular value decomposition, and the connection was revealed as early as Eckart and Yount (1936)]" It is surprising that it took until the 90s for the problem to bite someone and for explicit algorithms to emerge. Also, I am amazed that the body of the literature seems to come from the data sciences, as I arrived at the problem from studying discrete dynamical systems. I am hoping to write up a paper on the connection between them and NKP soon. Would anyone know where I can find a python, C++, or Haskell implementation easily? Thanks again for the help. [1993] https://www.cs.cornell.edu/cv/ResearchPDF/Approximation%20with%20Kronecker%20Products,%20from%20Linear%20Algebra%20for%20Large%20Scale%20and%20Real-%20Time%20Applications.pdf [KoPA] https://arxiv.org/pdf/1912.02392.pdf
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