Thank you for your response and explanation in Q&A session! It’s clear to me. Thank you!

Thank you very much for your question! The idea is that the matrix of the WFA needs to have the Hankel property. When you truncate the SVD the result might not be Hankel, so you need to find a way to preserve the property when doing approximation. This can be done in several ways. An alternative example can be found in Balle 2019 (Singular Value Automaton and Approximate Minimization), where the authors truncate a canonical form of WFA instead of the corresponding Hankel matrix. The tools we use, in particular AAK Theory, are needed to guarantee that this property is preserved and that the matrix obtained is still Hankel. In fact, AAK Theorem “returns” a Hankel matrix. On the other hand, when you apply spectral methods to recover WFAs from Hankel matrices, you start from a Hankel matrix, and use the truncated SVD and its relation with rank factorization in order to compute the parameters of the WFA. Note that the truncated SVD in the case of spectral methods is not the full Hankel matrix of the extracted WFA. I hope that I understood your question well and that I answered clearly, please don’t hesitate asking again if I didn’t.

There are two important challenges to extend the work. The main challenge will be to adapt results from harmonic analysis to the case of non abelian structures. While some work as been done in that direction (Popescu 2003), it is still unclear how we can transfer these results to the setting of multi-letter alphabets. A second major obstacle is that the generalization done by Popescu leads to a proof of AAK Theory that is not constructive, making it difficult to develop an algorithm to optimal approximation.