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Towards an AAK Theory Approach to Approximate Minimization in the Multi-Letter Case | | Clara Lacroce
; Prakash Panangaden
; Guillaume Rabusseau
; | | Date: |
1 Jun 2022 | | Abstract: | We study the approximate minimization problem of weighted finite automata
(WFAs): given a WFA, we want to compute its optimal approximation when
restricted to a given size. We reformulate the problem as a rank-minimization
task in the spectral norm, and propose a framework to apply Adamyan-Arov-Krein
(AAK) theory to the approximation problem. This approach has already been
successfully applied to the case of WFAs and language modelling black boxes
over one-letter alphabets citep{AAK-WFA,AAK-RNN}. Extending the result to
multi-letter alphabets requires solving the following two steps. First, we need
to reformulate the approximation problem in terms of noncommutative Hankel
operators and noncommutative functions, in order to apply results from
multivariable operator theory. Secondly, to obtain the optimal approximation we
need a version of noncommutative AAK theory that is constructive. In this
paper, we successfully tackle the first step, while the second challenge
remains open. | | Source: | arXiv, 2206.00172 | | Services: | Forum | Review | PDF | Favorites | |
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