I leafed through Quirk et al's Comprehensive Grammar, and it seems that this might be an example of what they call subjectless supplementive clause.
They provide the following ambiguous example:
I saw Pam going home.
If we read it as "I saw [Pam going home]", then going home supplements Pam. "Pam" is the "overt subject" of "going home".
If we read it as "Going home, I saw Pam", then going home is a "subjectless supplementive clause" with the pronoun I as its "implied subject".
I don't know how other grammars treat such constructions, and "supplementive clause" is a rarely-used term.
Quirk et al., "A Comprehensive Grammar of the English Language", 15.62, 'Supplementive Clauses in Final Position'.
Their definition of a supplemental clause:
Supplemental clauses: adverbial participle and verbless clauses without a subordinator.
P.S. In CGEL, such constructions are discussed in Chapter 14, Part 9: "Non-finite clauses as modifiers and supplements"