  As part of the effort to understand implicit bias of gradient descent in
overparametrized models, several results have shown how the training trajectory
on the overparametrized model can be understood as mirror descent on a
different objective. The main result here is a characterization of this
phenomenon under a notion termed commuting parametrization, which encompasses
all the previous results in this setting. It is shown that gradient flow with
any commuting parametrization is equivalent to continuous mirror descent with a
related Legendre function. Conversely, continuous mirror descent with any
Legendre function can be viewed as gradient flow with a related commuting
parametrization. The latter result relies upon Nash's embedding theorem.
