  There has been a recent surge of interest in coupling methods for Markov
chain Monte Carlo algorithms: they facilitate convergence quantification and
unbiased estimation, while exploiting embarrassingly parallel computing
capabilities. Motivated by these, we consider the design and analysis of
couplings of the random walk Metropolis algorithm which scale well with the
dimension of the target measure. Methodologically, we introduce a low-rank
modification of the synchronous coupling that is provably optimally contractive
in standard high-dimensional asymptotic regimes. We expose a shortcoming of the
reflection coupling, the status quo at time of writing, and we propose a
modification which mitigates the issue. Our analysis bridges the gap to the
optimal scaling literature and builds a framework of asymptotic optimality
which may be of independent interest. We illustrate the applicability of our
proposed couplings, and the potential for extending our ideas, with various
numerical experiments.
