  Manifold Markov chain Monte Carlo algorithms have been introduced to sample
more effectively from challenging target densities exhibiting multiple modes or
strong correlations. Such algorithms exploit the local geometry of the
parameter space, thus enabling chains to achieve a faster convergence rate when
measured in number of steps. However, acquiring local geometric information can
often increase computational complexity per step to the extent that sampling
from high-dimensional targets becomes inefficient in terms of total
computational time. This paper analyzes the computational complexity of
manifold Langevin Monte Carlo and proposes a geometric adaptive Monte Carlo
sampler aimed at balancing the benefits of exploiting local geometry with
computational cost to achieve a high effective sample size for a given
computational cost. The suggested sampler is a discrete-time stochastic process
in random environment. The random environment allows to switch between local
geometric and adaptive proposal kernels with the help of a schedule. An
exponential schedule is put forward that enables more frequent use of geometric
information in early transient phases of the chain, while saving computational
time in late stationary phases. The average complexity can be manually set
depending on the need for geometric exploitation posed by the underlying model.
