  We present approximate algorithms for performing smoothing in a class of
high-dimensional state-space models via sequential Monte Carlo methods
("particle filters"). In high dimensions, a prohibitively large number of Monte
Carlo samples ("particles") -- growing exponentially in the dimension of the
state space -- is usually required to obtain a useful smoother. Using blocking
strategies as in Rebeschini and Van Handel (2015) (and earlier pioneering work
on blocking), we exploit the spatial ergodicity properties of the model to
circumvent this curse of dimensionality. We thus obtain approximate smoothers
that can be computed recursively in time and in parallel in space. First, we
show that the bias of our blocked smoother is bounded uniformly in the time
horizon and in the model dimension. We then approximate the blocked smoother
with particles and derive the asymptotic variance of idealised versions of our
blocked particle smoother to show that variance is no longer adversely effected
by the dimension of the model. Finally, we employ our method to successfully
perform maximum-likelihood estimation via stochastic gradient-ascent and
stochastic expectation--maximisation algorithms in a 100-dimensional
state-space model.
