  We consider the problem of high-dimensional filtering of state-space models
(SSMs) at discrete times. This problem is particularly challenging as
analytical solutions are typically not available and many numerical
approximation methods can have a cost that scales exponentially with the
dimension of the hidden state. Inspired by lag-approximation methods for the
smoothing problem, we introduce a lagged approximation of the smoothing
distribution that is necessarily biased. For certain classes of SSMs,
particularly those that forget the initial condition exponentially fast in
time, the bias of our approximation is shown to be uniformly controlled in the
dimension and exponentially small in time. We develop a sequential Monte Carlo
(SMC) method to recursively estimate expectations with respect to our biased
filtering distributions. Moreover, we prove for a class of class of SSMs that
can contain dependencies amongst coordinates that as the dimension
$d\rightarrow\infty$ the cost to achieve a stable mean square error in
estimation, for classes of expectations, is of $\mathcal{O}(Nd^2)$ per-unit
time, where $N$ is the number of simulated samples in the SMC algorithm. Our
methodology is implemented on several challenging high-dimensional examples
including the conservative shallow-water model.
