  Feedback particle filter (FPF) is a numerical algorithm to approximate the
solution of the nonlinear filtering problem in continuous-time settings. In any
numerical implementation of the FPF algorithm, the main challenge is to
numerically approximate the so-called gain function. A numerical algorithm for
gain function approximation is the subject of this paper. The exact gain
function is the solution of a Poisson equation involving a probability-weighted
Laplacian $\Delta_\rho$. The numerical problem is to approximate this solution
using {\em only} finitely many particles sampled from the probability
distribution $\rho$. A diffusion map-based algorithm was proposed by the
authors in a prior work to solve this problem. The algorithm is named as such
because it involves, as an intermediate step, a diffusion map approximation of
the exact semigroup $e^{\Delta_\rho}$. The original contribution of this paper
is to carry out a rigorous error analysis of the diffusion map-based algorithm.
The error is shown to include two components: bias and variance. The bias
results from the diffusion map approximation of the exact semigroup. The
variance arises because of finite sample size. Scalings and upper bounds are
derived for bias and variance. These bounds are then illustrated with numerical
experiments that serve to emphasize the effects of problem dimension and sample
size. The proposed algorithm is applied to two filtering examples and
comparisons provided with the sequential importance resampling (SIR) particle
filter.
