  Heavy tails is a common feature of filtering distributions that results from
the nonlinear dynamical and observation processes as well as the uncertainty
from physical sensors. In these settings, the Kalman filter and its ensemble
version - the ensemble Kalman filter (EnKF) - that have been designed under
Gaussian assumptions result in degraded performance. t-distributions are a
parametric family of distributions whose tail-heaviness is modulated by a
degree of freedom $\nu$. Interestingly, Cauchy and Gaussian distributions
correspond to the extreme cases of a t-distribution for $\nu = 1$ and $\nu =
\infty$, respectively. Leveraging tools from measure transport (Spantini et
al., SIAM Review, 2022), we present a generalization of the EnKF whose
prior-to-posterior update leads to exact inference for t-distributions. We
demonstrate that this filter is less sensitive to outlying synthetic
observations generated by the observation model for small $\nu$. Moreover, it
recovers the Kalman filter for $\nu = \infty$. For nonlinear state-space models
with heavy-tailed noise, we propose an algorithm to estimate the
prior-to-posterior update from samples of joint forecast distribution of the
states and observations. We rely on a regularized expectation-maximization (EM)
algorithm to estimate the mean, scale matrix, and degree of freedom of
heavy-tailed \textit{t}-distributions from limited samples (Finegold and Drton,
arXiv preprint, 2014). Leveraging the conditional independence of the joint
forecast distribution, we regularize the scale matrix with an $l1$
sparsity-promoting penalization of the log-likelihood at each iteration of the
EM algorithm. By sequentially estimating the degree of freedom at each analysis
step, our filter can adapt its prior-to-posterior update to the tail-heaviness
of the data. We demonstrate the benefits of this new ensemble filter on
challenging filtering problems.
