  Many problems in machine learning can be formulated as optimizing a convex
functional over a vector space of measures. This paper studies the convergence
of the mirror descent algorithm in this infinite-dimensional setting. Defining
Bregman divergences through directional derivatives, we derive the convergence
of the scheme for relatively smooth and convex pairs of functionals. Such
assumptions allow to handle non-smooth functionals such as the
Kullback--Leibler (KL) divergence. Applying our result to joint distributions
and KL, we show that Sinkhorn's primal iterations for entropic optimal
transport in the continuous setting correspond to a mirror descent, and we
obtain a new proof of its (sub)linear convergence. We also show that
Expectation Maximization (EM) can always formally be written as a mirror
descent. When optimizing only on the latent distribution while fixing the
mixtures parameters -- which corresponds to the Richardson--Lucy deconvolution
scheme in signal processing -- we derive sublinear rates of convergence.
