  Unbalanced optimal transport (UOT) extends optimal transport (OT) to take
into account mass variations to compare distributions. This is crucial to make
OT successful in ML applications, making it robust to data normalization and
outliers. The baseline algorithm is Sinkhorn, but its convergence speed might
be significantly slower for UOT than for OT. In this work, we identify the
cause for this deficiency, namely the lack of a global normalization of the
iterates, which equivalently corresponds to a translation of the dual OT
potentials. Our first contribution leverages this idea to develop a provably
accelerated Sinkhorn algorithm (coined 'translation invariant Sinkhorn') for
UOT, bridging the computational gap with OT. Our second contribution focusses
on 1-D UOT and proposes a Frank-Wolfe solver applied to this translation
invariant formulation. The linear oracle of each steps amounts to solving a 1-D
OT problems, resulting in a linear time complexity per iteration. Our last
contribution extends this method to the computation of UOT barycenter of 1-D
measures. Numerical simulations showcase the convergence speed improvement
brought by these three approaches.
