As the problem of minimizing functionals on the Wasserstein space encompasses
many applications in machine learning, different optimization algorithms on
$\mathbb{R}^d$ have received their counterpart analog on the Wasserstein space.
We focus here on lifting two explicit algorithms: mirror descent and
preconditioned gradient descent. These algorithms have been introduced to
better capture the geometry of the function to minimize and are provably
convergent under appropriate (namely relative) smoothness and convexity
conditions. Adapting these notions to the Wasserstein space, we prove
guarantees of convergence of some Wasserstein-gradient-based discrete-time
schemes for new pairings of objective functionals and regularizers. The
difficulty here is to carefully select along which curves the functionals
should be smooth and convex. We illustrate the advantages of adapting the
geometry induced by the regularizer on ill-conditioned optimization tasks, and
showcase the improvement of choosing different discrepancies and geometries in
a computational biology task of aligning single-cells.

通过在 Wasserstein 空间上引入测度梯度的离散时间方案，证明了一些目标函数和正则化器的收敛性，同时展示了在计算生物学中选择不同的差异度和几何结构的优势。