Non-asymptotic convergence bounds for Wasserstein approximation using point clouds
This addresses the challenge of generating discrete data from continuous distributions in machine learning and inverse problems, offering theoretical guarantees for an incremental improvement in optimization methods.
The paper tackles the non-convex problem of approximating a model probability distribution by minimizing the Wasserstein distance using a discrete set of points, and provides explicit upper bounds for the convergence speed of a Lloyd-type algorithm starting from well-separated points.
Several issues in machine learning and inverse problems require to generate discrete data, as if sampled from a model probability distribution. A common way to do so relies on the construction of a uniform probability distribution over a set of $N$ points which minimizes the Wasserstein distance to the model distribution. This minimization problem, where the unknowns are the positions of the atoms, is non-convex. Yet, in most cases, a suitably adjusted version of Lloyd's algorithm -- in which Voronoi cells are replaced by Power cells -- leads to configurations with small Wasserstein error. This is surprising because, again, of the non-convex nature of the problem, as well as the existence of spurious critical points. We provide explicit upper bounds for the convergence speed of this Lloyd-type algorithm, starting from a cloud of points sufficiently far from each other. This already works after one step of the iteration procedure, and similar bounds can be deduced, for the corresponding gradient descent. These bounds naturally lead to a modified Poliak-Lojasiewicz inequality for the Wasserstein distance cost, with an error term depending on the distances between Dirac masses in the discrete distribution.