Sparse optimization on measures with over-parameterized gradient descent

Minimizing a convex function of measure with sparsity-inducing penalty is typical problem arising, e.g., in sparse spikes deconvolution or two-layer neural networks training. We show that this can be solved by discretizing the and running non-convex gradient descent on positions weights particles. For measures d-dimensional manifold under some non-degeneracy assumptions, leads to global optimization algorithm complexity scaling as $$\log (1/\epsilon )$$ desired accuracy $$\epsilon $$ , instead ^{-d}$$ for methods. The key theoretical tools are local convergence analysis Wasserstein space an perturbed mirror measures. Our bounds involve quantities exponential d which unavoidable our assumptions.