Submitted Paper

Inserted: 31 mar 2026
Last Updated: 31 mar 2026

Pages: 22
Year: 2026

Abstract:

One of the most popular approaches for solving total variation-regularized optimization prob- lems in the space of measures are Particle Gradient Flows (PGFs). These restrict the problem to linear combinations of Dirac deltas and then perform a Euclidean gradient flow in the weights and positions, significantly reducing the computational cost while still decreasing the energy. In this work, we generalize PGFs to convex optimization problems in arbitrary Banach spaces, which we call Atomic Gradient Flows (AGFs). To this end, the crucial ingredient turns out to be the right notion of particles, or atoms, chosen here as the extremal points of the unit ball of the regularizer. This choice is motivated by the Krein–Milman theorem, which ensures that minimizers can be approximated by linear combi- nations of extremal points or, as we call them, sparse representations. We investigate metric gradient flows of the optimization problem when restricted to such sparse representations, for which we define a suitable discretized functional that we show to be to be consistent with the original problem via the means of Γ-convergence. We prove that the resulting evolution of the latter is well-defined using a minimizing movement scheme, and we establish conditions ensur- ing λ-convexity and uniqueness of the flow. These conditions crucially depend on the geometric properties of the set of extremal points as a metric space. Then, using Choquet’s theorem, we lift the problem into the Wasserstein space on weights and extremal points, and consider Wasserstein gradient flows in this lifted setting. As observed for PGFs, this lifted perspective is essential for understanding stability and convergence properties of AGFs. Our main result is that the lifting of the AGF evolution is again a metric gradi- ent flow in the Wasserstein space, verifying the consistency of the approach with respect to a Wasserstein-type dynamic. Finally, we illustrate the applicability of AGFs to several relevant infinite-dimensional problems, including optimization of functions of bounded variation and curves of measures regularized by Optimal Transport-type penalties.