Extremal points and sparse optimization for generalized Kantorovich-Rubinstein norms

TL;DR

This work addresses inverse-type problems regularized by a generalized Kantorovich-Rubinstein norm to model unbalanced transport between signed Radon measures. The authors derive a precise extremal-point characterization of the KR_p^{\alpha,\beta} unit ball, showing extremals are Dirac masses and rescaled dipoles (δ_x - δ_y)/(β+|x-y|^p) subject to 0<|x-y|^p<2α−β, which enables sharp first-order optimality conditions. Building on this, they develop an accelerated generalized conditional gradient (AGCG) method tailored to infinite-dimensional settings and prove sublinear convergence, with linear convergence under stronger assumptions. Numerical experiments illustrate the method's ability to promote transport near a reference measure while allowing mass creation elsewhere, confirming the theoretical findings. The framework offers a scalable, provably convergent approach to KR-based regularization with potential applications in sparse optimal design and unbalanced transport in imaging and inverse problems.

Abstract

A precise characterization of the extremal points of sublevel sets of nonsmooth penalties provides both detailed information about minimizers, and optimality conditions in general classes of minimization problems involving them. Moreover, it enables the application of accelerated generalized conditional gradient methods for their efficient solution. In this manuscript, this program is adapted to the minimization of a smooth convex fidelity term which is augmented with an unbalanced transport regularization term given in the form of a generalized Kantorovich-Rubinstein norm for Radon measures. More precisely, we show that the extremal points associated to the latter are given by all Dirac delta functionals supported in the spatial domain as well as certain dipoles, i.e., pairs of Diracs with the same mass but with different signs. Subsequently, this characterization is used to derive precise first-order optimality conditions as well as an efficient solution algorithm for which linear convergence is proved under natural assumptions. This behaviour is also reflected in numerical examples for a model problem.

Figures (6)

• Figure 1: Reconstruction of a minimizer of \ref{['def:problemnum2']} by the application of Algorithm \ref{['alg:accgcg']}, with the crosses depicting $K \mu_{\bar{k}}$.
• Figure 2: Plot of the approximate residual $\hat{r}(\mu_k)$ in logarithmic scale.
• Figure 3: On the left the rescaled dual variable $\bar{q}(z)/\alpha$ computed in the output of Algorithm \ref{['alg:accgcg']}. On the right $\bar{q}(z)/\alpha$ and its second derivative. The vertical dotted lines, both on the left and on the right figure, represent the location of the Dirac deltas appearing in the output of Algorithm \ref{['alg:accgcg']}.
• Figure 4: On the left the plot of $\Psi_{\bar{q}}$. The dotted lines are drawn for each dipole in the output measure of the algorithm. On the right, to demonstrate assumptions $(\mathbf{B3})$ and $(\mathbf{B4})$, a table with the location of each dipole and the corresponding $\operatorname{det}\left( \nabla^2 \Psi_{\bar{q}}\right)$, and the singular values of the matrix constructed from $\left\{K\left(\delta_{\bar{z}_i}\right)\right\}^{\bar{N}_1}_{i=1}$ and $\left\{K\left(\mathcal{D}_\beta(\bar{x}_j, \bar{y}_j)\right)\right\}^{\bar{N}_2}_{j=1}$.
• Figure 5: Reconstruction of a minimizer of \ref{['def:problemnum2']} for infinite dimensional measurements by the application of Algorithm \ref{['alg:accgcg']}, with a continuous plot showing $K \mu_{\bar{k}}$.

Theorems & Definitions (69)

• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Theorem 2.5
• Proposition 2.6