arXiv subjects

Giacomo Cristinelli

Publications and source records attributed to Giacomo Cristinelli.

3 recordsLinked to original sources

arXiv2026

Linear convergence of a one-cut conditional gradient method for total variation regularization

We introduce a fully-corrective generalized conditional gradient method for convex minimization problems involving total variation regularization on multidimensional domains. It relies on alternatively updating an active set of subsets of the spatial domain and an iterate given by a conic combination of the associated characteristic functions. Different to previous approaches in the same spirit, the computation of a new candidate set only requires the solution of one prescribed mean curvature problem, instead of the resolution of a fractional minimization task analogous to finding a generalized Cheeger set. After discretization, the former can be realized by a single run of a graph cut algorithm, leading to a significant speedup in practice. We prove the global sublinear convergence of the resulting method, under mild assumptions, and its asymptotic linear convergence in a more restrictive two-dimensional setting which uses results of stability of surfaces of prescribed mean curvature under perturbations of the curvature. Finally, we numerically demonstrate this convergence behavior in some model PDE-constrained minimization problems.

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arXiv2026

Towards sparse optimization over convex loops: Equivalence of Square Root Velocity distance and Wasserstein-Fisher-Rao

The Wasserstein-Fisher-Rao (WFR) distance on $S^{2}$ has recently been shown to coincide with a classical elastic distance between $S^{2}$-immersions in the theory of Riemannian shape analysis. While this correspondence holds in dimension $2$, the analogous statement fails in general on $S^{1}$ and, in the case of convex curves, it cannot be derived from existing two-dimensional arguments. In this paper, we establish that for convex absolutely continuous immersions of $S^{1}$ in the plane, the shape distance induced by the square root velocity transformation (SRVT) is indeed equivalent to the WFR distance acting on their associated length measures. The proof exploits a monotonicity principle for optimal transport on the universal cover of the circle, which in turn guarantees the existence of an optimal reparametrization achieving the SRVT infimum and enables a one-dimensional unbalanced optimal transport reformulation. Motivated by this equivalence, we further investigate the role of sparsity in shape optimization problems formulated in terms of length measures and regularized by the WFR distance. We study linear optimization over the corresponding balls, for which we prove a finiteness result when the reference measure is discrete, and propose a convex, positively one-homogeneous regularizer suitable for conditional gradient algorithms.

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arXiv2024

Conditional gradients for total variation regularization with PDE constraints: a graph cuts approach

Total variation regularization has proven to be a valuable tool in the context of optimal control of differential equations. This is particularly attributed to the observation that TV-penalties often favor piecewise constant minimizers with well-behaved jumpsets. On the downside, their intricate properties significantly complicate every aspect of their analysis, from the derivation of first-order optimality conditions to their discrete approximation and the choice of a suitable solution algorithm. In this paper, we investigate a general class of minimization problems with TV-regularization, comprising both continuous and discretized control spaces, from a convex geometry perspective. This leads to a variety of novel theoretical insights on minimization problems with total variation regularization as well as tools for their practical realization. First, by studying the extremal points of the respective total variation unit balls, we enable their efficient solution by geometry exploiting algorithms, e.g. fully-corrective generalized conditional gradient methods. We give a detailed account on the practical realization of such a method for piecewise constant finite element approximations of the control on triangulations of the spatial domain. Second, in the same setting and for suitable sequences of uniformly refined meshes, it is shown that minimizers to discretized PDE-constrained optimal control problems approximate solutions to a continuous limit problem involving an anisotropic total variation reflecting the fine-scale geometry of the mesh.

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