Gradient flows

Santambrogio, Filippo

doi:10.1007/978-3-319-20828-2_8

Cited by 70 publications

(129 citation statements)

References 40 publications

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“…In this section, we provide simple examples of submodular functions. We will use as running examples throughout the paper: submodular set-functions defined on {0, 1} n (to show that our new results directly extend the ones for set-functions), modular functions (because they provide a very simple example of the concepts we introduce), and functions that are sums of terms ϕ ij (x i − x j ) where ϕ ij is convex (for the link with Wasserstein distances between probability measures [54,46]).…”

Section: Examplesmentioning

confidence: 99%

“…In order to prove this equivalence, there are three potential proofs: (a) based on convex duality, by exhibiting primal and dual candidates (this is exactly the traditional proof for submodular setfunctions [36,13], which is cumbersome for continuous domains, but which we will follow in Section 4 for finite sets); (b) based on Hardy-Littlewood's inequalities [35]; or (c) using properties of optimal transport. We consider the third approach (based on the two-marginal proof of [46]) and use four steps:…”

Section: Equivalence Between the Two Extensions Through One-dimensionmentioning

confidence: 99%

“…In this paper, we assume basic knowledge of convex analysis (see, e.g., [6,5]), while the relevant notions of submodular analysis (see, e.g., [19,1]) and optimal transport (see, e.g., [54,46]) will be rederived as needed.…”

Section: Introductionmentioning

confidence: 99%

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Submodular functions: from discrete to continuous domains

Bach

2018

Math. Program.

134

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Submodular set-functions have many applications in combinatorial optimization, as they can be minimized and approximately maximized in polynomial time. A key element in many of the algorithms and analyses is the possibility of extending the submodular set-function to a convex function, which opens up tools from convex optimization. Submodularity goes beyond set-functions and has naturally been considered for problems with multiple labels or for functions defined on continuous domains, where it corresponds essentially to cross second-derivatives being nonpositive. In this paper, we show that most results relating submodularity and convexity for set-functions can be extended to all submodular functions. In particular, (a) we naturally define a continuous extension in a set of probability measures, (b) show that the extension is convex if and only if the original function is submodular, (c) prove that the problem of minimizing a submodular function is equivalent to a typically non-smooth convex optimization problem, and (d) propose another convex optimization problem with better computational properties (e.g., a smooth dual problem). Most of these extensions from the set-function situation are obtained by drawing links with the theory of multi-marginal optimal transport, which provides also a new interpretation of existing results for set-functions. We then provide practical algorithms to minimize generic submodular functions on discrete domains, with associated convergence rates.

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Section: Examplesmentioning

confidence: 99%

Section: Equivalence Between the Two Extensions Through One-dimensionmentioning

confidence: 99%

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Submodular functions: from discrete to continuous domains

Bach

2018

Math. Program.

134

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“…Optimal potentials in the problem above are called Kantorovich potentials between ρ 0 and ρ 1 , they can be taken semi-concave and their existence is well-known (see [17,18,2,16]). If ρ 0 ∈ P ac 2 (Ω) a celebrated result of Brenier [3] states that there is a unique optimal plan γ between ρ 0 and ρ 1 and it is induced by a transport map T i.e.…”

Section: Wasserstein Spacementioning

confidence: 99%

A splitting method for nonlinear diffusions with nonlocal, nonpotential drifts

Carlier

Laborde

2017

Nonlinear Analysis: Theory, Methods & Applications

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We prove an existence result for nonlinear diffusion equations in the presence of a nonlocal densitydependent drift which is not necessarily potential. The proof is constructive and based on the Helmholtz decomposition of the drift and a splitting scheme. The splitting scheme combines transport steps by the divergence-free part of the drift and semi-implicit minimization steps à la Jordan-Kinderlherer Otto to deal with the potential part.

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“…One may ask whether the relaxed solution in the extended space has certain regularity properties, for example whether it is the graph of a (sufficiently regular) map and thus can be considered a solution to the original ("unlifted") problem. In the case of optimal transport, such regularity theory can be guaranteed under some assumptions [63,52]. Establishing existence and regularity for problems in which the cost additionally depends on the Jacobian (for example minimal surface problems) has been a driving factor in the development of geometric measure theory, see [44] for an introduction.…”

Section: Introductionmentioning

confidence: 99%

Lifting Vectorial Variational Problems: A Natural Formulation Based on Geometric Measure Theory and Discrete Exterior Calculus

Möllenhoff

Cremers²

2019

2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)

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Numerous tasks in imaging and vision can be formulated as variational problems over vector-valued maps. We approach the relaxation and convexification of such vectorial variational problems via a lifting to the space of currents. To that end, we recall that functionals with polyconvex Lagrangians can be reparametrized as convex onehomogeneous functionals on the graph of the function. This leads to an equivalent shape optimization problem over oriented surfaces in the product space of domain and codomain. A convex formulation is then obtained by relaxing the search space from oriented surfaces to more general currents. We propose a discretization of the resulting infinite-dimensional optimization problem using Whitney forms, which also generalizes recent "sublabel-accurate" multilabeling approaches. 1 We refer the interested reader to the historical remarks in L. C. Young's book on the calculus of variations [69, pp. 122-123].

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Gradient flows

Cited by 70 publications

References 40 publications

Submodular functions: from discrete to continuous domains

Submodular functions: from discrete to continuous domains

A splitting method for nonlinear diffusions with nonlocal, nonpotential drifts

Lifting Vectorial Variational Problems: A Natural Formulation Based on Geometric Measure Theory and Discrete Exterior Calculus

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