Jonas Kahn and Pierre Weiss (pierre.armand.weiss@gmail.com)October 9, 2015

Internship proposal

Variational analysis on measure spaces.

Context

The internship will take place either at the Mathematics Institute of Toulouse (IMT - http://www.math.univ-toulouse.fr/) or the Institut des Technologies Avancées du Vivant(ITAV - http://www.itav-recherche.fr/index.php/fr/) an interdisciplinary laboratorycomposed of researchers from biology, chemistry, optics and mathematics.

The advisors work in close relationship with researchers from dierent disciplines such ascancer biology, physics (optics, electromagnetism, Magnetic Resonance Imaging), mechan-ics (uid and structure) and signal/image processing. It aims at solving real-life problemswith advanced mathematics.

This internship is intended to lead to a PhD thesis.

Topic

We recently proposed a projection algorithm on measures sets [1]. A typical problemthat can be tackled with this approach is image halftoning (i.e. approximate an imagewith a nite sum of Diracs). This type of problem is a great challenge in mathematics(7th problem of Smale), computation (nd approximate solutions in decent times) andapplications (biology, nance, image processing,...).

In a nutshell, we proposed an algorithm to solve the following generic problem:

minπ∈M

‖h ? (π − µ)‖22, (1)

where M is a set of probability measures on a bounded set Ω ⊂ Rd, h is a continuousfunction on Ω and µ is a target probability measure. The numerical resolution of (1) isbased on a dynamical system. A practical example of this system is shown on Figure 1. Onthis example, we project the image of a lion on the set of push-forward measures associatedto a subset of parameterized curves.

The distance ‖h?(π−µ)‖22 is quadratic and hence interesting from a computational point ofview. In many applications, the transportation (or Wasserstein) distance is more natural.Recently, new computational tools [2] based on entropic regularization have allowed toevaluate this distance eciently. The main goal of the internship is to develop a theoryand a numerical approach to minimize:

minπ∈M

Wp(π, µ), (2)

where Wp(·, ·) is the p-Wasserstein distance [3].

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Figure 1: Dynamics of a gradient descent: whatever the initial point, the sequence ofiterates seems to weakly converge to the target image.

This topic is linked both to deep mathematical theories and a wide array of applications.

Depending on the candidate's interest, this internship will lead to works of a theoretical na-

ture (mean eld limits), computational nature (multipole methods, parallel programming),application nature (system biology, computer graphics, compressive sampling).

Practical aspects

This internship will take place either at ITAV or at IMT (in Toulouse, France), dependingon the candidate's interest. It will be co-supervised by Jonas Kahn (CR Toulouse) andPierre Weiss (CR Toulouse).

Ideally, this internship will be continued by a PhD thesis. The funding will be deliv-ered by Labex CIMI (http://www.cimi.univ-toulouse.fr/en) or the doctoral school ofmathematics from Toulouse.

Do not hesitate to contact us for more information.

Bibliography

[1] Nicolas Chauert, Philippe Ciuciu, Jonas Kahn, and Pierre Weiss. A projection algo-rithm on measures sets. arXiv preprint arXiv:1509.00229, 2015.

[2] Marco Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport. InAdvances in Neural Information Processing Systems, pages 22922300, 2013.

[3] Cédric Villani. Optimal transport: old and new, volume 338. Springer Science &Business Media, 2008.

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