Titre: Le transport optimal numérique et ses applications

Abstract: Optimal  transport (OT) has become  a fundamental mathematical
tool  at   the  interface   between  calculus  of   variations,  partial
differential  equations  and  probability.  It took  however  much  more
time for  this notion  to become  mainstream in  numerical applications.
This  situation is  in large  part due  to the  high computational  cost
of  the underlying  optimization  problems. There  is  however a  recent
wave  of  activity  on  the  use of  OT-related  methods  in  fields  as
diverse as  computer vision,  computer graphics,  statistical inference,
machine learning  and image processing. In  this talk, I will  review an
emerging class of numerical approaches for the approximate resolution of
OT-based optimization  problems. These methods  make use of  an entropic
regularization of the  functionals to be minimized, in  order to unleash
the  power  of  optimization  algorithms  based  on  Bregman-divergences
geometry (see [2] for a  theoretical analysis and a litterature review).
This results  in fast, simple  and highly parallelizable  algorithms, in
sharp contrast with traditional solvers  based on the geometry of linear
programming.  For instance,  they allow  for the  first time  to compute
barycenters  (according to  OT distances)  of probability  distributions
discretized on computational  2-D and 3-D grids with  millions of points
[1]. This offers a new perspective  for the application of OT in machine
learning  (to perform  clustering or  classification of  bag-of-features
data representations)  and imaging  sciences (to perform  color transfer
or  shape  and  texture  morphing [6]).  These  algorithms  also  enable
the  computation of  gradient  flows for  the OT  metric,  and can  thus
for  instance  be applied  to  simulate  crowd motions  with  congestion
constraints [4]. We will also discus various extensions of classical OT,
such as  handling unbalanced  transportation between  arbitrary positive
measures [3] (the so-called Hellinger-Kantorovich/Wasserstein-Fisher-Rao
problem) and the computation of  OT between different metric spaces (the
so-called Gromov-Wasserstein problem) [7, 5].

References:
[1] J-D. Benamou,  G. Carlier, M. Cuturi, L. Nenna,  G. Peyré. Iterative
Bregman  Projections  for   Regularized  Transportation  Problems.  SIAM
Journal on Scientific Computing, 37(2), pp. A1111-A1138, 2015.
[2]  G.  Carlier, V.  Duval,  G.  Peyré,  B. Schmitzer.  Convergence  of
Entropic Schemes for Optimal Transport  and Gradient Flows. to appear in
SIAM Journal on Mathematical Analysis, 2017.
[3] L. Chizat, G. Peyré,  B. Schmitzer, F-X. Vialard. Scaling Algorithms
for Unbalanced Transport Problems. Preprint Arxiv:1607.05816, 2016.
[4] G. Peyré. Entropic Approximation of Wasserstein Gradient Flows. SIAM
Journal on Imaging Sciences, 8(4), pp. 2323-2351, 2015.
[5] G.  Peyré, M.  Cuturi, J.  Solomon. Gromov-Wasserstein  Averaging of
Kernel and Distance Matrices. In Proc. ICML¿16, pp. 2664-2672, 2016.
[6] J. Solomon, F. de Goes, G. Peyré, M. Cuturi, A. Butscher, A. Nguyen,
T. Du, L. Guibas. Convolutional Wasserstein Distances: Efficient Optimal
Transportation on Geometric Domains. ACM Transactions on Graphics (Proc.
SIGGRAPH 2015), 34(4), pp. 66:1-66:11, 2015.
[7] J. Solomon, G. Peyré, V.  Kim, S. Sra. Entropic Metric Alignment for
Correspondence Problems.  ACM Transactions  on Graphics  (Proc. SIGGRAPH
2016), 35(4), pp. 72:1-72:13, 2016.