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BEGIN:VEVENT
SUMMARY:Luca Tamanini (Universite Paris Dauphine)
DTSTART:20210621T150000Z
DTEND:20210621T154000Z
DTSTAMP:20260404T042017Z
UID:BIRS-21w5120/1
DESCRIPTION:Title: Small-time asymptotics of the metric Schrödinger problem\
nby Luca Tamanini (Universite Paris Dauphine) as part of BIRS workshop: En
tropic Regularization of Optimal Transport and Applications\n\n\nAbstract\
nThe Schrödinger problem as "noised" optimal transport is by now a well-e
stablished interpretation. From this perspective several natural questions
stem\, as for instance the convergence rate as the noise parameter vanish
es of many quantities: optimal value\, Schrödinger bridges and potentials
... As for the optimal value\, after the works of Erbar-Maas-Renger and Pa
l a first-order Taylor expansion is available. First aim of this talk is
to improve this result in a twofold sense: from the first to the second or
der and from the Euclidean to the Riemannian setting (and actually far bey
ond). From the proof it will be clear that the statement is in fact a part
icular instance of a more general result. For this reason\, in the second
part of the talk we introduce a large class of dynamical variational probl
ems\, extending far beyond the classical Schrödinger problem\, and for th
em we prove $\\Gamma$-convergence towards the geodesic problem and a suita
ble generalization of the second-order Taylor expansion. (based on joint
works with G. Conforti\, L. Monsaingeon and D. Vorotnikov)\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5120/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luca Nenna (Université Paris-Saclay)
DTSTART:20210621T155000Z
DTEND:20210621T163000Z
DTSTAMP:20260404T042017Z
UID:BIRS-21w5120/2
DESCRIPTION:Title: From Schrödinger to Lasry-Lions\nby Luca Nenna (Universit
é Paris-Saclay) as part of BIRS workshop: Entropic Regularization of Opti
mal Transport and Applications\n\n\nAbstract\nThe minimization of a relati
ve entropy (with respect to the Wiener measure) is a very old problem whic
h dates back to Schrödinger. C. Léonard has established strong connectio
ns and analogies between this problem and the Monge-Kantorovich problem wi
th quadratic cost (namely the standard Optimal Transport problem). In part
icular\, the entropic interpolation leads to a system of PDEs which presen
t strong analogies with the Mean Field Game system with a quadratic Hamilt
onian. In this talk\, we will explain how such systems can indeed be obtai
ned by minimization of a relative entropy at the level of measures on path
s with an additional term involving the marginal in time. If time permitte
d we will also show the multi-population case and its connection with some
equations in Quantum Mechanics.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5120/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Young-Heon Kim (University of British Columbia)
DTSTART:20210621T164000Z
DTEND:20210621T172000Z
DTSTAMP:20260404T042017Z
UID:BIRS-21w5120/3
DESCRIPTION:Title: Optimal transport in Brownian motion stopping\nby Young-He
on Kim (University of British Columbia) as part of BIRS workshop: Entropic
Regularization of Optimal Transport and Applications\n\n\nAbstract\nWe co
nsider an optimal transport problem arising from stopping the Brownian mot
ion from a given distribution to get a fixed or free target distribution\;
the fixed target case is often called the optimal Skorokhod embedding pro
blem in the literature\, a popular topic in math finance pioneered by many
people. Our focus is on the case of general dimensions\, which has not be
en well understood. We explain that under certain natural assumptions on t
he transportation cost\, the optimal stopping time is given by the hitting
time to a barrier\, which is determined by the solution to the dual optim
ization problem. In the free target case\, the problem is related to the S
tefan problem\, that is\, a free boundary problem for the heat equation. W
e obtain analytical information on the optimal solutions\, including certa
in BV estimates. The fixed target case is mainly from the joint work with
Nassif Ghoussoub and Aaron Palmer at UBC\, while the free target case is t
he recent joint work (in-progress) with Inwon Kim at UCLA.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5120/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert McCann (University of Toronto)
DTSTART:20210621T173000Z
DTEND:20210621T181000Z
DTSTAMP:20260404T042017Z
UID:BIRS-21w5120/4
DESCRIPTION:Title: Inscribed radius bounds for lower Ricci bounded metric measure
spaces with mean convex boundary\nby Robert McCann (University of Tor
onto) as part of BIRS workshop: Entropic Regularization of Optimal Transpo
rt and Applications\n\n\nAbstract\nInscribed radius bounds for lower Ricci
bounded metricConsider an essentially nonbranching metric measure space w
ith the measure contraction property of Ohta and Sturm. We prove a sharp u
pper bound on the inscribed radius of any subset whose boundary has a suit
ably signed lower bound on its generalized mean curvature. This provides a
nonsmooth analog of results dating back to Kasue (1983) and subsequent au
thors. We prove a stability statement concerning such bounds and --- in th
e Riemannian curvature-dimension (RCD) setting --- characterize the cases
of equality. This represents joint work with Annegret Burtscher\, Christia
n Ketterer and Eric Woolgar.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5120/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yongxin Chen (Georgia Tech)
DTSTART:20210621T203000Z
DTEND:20210621T211000Z
DTSTAMP:20260404T042017Z
UID:BIRS-21w5120/5
DESCRIPTION:Title: Graphical Optimal Transport and its Applications\nby Yongx
in Chen (Georgia Tech) as part of BIRS workshop: Entropic Regularization o
f Optimal Transport and Applications\n\n\nAbstract\nMulti-marginal optimal
transport (MOT) is a generalization of optimal transport theory to settin
gs with possibly more than two marginals. The computation of the solutions
to MOT problems has been a longstanding challenge. In this talk\, we intr
oduce graphical optimal transport\, a special class of MOT problems. We co
nsider MOT problems from a probabilistic graphical model perspective and p
oint out an elegant connection between the two when the underlying cost fo
r optimal transport allows a graph structure. In particular\, an entropy r
egularized MOT is equivalent to a Bayesian marginal inference problem for
probabilistic graphical models with the additional requirement that some o
f the marginal distributions are specified. This relation on the one hand
extends the optimal transport as well as the probabilistic graphical model
theories\, and on the other hand leads to fast algorithms for MOT by leve
raging the well-developed algorithms in Bayesian inference. We will cover
recent developments of graphical optimal transport in theory and algorithm
s. We will also go over several applications in aggregate filtering and me
an field games.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5120/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gabriel Peyré (CNRS and Ecole Normale Supérieure)
DTSTART:20210622T150000Z
DTEND:20210622T154000Z
DTSTAMP:20260404T042017Z
UID:BIRS-21w5120/6
DESCRIPTION:Title: Scaling Optimal Transport for High dimensional Learning\nb
y Gabriel Peyré (CNRS and Ecole Normale Supérieure) as part of BIRS work
shop: Entropic Regularization of Optimal Transport and Applications\n\n\nA
bstract\nOptimal transport (OT) has recently gained lot of interest in mac
hine learning. It is a natural tool to compare in a geometrically faithful
way probability distributions. It finds applications in both supervised l
earning (using geometric loss functions) and unsupervised learning (to per
form generative model fitting). OT is however plagued by the curse of dime
nsionality\, since it might require a number of samples which grows expone
ntially with the dimension. In this talk\, I will explain how to leverage
entropic regularization methods to define computationally efficient loss f
unctions\, approximating OT with a better sample complexity. More informat
ion and references can be found on the website of our book "Computational
Optimal Transport" https://optimaltransport.github.io/\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5120/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anna Korba (École Nationale de la Statistique et de l'Administrat
ion Économique)
DTSTART:20210622T155000Z
DTEND:20210622T163000Z
DTSTAMP:20260404T042017Z
UID:BIRS-21w5120/7
DESCRIPTION:Title: Wasserstein Proximal Gradient\nby Anna Korba (École Natio
nale de la Statistique et de l'Administration Économique) as part of BIRS
workshop: Entropic Regularization of Optimal Transport and Applications\n
\n\nAbstract\nWasserstein gradient flows are continuous time dynamics that
define curves of steepest descent to minimize an objective function over
the space of probability measures (i.e.\, the Wasserstein space). This obj
ective is typically a divergence w.r.t. a fixed target distribution. In re
cent years\, these continuous time dynamics have been used to study the co
nvergence of machine learning algorithms aiming at approximating a probabi
lity distribution. However\, the discrete-time behavior of these algorithm
s might differ from the continuous time dynamics. Besides\, although discr
etized gradient flows have been proposed in the literature\, little is kno
wn about their minimization power. In this work\, we propose a Forward Bac
kward (FB) discretization scheme that can tackle the case where the object
ive function is the sum of a smooth and a nonsmooth geodesically convex te
rms. Using techniques from convex optimization and optimal transport\, we
analyze the FB scheme as a minimization algorithm on the Wasserstein space
. More precisely\, we show under mild assumptions that the FB scheme has c
onvergence guarantees similar to the proximal gradient algorithm in Euclid
ean spaces.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5120/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonathan Niles-Weed (New York University)
DTSTART:20210622T164000Z
DTEND:20210622T172000Z
DTSTAMP:20260404T042017Z
UID:BIRS-21w5120/8
DESCRIPTION:Title: Asymptotics for semi-discrete entropic optimal transport\n
by Jonathan Niles-Weed (New York University) as part of BIRS workshop: Ent
ropic Regularization of Optimal Transport and Applications\n\nAbstract: TB
A\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5120/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zaid Harchaoui
DTSTART:20210622T173000Z
DTEND:20210622T181000Z
DTSTAMP:20260404T042017Z
UID:BIRS-21w5120/9
DESCRIPTION:Title: chrödinger Bridge with Entropic Regularization: two-sample te
st\, chaos decomposition\, and large-sample limits\nby Zaid Harchaoui
as part of BIRS workshop: Entropic Regularization of Optimal Transport and
Applications\n\n\nAbstract\nWe consider an entropy-regularized statistic
that allows one to compare two data samples drawn from possibly different
distributions. The statistic admits an expression as a weighted average of
Monge couplings with respect to a Gibbs measure. This coupling can be rel
ated to the static Schrödinger bridge given a finite number of particles.
We establish the asymptotic consistency as the sample sizes go to infinit
y of the statistic and show that the population limit is the solution of F
öllmer's entropy-regularized optimal transport. The proof technique relie
s on a chaos decomposition for paired samples. This is joint work with Lan
g Liu and Soumik Pal.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5120/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Promit Ghosal (MIT)
DTSTART:20210622T203000Z
DTEND:20210622T211000Z
DTSTAMP:20260404T042017Z
UID:BIRS-21w5120/10
DESCRIPTION:Title: Geometry and large deviation of entropic optimal transport\nby Promit Ghosal (MIT) as part of BIRS workshop: Entropic Regularizatio
n of Optimal Transport and Applications\n\n\nAbstract\nOptimal transport (
OT) theory has flourished due to its connections with geometry\, analysis\
, probability theory\, and other fields in mathematics. A renewed interest
in OT stems from applied fields such as machine learning\, image processi
ng and statistics through the introduction of entropic regularization. In
this talk\, we will discuss the convergence of entropically regularized op
timal transport. Our first result is about a large deviation principle of
the associated optimizers in entropic OT and the second result is about t
he stability of the optimizers under weak convergence. To prove these resu
lts\, we will introduce a new notion called 'cyclical invariance' of meas
ures. This is a joint work with Marcel Nutz and Espen Bernton.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5120/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Beatrice Acciaio (ETH Zürich)
DTSTART:20210623T150000Z
DTEND:20210623T154000Z
DTSTAMP:20260404T042017Z
UID:BIRS-21w5120/11
DESCRIPTION:Title: PQ-GAN: a market generation model consistent with observed sp
ot prices and derivative price\nby Beatrice Acciaio (ETH Zürich) as p
art of BIRS workshop: Entropic Regularization of Optimal Transport and App
lications\n\n\nAbstract\nOptimal transport (OT) theory has flourished due
to its connections with geometry\, analysis\, probability theory\, and oth
er fields in mathematics. A renewed interest in OT stems from applied fiel
ds such as machine learning\, image processing and statistics through the
introduction of entropic regularization. In this talk\, we will discuss th
e convergence of entropically regularized optimal transport. Our first re
sult is about a large deviation principle of the associated optimizers in
entropic OT and the second result is about the stability of the optimizers
under weak convergence. To prove these results\, we will introduce a new
notion called 'cyclical invariance' of measures. This is a joint work wi
th Marcel Nutz and Espen Bernton.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5120/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alfred Galichon (New York University)
DTSTART:20210623T155000Z
DTEND:20210623T163000Z
DTSTAMP:20260404T042017Z
UID:BIRS-21w5120/12
DESCRIPTION:Title: Dynamic Matching Problems (joint w Pauline Corblet and Jeremy
Fox)\nby Alfred Galichon (New York University) as part of BIRS worksh
op: Entropic Regularization of Optimal Transport and Applications\n\n\nAbs
tract\nFor the purposes of economics applications\, we formulate a class o
f dynamic matching problems. We investigate in particular the stationary c
ase\, and computation and estimation issues are investigated.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5120/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ting-Kam Leonard Wong (University of Toronto)
DTSTART:20210623T164000Z
DTEND:20210623T172000Z
DTSTAMP:20260404T042017Z
UID:BIRS-21w5120/13
DESCRIPTION:Title: Logarithmic divergences and statistical applications\nby
Ting-Kam Leonard Wong (University of Toronto) as part of BIRS workshop: En
tropic Regularization of Optimal Transport and Applications\n\n\nAbstract\
nWe consider the Dirichlet optimal transport which is a multiplicative ana
logue of the Wasserstein transport and is deeply connected to the Dirichle
t distribution. The log-likelihood of this distribution defines a logarith
mic divergence\, in the same way that the square loss comes from the norma
l distribution. Using this divergence\, which can be extended to a family
of generalized exponential families\, we consider statistical methodologie
s including clustering and nonlinear principal component analysis. Our app
roach extends a well-known duality between exponential family and Bregman
divergence. Joint work with Zhixu Tao\, Jiaowen Yang and Jun Zhang.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5120/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Martin Huesmann (Universität Münster)
DTSTART:20210624T150000Z
DTEND:20210624T154000Z
DTSTAMP:20260404T042017Z
UID:BIRS-21w5120/14
DESCRIPTION:Title: Fluctuations in the optimal matching problems\nby Martin
Huesmann (Universität Münster) as part of BIRS workshop: Entropic Regula
rization of Optimal Transport and Applications\n\n\nAbstract\nThe optimal
matching problem is one of the classical random optimization problems. Whi
le the asymptotic behavior of the expected cost is well understood only li
ttle is known for the asymptotic behavior of the optimal couplings - the s
olutions to the optimal matching problem. In this talk we show that at all
mesoscopic scales the displacement under the optimal coupling converges i
n suitable Sobolev spaces to a Gaussian field which can be identified as t
he curl-free part of a vector Gaussian free field. (based on joint work w
ith Michael Goldman)\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5120/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mathias Beiglböck (University of Vienna)
DTSTART:20210624T155000Z
DTEND:20210624T163000Z
DTSTAMP:20260404T042017Z
UID:BIRS-21w5120/15
DESCRIPTION:Title: The Wasserstein space of stochastic processes\nby Mathias
Beiglböck (University of Vienna) as part of BIRS workshop: Entropic Regu
larization of Optimal Transport and Applications\n\n\nAbstract\nWasserstei
n distance induces a natural Riemannian structure for the probabilities on
the Euclidean space. This insight of classical transport theory is fundam
ental for tremendous applications in various fields of pure and applied ma
thematics. We believe that an appropriate probabilistic variant\, the adap
ted Wasserstein distance AW\, can play a similar role for the class FP of
filtered processes\, i.e. stochastic processes together with a filtration.
In contrast to other topologies for stochastic processes\, probabilistic
operations such as the Doob-decomposition\, optimal stopping and stochasti
c control are continuous w.r.t. AW. We also show that (FP\,AW) is a geodes
ic space\, isometric to a classical Wasserstein space\, and that martingal
es form a closed geodesically convex subspace.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5120/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anna Kausamo
DTSTART:20210624T164000Z
DTEND:20210624T172000Z
DTSTAMP:20260404T042017Z
UID:BIRS-21w5120/16
DESCRIPTION:Title: Multi-marginal entropy-regularized optimal transportation for
singular cost functions\nby Anna Kausamo as part of BIRS workshop: En
tropic Regularization of Optimal Transport and Applications\n\n\nAbstract\
nI will introduce multi-marginal optimal transportation (MOT) for singular
cost functions and mention some of its applications. Then I move on to th
e entropy-regularised framework\, focusing on the Gamma-convergence proof
of the regularized minimizers for the singular MOT problem towards a non-r
egularised solution when the regularisation parameter goes to zero. When o
ne goes from two to many marginals and from attractive to singular cost fu
nction\, different levels of difficulty are introduced. One of the aims of
my talk is to show how these difficulties can be tackled.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5120/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giovanni Conforti (Ecole Polytechnique Paris – Mathematics)
DTSTART:20210624T173000Z
DTEND:20210624T181000Z
DTSTAMP:20260404T042017Z
UID:BIRS-21w5120/17
DESCRIPTION:Title: Hamilton Jacobi equations for controlled gradient flows: the
comparison principle\nby Giovanni Conforti (Ecole Polytechnique Paris
– Mathematics) as part of BIRS workshop: Entropic Regularization of Opti
mal Transport and Applications\n\n\nAbstract\nThis talk is devoted to the
study of a class of Hamilton-Jacobi equations on the space of probability
measures that arises naturally in connection with the study of a general
form of the Schrödinger problem for interacting particle systems. After
presenting the equations and their geometrical interpretation\, I will mov
e on to illustrate the main ideas behind a general strategy for to prove u
niqueness of viscosity solutions\, i.e. the comparison principle. Joint wo
rk with D.Tonon (U. Padova) and R.Kraaij (TU Delft).\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5120/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Geoffrey Schiebinger (University of British Columbia)
DTSTART:20210624T203000Z
DTEND:20210624T211000Z
DTSTAMP:20260404T042017Z
UID:BIRS-21w5120/18
DESCRIPTION:Title: Towards a mathematical theory of development\nby Geoffrey
Schiebinger (University of British Columbia) as part of BIRS workshop: En
tropic Regularization of Optimal Transport and Applications\n\n\nAbstract\
nNew measurement technologies like single-cell RNA sequencing are bringing
'big data' to biology. My group develops mathematical tools for analyzing
time-courses of high-dimensional gene expression data\, leveraging tools
from probability and optimal transport. We aim to develop a mathematical t
heory to answer questions like How does a stem cell transform into a muscl
e cell\, a skin cell\, or a neuron? How can we reprogram a skin cell into
a neuron? We model a developing population of cells with a curve in the s
pace of probability distributions on a high-dimensional gene expression sp
ace. We design algorithms to recover these curves from samples at various
time-points and we collaborate closely with experimentalists to test these
ideas on real data.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5120/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Max von Renesse (Universitaet Leipzig)
DTSTART:20210625T150000Z
DTEND:20210625T154000Z
DTSTAMP:20260404T042017Z
UID:BIRS-21w5120/19
DESCRIPTION:Title: On Overrelaxation in the Sinkhorn Algorithm\nby Max von R
enesse (Universitaet Leipzig) as part of BIRS workshop: Entropic Regulariz
ation of Optimal Transport and Applications\n\n\nAbstract\nWe discuss a si
mple but potent modification of the Sinkhorn algorithm based on overrelaxa
tion. We provide an a priori estimate for the crucial overrelaxation param
eter which guarantees both global and improved local convergence.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5120/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Flavien Léger (Sciences Po Paris)
DTSTART:20210625T155000Z
DTEND:20210625T163000Z
DTSTAMP:20260404T042017Z
UID:BIRS-21w5120/20
DESCRIPTION:Title: Taylor expansions for the regularized optimal transport probl
em\nby Flavien Léger (Sciences Po Paris) as part of BIRS workshop: En
tropic Regularization of Optimal Transport and Applications\n\n\nAbstract\
nWe prove Taylor expansions of the regularized optimal transport problem w
ith general cost as the temperature goes to zero. \nOur first contribution
is a multivariate Laplace expansion formula. We show that the first-order
terms involve the scalar curvature in the corresponding Hessian geometry.
\nWe then obtain: \n - first-order expansion of the potentials\; \n - sec
ond-order expansion of the optimal transport value. \nJoint work with Pier
re Roussillon\, François-Xavier Vialard and Gabriel Peyré.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5120/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yunan Yang (New York University)
DTSTART:20210625T164000Z
DTEND:20210625T172000Z
DTSTAMP:20260404T042017Z
UID:BIRS-21w5120/21
DESCRIPTION:Title: Optimal transport-based objective function for physical inver
se problems\nby Yunan Yang (New York University) as part of BIRS works
hop: Entropic Regularization of Optimal Transport and Applications\n\n\nAb
stract\nWe have proposed the quadratic Wasserstein distance from optimal t
ransport theory for inverse problems\, including nonlinear medium reconstr
uction for waveform inversions and chaotic dynamical systems parameter ide
ntification. Traditional methods for both applications suffered from longs
tanding difficulties such as nonconvexity and noise sensitivity. As we adv
ance\, we discover that the advantages of using optimal transposed-based m
etrics apply in a broader class of data-fitting problems where the continu
ous dependence between the parameter and the data involves the change of d
ata phase or support of the data. The implicit regularization effects of t
he Wasserstein distance similar to a weak norm also help improve stability
of parameter identification.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5120/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Katy Craig (University of California Santa Barbara)
DTSTART:20210625T173000Z
DTEND:20210625T181000Z
DTSTAMP:20260404T042017Z
UID:BIRS-21w5120/22
DESCRIPTION:Title: A blob method for diffusion and applications to sampling and
two layer neural networks\nby Katy Craig (University of California San
ta Barbara) as part of BIRS workshop: Entropic Regularization of Optimal T
ransport and Applications\n\n\nAbstract\nGiven a desired target distributi
on and an initial guess of that distribution\, composed of finitely many s
amples\, what is the best way to evolve the locations of the samples so th
at they more accurately represent the desired distribution? A classical so
lution to this problem is to allow the samples to evolve according to Lang
evin dynamics\, the stochastic particle method corresponding to the Fokker
-Planck equation. In today’s talk\, I will contrast this classical appro
ach with a deterministic particle method corresponding to the porous mediu
m equation. This method corresponds exactly to the mean-field dynamics of
training a two layer neural network for a radial basis function activation
function. We prove that\, as the number of samples increases and the vari
ance of the radial basis function goes to zero\, the particle method conve
rges to a bounded entropy solution of the porous medium equation. As a con
sequence\, we obtain both a novel method for sampling probability distribu
tions as well as insight into the training dynamics of two layer neural ne
tworks in the mean field regime. This is joint work with Karthik Elamvazhu
thi (UCLA)\, Matt Haberland (Cal Poly)\, and Olga Turanova (Michigan State
).\n
LOCATION:https://stable.researchseminars.org/talk/BIRS-21w5120/22/
END:VEVENT
END:VCALENDAR