918 research outputs found
Matrix-valued Monge-Kantorovich Optimal Mass Transport
We formulate an optimal transport problem for matrix-valued density
functions. This is pertinent in the spectral analysis of multivariable
time-series. The "mass" represents energy at various frequencies whereas, in
addition to a usual transportation cost across frequencies, a cost of rotation
is also taken into account. We show that it is natural to seek the
transportation plan in the tensor product of the spaces for the two
matrix-valued marginals. In contrast to the classical Monge-Kantorovich
setting, the transportation plan is no longer supported on a thin zero-measure
set.Comment: 11 page
Matricial Wasserstein-1 Distance
In this note, we propose an extension of the Wasserstein 1-metric () for
matrix probability densities, matrix-valued density measures, and an unbalanced
interpretation of mass transport. The key is using duality theory, in
particular, a "dual of the dual" formulation of . This matrix analogue of
the Earth Mover's Distance has several attractive features including ease of
computation.Comment: 8 page
Transport Problems and Disintegration Maps
By disintegration of transport plans it is introduced the notion of transport
class. This allows to consider the Monge problem as a particular case of the
Kantorovich transport problem, once a transport class is fixed. The transport
problem constrained to a fixed transport class is equivalent to an abstract
Monge problem over a Wasserstein space of probability measures. Concerning
solvability of this kind of constrained problems, it turns out that in some
sense the Monge problem corresponds to a lucky case
Optimal transportation, topology and uniqueness
The Monge-Kantorovich transportation problem involves optimizing with respect
to a given a cost function. Uniqueness is a fundamental open question about
which little is known when the cost function is smooth and the landscapes
containing the goods to be transported possess (non-trivial) topology. This
question turns out to be closely linked to a delicate problem (# 111) of
Birkhoff [14]: give a necessary and sufficient condition on the support of a
joint probability to guarantee extremality among all measures which share its
marginals. Fifty years of progress on Birkhoff's question culminate in Hestir
and Williams' necessary condition which is nearly sufficient for extremality;
we relax their subtle measurability hypotheses separating necessity from
sufficiency slightly, yet demonstrate by example that to be sufficient
certainly requires some measurability. Their condition amounts to the vanishing
of the measure \gamma outside a countable alternating sequence of graphs and
antigraphs in which no two graphs (or two antigraphs) have domains that
overlap, and where the domain of each graph / antigraph in the sequence
contains the range of the succeeding antigraph (respectively, graph). Such
sequences are called numbered limb systems. We then explain how this
characterization can be used to resolve the uniqueness of Kantorovich solutions
for optimal transportation on a manifold with the topology of the sphere.Comment: 36 pages, 6 figure
A glimpse into the differential topology and geometry of optimal transport
This note exposes the differential topology and geometry underlying some of
the basic phenomena of optimal transportation. It surveys basic questions
concerning Monge maps and Kantorovich measures: existence and regularity of the
former, uniqueness of the latter, and estimates for the dimension of its
support, as well as the associated linear programming duality. It shows the
answers to these questions concern the differential geometry and topology of
the chosen transportation cost. It also establishes new connections --- some
heuristic and others rigorous --- based on the properties of the
cross-difference of this cost, and its Taylor expansion at the diagonal.Comment: 27 page
The Monge problem with vanishing gradient penalization: Vortices and asymptotic profile
We investigate the approximation of the Monge problem (minimizing
\int\_ |T (x) -- x| d(x) among the vector-valued maps T with
prescribed image measure T \# ) by adding a vanishing Dirichlet energy,
namely \int\_ |DT |^2. We study the -convergence as
0, proving a density result for Sobolev (or Lipschitz)
transport maps in the class of transport plans. In a certain two-dimensional
framework that we analyze in details, when no optimal plan is induced by an H
^1 map, we study the selected limit map, which is a new "special" Monge
transport, possibly different from the monotone one, and we find the precise
asymptotics of the optimal cost depending on , where the leading term
is of order | log |
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