5,236 research outputs found
A Minimal Set of Shannon-type Inequalities for Functional Dependence Structures
The minimal set of Shannon-type inequalities (referred to as elemental
inequalities), plays a central role in determining whether a given inequality
is Shannon-type. Often, there arises a situation where one needs to check
whether a given inequality is a constrained Shannon-type inequality. Another
important application of elemental inequalities is to formulate and compute the
Shannon outer bound for multi-source multi-sink network coding capacity. Under
this formulation, it is the region of feasible source rates subject to the
elemental inequalities and network coding constraints that is of interest.
Hence it is of fundamental interest to identify the redundancies induced
amongst elemental inequalities when given a set of functional dependence
constraints. In this paper, we characterize a minimal set of Shannon-type
inequalities when functional dependence constraints are present.Comment: 5 pagers, accepted ISIT201
The Inflation Technique for Causal Inference with Latent Variables
The problem of causal inference is to determine if a given probability
distribution on observed variables is compatible with some causal structure.
The difficult case is when the causal structure includes latent variables. We
here introduce the for tackling this problem. An
inflation of a causal structure is a new causal structure that can contain
multiple copies of each of the original variables, but where the ancestry of
each copy mirrors that of the original. To every distribution of the observed
variables that is compatible with the original causal structure, we assign a
family of marginal distributions on certain subsets of the copies that are
compatible with the inflated causal structure. It follows that compatibility
constraints for the inflation can be translated into compatibility constraints
for the original causal structure. Even if the constraints at the level of
inflation are weak, such as observable statistical independences implied by
disjoint causal ancestry, the translated constraints can be strong. We apply
this method to derive new inequalities whose violation by a distribution
witnesses that distribution's incompatibility with the causal structure (of
which Bell inequalities and Pearl's instrumental inequality are prominent
examples). We describe an algorithm for deriving all such inequalities for the
original causal structure that follow from ancestral independences in the
inflation. For three observed binary variables with pairwise common causes, it
yields inequalities that are stronger in at least some aspects than those
obtainable by existing methods. We also describe an algorithm that derives a
weaker set of inequalities but is more efficient. Finally, we discuss which
inflations are such that the inequalities one obtains from them remain valid
even for quantum (and post-quantum) generalizations of the notion of a causal
model.Comment: Minor final corrections, updated to match the published version as
closely as possibl
Geometric inequalities on Heisenberg groups
We establish geometric inequalities in the sub-Riemannian setting of the
Heisenberg group . Our results include a natural sub-Riemannian
version of the celebrated curvature-dimension condition of Lott-Villani and
Sturm and also a geodesic version of the Borell-Brascamp-Lieb inequality akin
to the one obtained by Cordero-Erausquin, McCann and Schmuckenschl\"ager. The
latter statement implies sub-Riemannian versions of the geodesic
Pr\'ekopa-Leindler and Brunn-Minkowski inequalities. The proofs are based on
optimal mass transportation and Riemannian approximation of
developed by Ambrosio and Rigot. These results refute a general point of view,
according to which no geometric inequalities can be derived by optimal mass
transportation on singular spaces.Comment: to appear in Calculus of Variations and Partial Differential
Equations (42 pages, 1 figure
Regularized Optimal Transport and the Rot Mover's Distance
This paper presents a unified framework for smooth convex regularization of
discrete optimal transport problems. In this context, the regularized optimal
transport turns out to be equivalent to a matrix nearness problem with respect
to Bregman divergences. Our framework thus naturally generalizes a previously
proposed regularization based on the Boltzmann-Shannon entropy related to the
Kullback-Leibler divergence, and solved with the Sinkhorn-Knopp algorithm. We
call the regularized optimal transport distance the rot mover's distance in
reference to the classical earth mover's distance. We develop two generic
schemes that we respectively call the alternate scaling algorithm and the
non-negative alternate scaling algorithm, to compute efficiently the
regularized optimal plans depending on whether the domain of the regularizer
lies within the non-negative orthant or not. These schemes are based on
Dykstra's algorithm with alternate Bregman projections, and further exploit the
Newton-Raphson method when applied to separable divergences. We enhance the
separable case with a sparse extension to deal with high data dimensions. We
also instantiate our proposed framework and discuss the inherent specificities
for well-known regularizers and statistical divergences in the machine learning
and information geometry communities. Finally, we demonstrate the merits of our
methods with experiments using synthetic data to illustrate the effect of
different regularizers and penalties on the solutions, as well as real-world
data for a pattern recognition application to audio scene classification
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