567 research outputs found
Topological Structures on DMC spaces
Two channels are said to be equivalent if they are degraded from each other.
The space of equivalent channels with input alphabet and output alphabet
can be naturally endowed with the quotient of the Euclidean topology by the
equivalence relation. A topology on the space of equivalent channels with fixed
input alphabet and arbitrary but finite output alphabet is said to be
natural if and only if it induces the quotient topology on the subspaces of
equivalent channels sharing the same output alphabet. We show that every
natural topology is -compact, separable and path-connected. On the
other hand, if , a Hausdorff natural topology is not Baire and it is
not locally compact anywhere. This implies that no natural topology can be
completely metrized if . The finest natural topology, which we call
the strong topology, is shown to be compactly generated, sequential and .
On the other hand, the strong topology is not first-countable anywhere, hence
it is not metrizable. We show that in the strong topology, a subspace is
compact if and only if it is rank-bounded and strongly-closed. We introduce a
metric distance on the space of equivalent channels which compares the noise
levels between channels. The induced metric topology, which we call the
noisiness topology, is shown to be natural. We also study topologies that are
inherited from the space of meta-probability measures by identifying channels
with their Blackwell measures. We show that the weak-* topology is exactly the
same as the noisiness topology and hence it is natural. We prove that if
, the total variation topology is not natural nor Baire, hence it is
not completely metrizable. Moreover, it is not locally compact anywhere.
Finally, we show that the Borel -algebra is the same for all Hausdorff
natural topologies.Comment: 43 pages, submitted to IEEE Trans. Inform. Theory and in part to
ISIT201
The persistent cosmic web and its filamentary structure I: Theory and implementation
We present DisPerSE, a novel approach to the coherent multi-scale
identification of all types of astrophysical structures, and in particular the
filaments, in the large scale distribution of matter in the Universe. This
method and corresponding piece of software allows a genuinely scale free and
parameter free identification of the voids, walls, filaments, clusters and
their configuration within the cosmic web, directly from the discrete
distribution of particles in N-body simulations or galaxies in sparse
observational catalogues. To achieve that goal, the method works directly over
the Delaunay tessellation of the discrete sample and uses the DTFE density
computed at each tracer particle; no further sampling, smoothing or processing
of the density field is required.
The idea is based on recent advances in distinct sub-domains of computational
topology, which allows a rigorous application of topological principles to
astrophysical data sets, taking into account uncertainties and Poisson noise.
Practically, the user can define a given persistence level in terms of
robustness with respect to noise (defined as a "number of sigmas") and the
algorithm returns the structures with the corresponding significance as sets of
critical points, lines, surfaces and volumes corresponding to the clusters,
filaments, walls and voids; filaments, connected at cluster nodes, crawling
along the edges of walls bounding the voids. The method is also interesting as
it allows for a robust quantification of the topological properties of a
discrete distribution in terms of Betti numbers or Euler characteristics,
without having to resort to smoothing or having to define a particular scale.
In this paper, we introduce the necessary mathematical background and
describe the method and implementation, while we address the application to 3D
simulated and observed data sets to the companion paper.Comment: A higher resolution version is available at
http://www.iap.fr/users/sousbie together with complementary material.
Submitted to MNRA
Continuity of Channel Parameters and Operations under Various DMC Topologies
We study the continuity of many channel parameters and operations under
various topologies on the space of equivalent discrete memoryless channels
(DMC). We show that mutual information, channel capacity, Bhattacharyya
parameter, probability of error of a fixed code, and optimal probability of
error for a given code rate and blocklength, are continuous under various DMC
topologies. We also show that channel operations such as sums, products,
interpolations, and Ar{\i}kan-style transformations are continuous.Comment: 31 pages. Submitted to IEEE Trans. Inform. Theory and in part to
ISIT201
A Characterization of the Shannon Ordering of Communication Channels
The ordering of communication channels was first introduced by Shannon. In
this paper, we aim to find a characterization of the Shannon ordering. We show
that contains if and only if is the skew-composition of with
a convex-product channel. This fact is used to derive a characterization of the
Shannon ordering that is similar to the Blackwell-Sherman-Stein theorem. Two
channels are said to be Shannon-equivalent if each one is contained in the
other. We investigate the topologies that can be constructed on the space of
Shannon-equivalent channels. We introduce the strong topology and the BRM
metric on this space. Finally, we study the continuity of a few channel
parameters and operations under the strong topology.Comment: 23 pages, presented in part at ISIT'17. arXiv admin note: text
overlap with arXiv:1702.0072
Pivotal tricategories and a categorification of inner-product modules
This article investigates duals for bimodule categories over finite tensor
categories. We show that finite bimodule categories form a tricategory and
discuss the dualities in this tricategory using inner homs. We consider
inner-product bimodule categories over pivotal tensor categories with
additional structure on the inner homs. Inner-product module categories are
related to Frobenius algebras and lead to the notion of -Morita equivalence
for pivotal tensor categories. We show that inner-product bimodule categories
form a tricategory with two duality operations and an additional pivotal
structure. This is work is motivated by defects in topological field theories.Comment: 64 pages, comments are welcom
On the Input-Degradedness and Input-Equivalence Between Channels
A channel is said to be input-degraded from another channel if
can be simulated from by randomization at the input. We provide a
necessary and sufficient condition for a channel to be input-degraded from
another one. We show that any decoder that is good for is also good for
. We provide two characterizations for input-degradedness, one of which is
similar to the Blackwell-Sherman-Stein theorem. We say that two channels are
input-equivalent if they are input-degraded from each other. We study the
topologies that can be constructed on the space of input-equivalent channels,
and we investigate their properties. Moreover, we study the continuity of
several channel parameters and operations under these topologies.Comment: 30 pages. Submitted to IEEE Trans. Inform. Theory and in part to
ISIT2017. arXiv admin note: substantial text overlap with arXiv:1701.0446
Algebraic Structures in Euclidean and Minkowskian Two-Dimensional Conformal Field Theory
We review how modular categories, and commutative and non-commutative
Frobenius algebras arise in rational conformal field theory. For Euclidean CFT
we use an approach based on sewing of surfaces, and in the Minkowskian case we
describe CFT by a net of operator algebras.Comment: 21 pages, contribution to proceedings for "Non-commutative Structures
in Mathematics and Physics" (Brussels, July 2008
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