66 research outputs found
Capacity of a Class of Broadcast Relay Channels
Consider the broadcast relay channel (BRC) which consists of a source sending
information over a two user broadcast
channel in presence of two relay nodes that help the transmission to the
destinations. Clearly, this network with
five nodes involves all the problems encountered in relay and broadcast
channels. New inner bounds on the capacity
region of this class of channels are derived. These results can be seen as a
generalization and hence unification of
previous work in this topic. Our bounds are based on the idea of
recombination of message bits and various effective
coding strategies for relay and broadcast channels. Capacity result is
obtained for the semi-degraded BRC-CR, where
one relay channel is degraded while the other one is reversely degraded. An
inner and upper bound is also presented
for the degraded BRC with common relay (BRC-CR), where both the relay and
broadcast channel are degraded which is
the capacity for the Gaussian case. Application of these results arise in the
context of opportunistic cooperation
of cellular networks.Comment: 5 pages, to appear in proc. IEEE ISIT, June 201
Cooperative Strategies for Simultaneous and Broadcast Relay Channels
Consider the \emph{simultaneous relay channel} (SRC) which consists of a set
of relay channels where the source wishes to transmit common and private
information to each of the destinations. This problem is recognized as being
equivalent to that of sending common and private information to several
destinations in presence of helper relays where each channel outcome becomes a
branch of the \emph{broadcast relay channel} (BRC). Cooperative schemes and
capacity region for a set with two memoryless relay channels are investigated.
The proposed coding schemes, based on \emph{Decode-and-Forward} (DF) and
\emph{Compress-and-Forward} (CF) must be capable of transmitting information
simultaneously to all destinations in such set.
Depending on the quality of source-to-relay and relay-to-destination
channels, inner bounds on the capacity of the general BRC are derived. Three
cases of particular interest are considered: cooperation is based on DF
strategy for both users --referred to as DF-DF region--, cooperation is based
on CF strategy for both users --referred to as CF-CF region--, and cooperation
is based on DF strategy for one destination and CF for the other --referred to
as DF-CF region--. These results can be seen as a generalization and hence
unification of previous works. An outer-bound on the capacity of the general
BRC is also derived. Capacity results are obtained for the specific cases of
semi-degraded and degraded Gaussian simultaneous relay channels. Rates are
evaluated for Gaussian models where the source must guarantee a minimum amount
of information to both users while additional information is sent to each of
them.Comment: 32 pages, 7 figures, To appear in IEEE Trans. on Information Theor
Broadcasting over the Relay Channel with Oblivious Cooperative Strategy
This paper investigates the problem of information transmission over the
simultaneous relay channel with two users (or two possible channel outcomes)
where for one of them the more suitable strategy is Decode-and-Forward (DF)
while for the other one is Compress-and-Forward (CF). In this setting, it is
assumed that the source wishes to send common and private informations to each
of the users (or channel outcomes). This problem is relevant to: (i) the
transmission of information over the broadcast relay channel (BRC) with
different relaying strategies and (ii) the transmission of information over the
conventional relay channel where the source is oblivious to the coding strategy
of relay. A novel coding that integrates simultaneously DF and CF schemes is
proposed and an inner bound on the capacity region is derived for the case of
general memoryless BRCs. As special case, the Gaussian BRC is studied where it
is shown that by means of the suggested broadcast coding the common rate can be
improved compared to existing strategies. Applications of these results arise
in broadcast scenarios with relays or in wireless scenarios where the source
does not know whether the relay is collocated with the source or with the
destination.Comment: 6 pages, presented at Allerton 201
On the Simultaneous Relay Channel with Collocated Relay and Destination Nodes
International audienceThe simultaneous relay channel with collocated relay and destination nodes is investigated. This models the scenario in which the source user is unaware of the channel controlling the communication but it knows the set of all possible relay channels. Our primary focus is the case where the relay node is physically near to the destination so that Compress-and-Forward (CF) coding is the adequate cooperative strategy. A broadcasting scheme for flexible user cooperation is developed. This enables the encoder to send part of the information regardless of which channel is present and additional information intended for each of the different channels. It can be seen that this problem is equivalent to that of sending common and private information to several destinations in presence of helper relays. This scenarios is identified as the broadcast relay channel (BRC). A general achievable rate region is derived and specific results are obtained for the case of Gaussian BRCs. The advantage of the proposed coding scheme is that the source can adapt its coding rate to the different channels that might be present during the communicatio
Algebraic Topological Networks via the Persistent Local Homology Sheaf
In this work, we introduce a novel approach based on algebraic topology to
enhance graph convolution and attention modules by incorporating local
topological properties of the data. To do so, we consider the framework of
sheaf neural networks, which has been previously leveraged to incorporate
additional structure into graph neural networks' features and construct more
expressive, non-isotropic messages. Specifically, given an input simplicial
complex (e.g. generated by the cliques of a graph or the neighbors in a point
cloud), we construct its local homology sheaf, which assigns to each node the
vector space of its local homology. The intermediate features of our networks
live in these vector spaces and we leverage the associated sheaf Laplacian to
construct more complex linear messages between them. Moreover, we extend this
approach by considering the persistent version of local homology associated
with a weighted simplicial complex (e.g., built from pairwise distances of
nodes embeddings). This i) solves the problem of the lack of a natural choice
of basis for the local homology vector spaces and ii) makes the sheaf itself
differentiable, which enables our models to directly optimize the topology of
their intermediate features.Comment: Symmetry and Geometry in Neural Representations - NeurReps Workshop @
NeurIPS 202
Selective Coding Strategy for Unicast Composite Networks
Consider a composite unicast relay network where the channel statistic is
randomly drawn from a set of conditional distributions indexed by a random
variable, which is assumed to be unknown at the source, fully known at the
destination and only partly known at the relays. Commonly, the coding strategy
at each relay is fixed regardless of its channel measurement. A novel coding
for unicast composite networks with multiple relays is introduced. This enables
the relays to select dynamically --based on its channel measurement-- the best
coding scheme between compress-and-forward (CF) and decode-and-forward (DF). As
a part of the main result, a generalization of Noisy Network Coding is shown
for the case of unicast general networks where the relays are divided between
those using DF and CF coding. Furthermore, the relays using DF scheme can
exploit the help of those based on CF scheme via offset coding. It is
demonstrated via numerical results that this novel coding, referred to as
Selective Coding Strategy (SCS), outperforms conventional coding schemes.Comment: To appear in International Symposium on Information Theory (ISIT)
201
Tight bounds on the mutual coherence of sensing matrices for Wigner D-functions on regular grids
Many practical sampling patterns for function approximation on the rotation
group utilizes regular samples on the parameter axes. In this paper, we relate
the mutual coherence analysis for sensing matrices that correspond to a class
of regular patterns to angular momentum analysis in quantum mechanics and
provide simple lower bounds for it. The products of Wigner d-functions, which
appear in coherence analysis, arise in angular momentum analysis in quantum
mechanics. We first represent the product as a linear combination of a single
Wigner d-function and angular momentum coefficients, otherwise known as the
Wigner 3j symbols. Using combinatorial identities, we show that under certain
conditions on the bandwidth and number of samples, the inner product of the
columns of the sensing matrix at zero orders, which is equal to the inner
product of two Legendre polynomials, dominates the mutual coherence term and
fixes a lower bound for it. In other words, for a class of regular sampling
patterns, we provide a lower bound for the inner product of the columns of the
sensing matrix that can be analytically computed. We verify numerically our
theoretical results and show that the lower bound for the mutual coherence is
larger than Welch bound. Besides, we provide algorithms that can achieve the
lower bound for spherical harmonics
Sensing Matrix Design and Sparse Recovery on the Sphere and the Rotation Group
In this paper, {the goal is to design deterministic sampling patterns on the
sphere and the rotation group} and, thereby, construct sensing matrices for
sparse recovery of band-limited functions. It is first shown that random
sensing matrices, which consists of random samples of Wigner D-functions,
satisfy the Restricted Isometry Property (RIP) with proper preconditioning and
can be used for sparse recovery on the rotation group. The mutual coherence,
however, is used to assess the performance of deterministic and regular sensing
matrices. We show that many of widely used regular sampling patterns yield
sensing matrices with the worst possible mutual coherence, and therefore are
undesirable for sparse recovery. Using tools from angular momentum analysis in
quantum mechanics, we provide a new expression for the mutual coherence, which
encourages the use of regular elevation samples. We construct low coherence
deterministic matrices by fixing the regular samples on the elevation and
minimizing the mutual coherence over the azimuth-polarization choice. It is
shown that once the elevation sampling is fixed, the mutual coherence has a
lower bound that depends only on the elevation samples. This lower bound,
however, can be achieved for spherical harmonics, which leads to new sensing
matrices with better coherence than other representative regular sampling
patterns. This is reflected as well in our numerical experiments where our
proposed sampling patterns perfectly match the phase transition of random
sampling patterns.Comment: IEEE Trans. on Signal Processin
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