320 research outputs found
Boolean functions: noise stability, non-interactive correlation distillation, and mutual information
Let be the noise operator acting on Boolean functions , where is the noise parameter. Given
and fixed mean , which Boolean function has the
largest -th moment ? This question has
close connections with noise stability of Boolean functions, the problem of
non-interactive correlation distillation, and Courtade-Kumar's conjecture on
the most informative Boolean function. In this paper, we characterize
maximizers in some extremal settings, such as low noise (
is close to 0), high noise ( is close to 1/2), as well as
when is large. Analogous results are also established in
more general contexts, such as Boolean functions defined on discrete torus
and the problem of noise stability in a tree
model.Comment: Corrections of some inaccuracie
Algebraic Network Coding Approach to Deterministic Wireless Relay Networks
The deterministic wireless relay network model, introduced by Avestimehr et
al., has been proposed for approximating Gaussian relay networks. This model,
known as the ADT network model, takes into account the broadcast nature of
wireless medium and interference. Avestimehr et al. showed that the Min-cut
Max-flow theorem holds in the ADT network.
In this paper, we show that the ADT network model can be described within the
algebraic network coding framework introduced by Koetter and Medard. We prove
that the ADT network problem can be captured by a single matrix, called the
"system matrix". We show that the min-cut of an ADT network is the rank of the
system matrix; thus, eliminating the need to optimize over exponential number
of cuts between two nodes to compute the min-cut of an ADT network.
We extend the capacity characterization for ADT networks to a more general
set of connections. Our algebraic approach not only provides the Min-cut
Max-flow theorem for a single unicast/multicast connection, but also extends to
non-multicast connections such as multiple multicast, disjoint multicast, and
two-level multicast. We also provide sufficiency conditions for achievability
in ADT networks for any general connection set. In addition, we show that the
random linear network coding, a randomized distributed algorithm for network
code construction, achieves capacity for the connections listed above.
Finally, we extend the ADT networks to those with random erasures and cycles
(thus, allowing bi-directional links). Note that ADT network was proposed for
approximating the wireless networks; however, ADT network is acyclic.
Furthermore, ADT network does not model the stochastic nature of the wireless
links. With our algebraic framework, we incorporate both cycles as well as
random failures into ADT network model.Comment: 9 pages, 12 figures, submitted to Allerton Conferenc
On the Non-Coherent Wideband Multipath Fading Relay Channel
We investigate the multipath fading relay channel in the limit of a large
bandwidth, and in the non-coherent setting, where the channel state is unknown
to all terminals, including the relay and the destination. We propose a
hypergraph model of the wideband multipath fading relay channel, and show that
its min-cut is achieved by a non-coherent peaky frequency binning scheme. The
so-obtained lower bound on the capacity of the wideband multipath fading relay
channel turns out to coincide with the block-Markov lower bound on the capacity
of the wideband frequency-division Gaussian (FD-AWGN) relay channel. In certain
cases, this achievable rate also meets the cut-set upper-bound, and thus
reaches the capacity of the non-coherent wideband multipath fading relay
channel.Comment: 8 pages, 4 figures, longer version (including proof) of the paper in
Proc. of IEEE ISIT 201
Communication Cost for Updating Linear Functions when Message Updates are Sparse: Connections to Maximally Recoverable Codes
We consider a communication problem in which an update of the source message
needs to be conveyed to one or more distant receivers that are interested in
maintaining specific linear functions of the source message. The setting is one
in which the updates are sparse in nature, and where neither the source nor the
receiver(s) is aware of the exact {\em difference vector}, but only know the
amount of sparsity that is present in the difference-vector. Under this
setting, we are interested in devising linear encoding and decoding schemes
that minimize the communication cost involved. We show that the optimal
solution to this problem is closely related to the notion of maximally
recoverable codes (MRCs), which were originally introduced in the context of
coding for storage systems. In the context of storage, MRCs guarantee optimal
erasure protection when the system is partially constrained to have local
parity relations among the storage nodes. In our problem, we show that optimal
solutions exist if and only if MRCs of certain kind (identified by the desired
linear functions) exist. We consider point-to-point and broadcast versions of
the problem, and identify connections to MRCs under both these settings. For
the point-to-point setting, we show that our linear-encoder based achievable
scheme is optimal even when non-linear encoding is permitted. The theory is
illustrated in the context of updating erasure coded storage nodes. We present
examples based on modern storage codes such as the minimum bandwidth
regenerating codes.Comment: To Appear in IEEE Transactions on Information Theor
Scalar-linear Solvability of Matroidal Networks Associated with Representable Matroids
We study matroidal networks introduced by Dougherty et al. We prove the
converse of the following theorem: If a network is scalar-linearly solvable
over some finite field, then the network is a matroidal network associated with
a representable matroid over a finite field. It follows that a network is
scalar-linearly solvable if and only if the network is a matroidal network
associated with a representable matroid over a finite field. We note that this
result combined with the construction method due to Dougherty et al. gives a
method for generating scalar-linearly solvable networks. Using the converse
implicitly, we demonstrate scalar-linear solvability of two classes of
matroidal networks: networks constructed from uniform matroids and those
constructed from graphic matroids.Comment: 5 pages, submitted to IEEE ISIT 201
Computing Bounds on Network Capacity Regions as a Polytope Reconstruction Problem
We define a notion of network capacity region of networks that generalizes
the notion of network capacity defined by Cannons et al. and prove its notable
properties such as closedness, boundedness and convexity when the finite field
is fixed. We show that the network routing capacity region is a computable
rational polytope and provide exact algorithms and approximation heuristics for
computing the region. We define the semi-network linear coding capacity region,
with respect to a fixed finite field, that inner bounds the corresponding
network linear coding capacity region, show that it is a computable rational
polytope, and provide exact algorithms and approximation heuristics. We show
connections between computing these regions and a polytope reconstruction
problem and some combinatorial optimization problems, such as the minimum cost
directed Steiner tree problem. We provide an example to illustrate our results.
The algorithms are not necessarily polynomial-time.Comment: Appeared in the 2011 IEEE International Symposium on Information
Theory, 5 pages, 1 figur
On Non-coherent MIMO Channels in the Wideband Regime: Capacity and Reliability
We consider a multiple-input, multiple-output (MIMO) wideband Rayleigh block
fading channel where the channel state is unknown to both the transmitter and
the receiver and there is only an average power constraint on the input. We
compute the capacity and analyze its dependence on coherence length, number of
antennas and receive signal-to-noise ratio (SNR) per degree of freedom. We
establish conditions on the coherence length and number of antennas for the
non-coherent channel to have a "near coherent" performance in the wideband
regime. We also propose a signaling scheme that is near-capacity achieving in
this regime.
We compute the error probability for this wideband non-coherent MIMO channel
and study its dependence on SNR, number of transmit and receive antennas and
coherence length. We show that error probability decays inversely with
coherence length and exponentially with the product of the number of transmit
and receive antennas. Moreover, channel outage dominates error probability in
the wideband regime. We also show that the critical as well as cut-off rates
are much smaller than channel capacity in this regime
Beyond the Min-Cut Bound: Deterministic Network Coding for Asynchronous Multirate Broadcast
In a single hop broadcast packet erasure network, we demonstrate that it is
possible to provide multirate packet delivery outside of what is given by the
network min-cut. This is achieved by using a deterministic non-block-based
network coding scheme, which allows us to sidestep some of the limitations put
in place by the block coding model used to determine the network capacity.
Under the network coding scheme we outline, the sender is able to transmit
network coded packets above the channel rate of some receivers, while ensuring
that they still experience nonzero delivery rates. Interestingly, in this
generalised form of asynchronous network coded broadcast, receivers are not
required to obtain knowledge of all packets transmitted so far. Instead, causal
feedback from the receivers about packet erasures is used by the sender to
determine a network coded transmission that will allow at least one, but often
multiple receivers, to deliver their next needed packet.
Although the analysis of deterministic coding schemes is generally a
difficult problem, by making some approximations we are able to obtain
tractable estimates of the receivers' delivery rates, which are shown to match
reasonably well with simulation. Using these estimates, we design a fairness
algorithm that allocates the sender's resources so all receivers will
experience fair delivery rate performance
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