395 research outputs found
A new graph perspective on max-min fairness in Gaussian parallel channels
In this work we are concerned with the problem of achieving max-min fairness
in Gaussian parallel channels with respect to a general performance function,
including channel capacity or decoding reliability as special cases. As our
central results, we characterize the laws which determine the value of the
achievable max-min fair performance as a function of channel sharing policy and
power allocation (to channels and users). In particular, we show that the
max-min fair performance behaves as a specialized version of the Lovasz
function, or Delsarte bound, of a certain graph induced by channel sharing
combinatorics. We also prove that, in addition to such graph, merely a certain
2-norm distance dependent on the allowable power allocations and used
performance functions, is sufficient for the characterization of max-min fair
performance up to some candidate interval. Our results show also a specific
role played by odd cycles in the graph induced by the channel sharing policy
and we present an interesting relation between max-min fairness in parallel
channels and optimal throughput in an associated interference channel.Comment: 41 pages, 8 figures. submitted to IEEE Transactions on Information
Theory on August the 6th, 200
Limits on Support Recovery with Probabilistic Models: An Information-Theoretic Framework
The support recovery problem consists of determining a sparse subset of a set
of variables that is relevant in generating a set of observations, and arises
in a diverse range of settings such as compressive sensing, and subset
selection in regression, and group testing. In this paper, we take a unified
approach to support recovery problems, considering general probabilistic models
relating a sparse data vector to an observation vector. We study the
information-theoretic limits of both exact and partial support recovery, taking
a novel approach motivated by thresholding techniques in channel coding. We
provide general achievability and converse bounds characterizing the trade-off
between the error probability and number of measurements, and we specialize
these to the linear, 1-bit, and group testing models. In several cases, our
bounds not only provide matching scaling laws in the necessary and sufficient
number of measurements, but also sharp thresholds with matching constant
factors. Our approach has several advantages over previous approaches: For the
achievability part, we obtain sharp thresholds under broader scalings of the
sparsity level and other parameters (e.g., signal-to-noise ratio) compared to
several previous works, and for the converse part, we not only provide
conditions under which the error probability fails to vanish, but also
conditions under which it tends to one.Comment: Accepted to IEEE Transactions on Information Theory; presented in
part at ISIT 2015 and SODA 201
On visualisation scaling, subeigenvectors and Kleene stars in max algebra
The purpose of this paper is to investigate the interplay arising between max
algebra, convexity and scaling problems. The latter, which have been studied in
nonnegative matrix theory, are strongly related to max algebra. One problem is
strict visualisation scaling, which means finding, for a given nonnegative
matrix A, a diagonal matrix X such that all elements of X^{-1}AX are less than
or equal to the maximum cycle geometric mean of A, with strict inequality for
the entries which do not lie on critical cycles. In this paper such scalings
are described by means of the max-algebraic subeigenvectors and Kleene stars of
nonnegative matrices as well as by some concepts of convex geometry.Comment: 22 page
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