4,075 research outputs found
Network Coding for Computing: Cut-Set Bounds
The following \textit{network computing} problem is considered. Source nodes
in a directed acyclic network generate independent messages and a single
receiver node computes a target function of the messages. The objective is
to maximize the average number of times can be computed per network usage,
i.e., the ``computing capacity''. The \textit{network coding} problem for a
single-receiver network is a special case of the network computing problem in
which all of the source messages must be reproduced at the receiver. For
network coding with a single receiver, routing is known to achieve the capacity
by achieving the network \textit{min-cut} upper bound. We extend the definition
of min-cut to the network computing problem and show that the min-cut is still
an upper bound on the maximum achievable rate and is tight for computing (using
coding) any target function in multi-edge tree networks and for computing
linear target functions in any network. We also study the bound's tightness for
different classes of target functions. In particular, we give a lower bound on
the computing capacity in terms of the Steiner tree packing number and a
different bound for symmetric functions. We also show that for certain networks
and target functions, the computing capacity can be less than an arbitrarily
small fraction of the min-cut bound.Comment: Submitted to the IEEE Transactions on Information Theory (Special
Issue on Facets of Coding Theory: from Algorithms to Networks); Revised on
Aug 9, 201
Tightness of the maximum likelihood semidefinite relaxation for angular synchronization
Maximum likelihood estimation problems are, in general, intractable
optimization problems. As a result, it is common to approximate the maximum
likelihood estimator (MLE) using convex relaxations. In some cases, the
relaxation is tight: it recovers the true MLE. Most tightness proofs only apply
to situations where the MLE exactly recovers a planted solution (known to the
analyst). It is then sufficient to establish that the optimality conditions
hold at the planted signal. In this paper, we study an estimation problem
(angular synchronization) for which the MLE is not a simple function of the
planted solution, yet for which the convex relaxation is tight. To establish
tightness in this context, the proof is less direct because the point at which
to verify optimality conditions is not known explicitly.
Angular synchronization consists in estimating a collection of phases,
given noisy measurements of the pairwise relative phases. The MLE for angular
synchronization is the solution of a (hard) non-bipartite Grothendieck problem
over the complex numbers. We consider a stochastic model for the data: a
planted signal (that is, a ground truth set of phases) is corrupted with
non-adversarial random noise. Even though the MLE does not coincide with the
planted signal, we show that the classical semidefinite relaxation for it is
tight, with high probability. This holds even for high levels of noise.Comment: 2 figure
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