7,232 research outputs found
A note on Probably Certifiably Correct algorithms
Many optimization problems of interest are known to be intractable, and while
there are often heuristics that are known to work on typical instances, it is
usually not easy to determine a posteriori whether the optimal solution was
found. In this short note, we discuss algorithms that not only solve the
problem on typical instances, but also provide a posteriori certificates of
optimality, probably certifiably correct (PCC) algorithms. As an illustrative
example, we present a fast PCC algorithm for minimum bisection under the
stochastic block model and briefly discuss other examples
Sample Approximation-Based Deflation Approaches for Chance SINR Constrained Joint Power and Admission Control
Consider the joint power and admission control (JPAC) problem for a
multi-user single-input single-output (SISO) interference channel. Most
existing works on JPAC assume the perfect instantaneous channel state
information (CSI). In this paper, we consider the JPAC problem with the
imperfect CSI, that is, we assume that only the channel distribution
information (CDI) is available. We formulate the JPAC problem into a chance
(probabilistic) constrained program, where each link's SINR outage probability
is enforced to be less than or equal to a specified tolerance. To circumvent
the computational difficulty of the chance SINR constraints, we propose to use
the sample (scenario) approximation scheme to convert them into finitely many
simple linear constraints. Furthermore, we reformulate the sample approximation
of the chance SINR constrained JPAC problem as a composite group sparse
minimization problem and then approximate it by a second-order cone program
(SOCP). The solution of the SOCP approximation can be used to check the
simultaneous supportability of all links in the network and to guide an
iterative link removal procedure (the deflation approach). We exploit the
special structure of the SOCP approximation and custom-design an efficient
algorithm for solving it. Finally, we illustrate the effectiveness and
efficiency of the proposed sample approximation-based deflation approaches by
simulations.Comment: The paper has been accepted for publication in IEEE Transactions on
Wireless Communication
Statistical Compressed Sensing of Gaussian Mixture Models
A novel framework of compressed sensing, namely statistical compressed
sensing (SCS), that aims at efficiently sampling a collection of signals that
follow a statistical distribution, and achieving accurate reconstruction on
average, is introduced. SCS based on Gaussian models is investigated in depth.
For signals that follow a single Gaussian model, with Gaussian or Bernoulli
sensing matrices of O(k) measurements, considerably smaller than the O(k
log(N/k)) required by conventional CS based on sparse models, where N is the
signal dimension, and with an optimal decoder implemented via linear filtering,
significantly faster than the pursuit decoders applied in conventional CS, the
error of SCS is shown tightly upper bounded by a constant times the best k-term
approximation error, with overwhelming probability. The failure probability is
also significantly smaller than that of conventional sparsity-oriented CS.
Stronger yet simpler results further show that for any sensing matrix, the
error of Gaussian SCS is upper bounded by a constant times the best k-term
approximation with probability one, and the bound constant can be efficiently
calculated. For Gaussian mixture models (GMMs), that assume multiple Gaussian
distributions and that each signal follows one of them with an unknown index, a
piecewise linear estimator is introduced to decode SCS. The accuracy of model
selection, at the heart of the piecewise linear decoder, is analyzed in terms
of the properties of the Gaussian distributions and the number of sensing
measurements. A maximum a posteriori expectation-maximization algorithm that
iteratively estimates the Gaussian models parameters, the signals model
selection, and decodes the signals, is presented for GMM-based SCS. In real
image sensing applications, GMM-based SCS is shown to lead to improved results
compared to conventional CS, at a considerably lower computational cost
Constrained Overcomplete Analysis Operator Learning for Cosparse Signal Modelling
We consider the problem of learning a low-dimensional signal model from a
collection of training samples. The mainstream approach would be to learn an
overcomplete dictionary to provide good approximations of the training samples
using sparse synthesis coefficients. This famous sparse model has a less well
known counterpart, in analysis form, called the cosparse analysis model. In
this new model, signals are characterised by their parsimony in a transformed
domain using an overcomplete (linear) analysis operator. We propose to learn an
analysis operator from a training corpus using a constrained optimisation
framework based on L1 optimisation. The reason for introducing a constraint in
the optimisation framework is to exclude trivial solutions. Although there is
no final answer here for which constraint is the most relevant constraint, we
investigate some conventional constraints in the model adaptation field and use
the uniformly normalised tight frame (UNTF) for this purpose. We then derive a
practical learning algorithm, based on projected subgradients and
Douglas-Rachford splitting technique, and demonstrate its ability to robustly
recover a ground truth analysis operator, when provided with a clean training
set, of sufficient size. We also find an analysis operator for images, using
some noisy cosparse signals, which is indeed a more realistic experiment. As
the derived optimisation problem is not a convex program, we often find a local
minimum using such variational methods. Some local optimality conditions are
derived for two different settings, providing preliminary theoretical support
for the well-posedness of the learning problem under appropriate conditions.Comment: 29 pages, 13 figures, accepted to be published in TS
Joint power and admission control via p norm minimization deflation
In an interference network, joint power and admission control aims to support
a maximum number of links at their specified signal to interference plus noise
ratio (SINR) targets while using a minimum total transmission power. In our
previous work, we formulated the joint control problem as a sparse
-minimization problem and relaxed it to a -minimization
problem. In this work, we propose to approximate the -optimization
problem to a p norm minimization problem where , since intuitively p
norm will approximate 0 norm better than 1 norm. We first show that the
-minimization problem is strongly NP-hard and then derive a
reformulation of it such that the well developed interior-point algorithms can
be applied to solve it. The solution to the -minimization problem can
efficiently guide the link's removals (deflation). Numerical simulations show
the proposed heuristic outperforms the existing algorithms.Comment: 2013 IEEE International Conference on Acoustics, Speech, and Signal
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