1,278 research outputs found
Precoder design for space-time coded systems over correlated Rayleigh fading channels using convex optimization
A class of computationally efficient linear precoders for space-time block coded multiple-input multiple-output wireless systems is derived based on the minimization of the exact symbol error rate (SER) and its upper bound. Both correlations at the transmitter and receiver are assumed to be present, and only statistical channel state information in the form of the transmit and receive correlation matrices is assumed to be available at the transmitter. The convexity of the design based on SER minimization is established and exploited. The advantage of the developed technique is its low complexity. We also find various relationships of the proposed designs to the existing precoding techniques, and derive very simple closed-form precoders for special cases such as two or three receive antennas and constant receive correlation. The numerical simulations illustrate the excellent SER performance of the proposed precoders
Coordinate descent algorithms for nonconvex penalized regression, with applications to biological feature selection
A number of variable selection methods have been proposed involving nonconvex
penalty functions. These methods, which include the smoothly clipped absolute
deviation (SCAD) penalty and the minimax concave penalty (MCP), have been
demonstrated to have attractive theoretical properties, but model fitting is
not a straightforward task, and the resulting solutions may be unstable. Here,
we demonstrate the potential of coordinate descent algorithms for fitting these
models, establishing theoretical convergence properties and demonstrating that
they are significantly faster than competing approaches. In addition, we
demonstrate the utility of convexity diagnostics to determine regions of the
parameter space in which the objective function is locally convex, even though
the penalty is not. Our simulation study and data examples indicate that
nonconvex penalties like MCP and SCAD are worthwhile alternatives to the lasso
in many applications. In particular, our numerical results suggest that MCP is
the preferred approach among the three methods.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS388 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Immunizing Conic Quadratic Optimization Problems Against Implementation Errors
We show that the robust counterpart of a convex quadratic constraint with ellipsoidal implementation error is equivalent to a system of conic quadratic constraints. To prove this result we first derive a sharper result for the S-lemma in case the two matrices involved can be simultaneously diagonalized. This extension of the S-lemma may also be useful for other purposes. We extend the result to the case in which the uncertainty region is the intersection of two convex quadratic inequalities. The robust counterpart for this case is also equivalent to a system of conic quadratic constraints. Results for convex conic quadratic constraints with implementation error are also given. We conclude with showing how the theory developed can be applied in robust linear optimization with jointly uncertain parameters and implementation errors, in sequential robust quadratic programming, in Taguchi’s robust approach, and in the adjustable robust counterpart.Conic Quadratic Program;hidden convexity;implementation error;robust optimization;simultaneous diagonalizability;S-lemma
Simple Approximations of Semialgebraic Sets and their Applications to Control
Many uncertainty sets encountered in control systems analysis and design can
be expressed in terms of semialgebraic sets, that is as the intersection of
sets described by means of polynomial inequalities. Important examples are for
instance the solution set of linear matrix inequalities or the Schur/Hurwitz
stability domains. These sets often have very complicated shapes (non-convex,
and even non-connected), which renders very difficult their manipulation. It is
therefore of considerable importance to find simple-enough approximations of
these sets, able to capture their main characteristics while maintaining a low
level of complexity. For these reasons, in the past years several convex
approximations, based for instance on hyperrect-angles, polytopes, or
ellipsoids have been proposed. In this work, we move a step further, and
propose possibly non-convex approximations , based on a small volume polynomial
superlevel set of a single positive polynomial of given degree. We show how
these sets can be easily approximated by minimizing the L1 norm of the
polynomial over the semialgebraic set, subject to positivity constraints.
Intuitively, this corresponds to the trace minimization heuristic commonly
encounter in minimum volume ellipsoid problems. From a computational viewpoint,
we design a hierarchy of linear matrix inequality problems to generate these
approximations, and we provide theoretically rigorous convergence results, in
the sense that the hierarchy of outer approximations converges in volume (or,
equivalently, almost everywhere and almost uniformly) to the original set. Two
main applications of the proposed approach are considered. The first one aims
at reconstruction/approximation of sets from a finite number of samples. In the
second one, we show how the concept of polynomial superlevel set can be used to
generate samples uniformly distributed on a given semialgebraic set. The
efficiency of the proposed approach is demonstrated by different numerical
examples
Polychromatic X-ray CT Image Reconstruction and Mass-Attenuation Spectrum Estimation
We develop a method for sparse image reconstruction from polychromatic
computed tomography (CT) measurements under the blind scenario where the
material of the inspected object and the incident-energy spectrum are unknown.
We obtain a parsimonious measurement-model parameterization by changing the
integral variable from photon energy to mass attenuation, which allows us to
combine the variations brought by the unknown incident spectrum and mass
attenuation into a single unknown mass-attenuation spectrum function; the
resulting measurement equation has the Laplace integral form. The
mass-attenuation spectrum is then expanded into first order B-spline basis
functions. We derive a block coordinate-descent algorithm for constrained
minimization of a penalized negative log-likelihood (NLL) cost function, where
penalty terms ensure nonnegativity of the spline coefficients and nonnegativity
and sparsity of the density map. The image sparsity is imposed using
total-variation (TV) and norms, applied to the density-map image and
its discrete wavelet transform (DWT) coefficients, respectively. This algorithm
alternates between Nesterov's proximal-gradient (NPG) and limited-memory
Broyden-Fletcher-Goldfarb-Shanno with box constraints (L-BFGS-B) steps for
updating the image and mass-attenuation spectrum parameters. To accelerate
convergence of the density-map NPG step, we apply a step-size selection scheme
that accounts for varying local Lipschitz constant of the NLL. We consider
lognormal and Poisson noise models and establish conditions for biconvexity of
the corresponding NLLs. We also prove the Kurdyka-{\L}ojasiewicz property of
the objective function, which is important for establishing local convergence
of the algorithm. Numerical experiments with simulated and real X-ray CT data
demonstrate the performance of the proposed scheme
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