61,372 research outputs found
Robust Adaptive Beamforming for General-Rank Signal Model with Positive Semi-Definite Constraint via POTDC
The robust adaptive beamforming (RAB) problem for general-rank signal model
with an additional positive semi-definite constraint is considered. Using the
principle of the worst-case performance optimization, such RAB problem leads to
a difference-of-convex functions (DC) optimization problem. The existing
approaches for solving the resulted non-convex DC problem are based on
approximations and find only suboptimal solutions. Here we solve the non-convex
DC problem rigorously and give arguments suggesting that the solution is
globally optimal. Particularly, we rewrite the problem as the minimization of a
one-dimensional optimal value function whose corresponding optimization problem
is non-convex. Then, the optimal value function is replaced with another
equivalent one, for which the corresponding optimization problem is convex. The
new one-dimensional optimal value function is minimized iteratively via
polynomial time DC (POTDC) algorithm.We show that our solution satisfies the
Karush-Kuhn-Tucker (KKT) optimality conditions and there is a strong evidence
that such solution is also globally optimal. Towards this conclusion, we
conjecture that the new optimal value function is a convex function. The new
RAB method shows superior performance compared to the other state-of-the-art
general-rank RAB methods.Comment: 29 pages, 7 figures, 2 tables, Submitted to IEEE Trans. Signal
Processing on August 201
A note on preconditioning weighted linear least squares, with consequences for weakly-constrained variational data assimilation
The effect of preconditioning linear weighted least-squares using an
approximation of the model matrix is analyzed, showing the interplay of the
eigenstructures of both the model and weighting matrices. A small example is
given illustrating the resulting potential inefficiency of such
preconditioners. Consequences of these results in the context of the
weakly-constrained 4D-Var data assimilation problem are finally discussed.Comment: 10 pages, 2 figure
A second derivative SQP method: local convergence
In [19], we gave global convergence results for a second-derivative SQP method for minimizing the exact â„“1-merit function for a fixed value of the penalty parameter. To establish this result, we used the properties of the so-called Cauchy step, which was itself computed from the so-called predictor step. In addition, we allowed for the computation of a variety of (optional) SQP steps that were intended to improve the efficiency of the algorithm. \ud
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Although we established global convergence of the algorithm, we did not discuss certain aspects that are critical when developing software capable of solving general optimization problems. In particular, we must have strategies for updating the penalty parameter and better techniques for defining the positive-definite matrix Bk used in computing the predictor step. In this paper we address both of these issues. We consider two techniques for defining the positive-definite matrix Bk—a simple diagonal approximation and a more sophisticated limited-memory BFGS update. We also analyze a strategy for updating the penalty paramter based on approximately minimizing the ℓ1-penalty function over a sequence of increasing values of the penalty parameter.\ud
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Algorithms based on exact penalty functions have certain desirable properties. To be practical, however, these algorithms must be guaranteed to avoid the so-called Maratos effect. We show that a nonmonotone varient of our algorithm avoids this phenomenon and, therefore, results in asymptotically superlinear local convergence; this is verified by preliminary numerical results on the Hock and Shittkowski test set
The Matrix Ridge Approximation: Algorithms and Applications
We are concerned with an approximation problem for a symmetric positive
semidefinite matrix due to motivation from a class of nonlinear machine
learning methods. We discuss an approximation approach that we call {matrix
ridge approximation}. In particular, we define the matrix ridge approximation
as an incomplete matrix factorization plus a ridge term. Moreover, we present
probabilistic interpretations using a normal latent variable model and a
Wishart model for this approximation approach. The idea behind the latent
variable model in turn leads us to an efficient EM iterative method for
handling the matrix ridge approximation problem. Finally, we illustrate the
applications of the approximation approach in multivariate data analysis.
Empirical studies in spectral clustering and Gaussian process regression show
that the matrix ridge approximation with the EM iteration is potentially
useful
A new family of high-resolution multivariate spectral estimators
In this paper, we extend the Beta divergence family to multivariate power
spectral densities. Similarly to the scalar case, we show that it smoothly
connects the multivariate Kullback-Leibler divergence with the multivariate
Itakura-Saito distance. We successively study a spectrum approximation problem,
based on the Beta divergence family, which is related to a multivariate
extension of the THREE spectral estimation technique. It is then possible to
characterize a family of solutions to the problem. An upper bound on the
complexity of these solutions will also be provided. Simulations suggest that
the most suitable solution of this family depends on the specific features
required from the estimation problem
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