TY - JOUR

T1 - Adaptive constrained learning in reproducing kernel hilbert spaces

T2 - The robust beamforming case

AU - Slavakis, Konstantinos

AU - Theodoridis, Sergios

AU - Yamada, Isao

PY - 2009/12/1

Y1 - 2009/12/1

N2 - This paper establishes a new paradigm for convexly constrained adaptive learning in reproducing kernel Hilbert spaces (RKHS). Although the technique is of a general nature, we present it in the context of the beamforming problem. A priori knowledge, like beampattern specifications and constraints concerning robustness against steering vector errors, takes the form of multiple closed convex sets in a high (possibly infinite) dimensional RKHS. Every robustness constraint is shown to be equivalent to a min-max optimization task formed by means of the robust statistics ε -insensitive loss function. Such a multiplicity of specifications turns out to obtain a simple expression by using the rich frame of fixed-point sets of certain mappings defined in a Hilbert space. Moreover, the cost function, that the final solution has to optimize, is expressed as an infinite sequence of convex, nondifferentiable loss functions, springing from the sequence of the incoming training data. A novel adaptive beamforming design, of linear complexity with respect to the number of unknown parameters, to such a constrained nonlinear learning problem is derived by employing a very recently developed version of the adaptive projected subgradient method (APSM). The method produces a sequence that, under mild conditions, exhibits properties like the strong convergence to a beamformer that satisfies all of the imposed constraints, and in the meantime asymptotically minimizes the sequence of the loss functions imposed by the training data. The numerical examples demonstrate that the proposed method displays increased resolution in cases where the classical linear beamforming solutions collapse. Moreover, it leads to solutions, which are in agreement with the imposed a priori knowledge, as opposed to unconstrained online kernel regression techniques.

AB - This paper establishes a new paradigm for convexly constrained adaptive learning in reproducing kernel Hilbert spaces (RKHS). Although the technique is of a general nature, we present it in the context of the beamforming problem. A priori knowledge, like beampattern specifications and constraints concerning robustness against steering vector errors, takes the form of multiple closed convex sets in a high (possibly infinite) dimensional RKHS. Every robustness constraint is shown to be equivalent to a min-max optimization task formed by means of the robust statistics ε -insensitive loss function. Such a multiplicity of specifications turns out to obtain a simple expression by using the rich frame of fixed-point sets of certain mappings defined in a Hilbert space. Moreover, the cost function, that the final solution has to optimize, is expressed as an infinite sequence of convex, nondifferentiable loss functions, springing from the sequence of the incoming training data. A novel adaptive beamforming design, of linear complexity with respect to the number of unknown parameters, to such a constrained nonlinear learning problem is derived by employing a very recently developed version of the adaptive projected subgradient method (APSM). The method produces a sequence that, under mild conditions, exhibits properties like the strong convergence to a beamformer that satisfies all of the imposed constraints, and in the meantime asymptotically minimizes the sequence of the loss functions imposed by the training data. The numerical examples demonstrate that the proposed method displays increased resolution in cases where the classical linear beamforming solutions collapse. Moreover, it leads to solutions, which are in agreement with the imposed a priori knowledge, as opposed to unconstrained online kernel regression techniques.

KW - Adaptive learning

KW - Beamforming

KW - Convex analysis

KW - Fixed-point set

KW - Reproducing kernel Hilbert space (RKHS)

UR - http://www.scopus.com/inward/record.url?scp=70450233513&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70450233513&partnerID=8YFLogxK

U2 - 10.1109/TSP.2009.2027771

DO - 10.1109/TSP.2009.2027771

M3 - Article

AN - SCOPUS:70450233513

VL - 57

SP - 4744

EP - 4764

JO - IEEE Transactions on Signal Processing

JF - IEEE Transactions on Signal Processing

SN - 1053-587X

IS - 12

M1 - 5256321

ER -