TY - GEN

T1 - Signal processing in dual domain by adaptive projected subgradient method

AU - Yukawa, Masahiro

AU - Slavakis, Konstantinos

AU - Yamada., Isao

PY - 2009

Y1 - 2009

N2 - The goal of this paper is to establish a novel signal processing paradigm that enables us to find a point meeting time-variable specifications in dual domain (e.g., time and frequency domains) simultaneously. For this purpose, we define a new problem which we call adaptive split feasibility problem (ASFP). In the ASFP formulation, we have (i) a priori knowledge based convex constraints in the Euclidean spaces RN and RM and (ii) data-dependent convex sets in RN and RM; the latter are obtained in a sequential fashion. Roughly speaking, the problem is to find a common point of all the sets defined on RN such that its image under a given linear transformation is a common point of all the sets defined on RM, if such a point exists. We prove that the adaptive projected subgradient method (APSM) deals with the ASFP by employing (i) a projected gradient operator with respect to (w.r.t.) a 'fixed' proximity function reflecting the convex constraints and (ii) a subgradient projection w.r.t. 'time-varying' objective functions reflecting the data-dependent sets. The resulting algorithm requires no unwanted operations such as matrix inversion, therefore it is suitable for realtime implementation. A convergence analysis is presented and verified by numerical examples.

AB - The goal of this paper is to establish a novel signal processing paradigm that enables us to find a point meeting time-variable specifications in dual domain (e.g., time and frequency domains) simultaneously. For this purpose, we define a new problem which we call adaptive split feasibility problem (ASFP). In the ASFP formulation, we have (i) a priori knowledge based convex constraints in the Euclidean spaces RN and RM and (ii) data-dependent convex sets in RN and RM; the latter are obtained in a sequential fashion. Roughly speaking, the problem is to find a common point of all the sets defined on RN such that its image under a given linear transformation is a common point of all the sets defined on RM, if such a point exists. We prove that the adaptive projected subgradient method (APSM) deals with the ASFP by employing (i) a projected gradient operator with respect to (w.r.t.) a 'fixed' proximity function reflecting the convex constraints and (ii) a subgradient projection w.r.t. 'time-varying' objective functions reflecting the data-dependent sets. The resulting algorithm requires no unwanted operations such as matrix inversion, therefore it is suitable for realtime implementation. A convergence analysis is presented and verified by numerical examples.

KW - Adaptive projected subgradient method

KW - Convex feasibility problem

KW - Projected gradient

KW - Split feasibility problem

UR - http://www.scopus.com/inward/record.url?scp=70449553188&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70449553188&partnerID=8YFLogxK

U2 - 10.1109/ICDSP.2009.5201250

DO - 10.1109/ICDSP.2009.5201250

M3 - Conference contribution

AN - SCOPUS:70449553188

SN - 9781424432981

T3 - DSP 2009: 16th International Conference on Digital Signal Processing, Proceedings

BT - DSP 2009:16th International Conference on Digital Signal Processing, Proceedings

T2 - DSP 2009:16th International Conference on Digital Signal Processing

Y2 - 5 July 2009 through 7 July 2009

ER -