### Abstract

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 R^{N} and R^{M} and (ii) data-dependent convex sets in R^{N} and R^{M}; the latter are obtained in a sequential fashion. Roughly speaking, the problem is to find a common point of all the sets defined on R^{N} such that its image under a given linear transformation is a common point of all the sets defined on R^{M}, 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.

Original language | English |
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Title of host publication | DSP 2009: 16th International Conference on Digital Signal Processing, Proceedings |

DOIs | |

Publication status | Published - 2009 |

Externally published | Yes |

Event | DSP 2009:16th International Conference on Digital Signal Processing - Santorini, Greece Duration: 2009 Jul 5 → 2009 Jul 7 |

### Other

Other | DSP 2009:16th International Conference on Digital Signal Processing |
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Country | Greece |

City | Santorini |

Period | 09/7/5 → 09/7/7 |

### Fingerprint

### Keywords

- Adaptive projected subgradient method
- Convex feasibility problem
- Projected gradient
- Split feasibility problem

### ASJC Scopus subject areas

- Computer Vision and Pattern Recognition
- Signal Processing

### Cite this

*DSP 2009: 16th International Conference on Digital Signal Processing, Proceedings*[5201250] https://doi.org/10.1109/ICDSP.2009.5201250

**Signal processing in dual domain by adaptive projected subgradient method.** / Yukawa, Masahiro; Slavakis, Konstantinos; Yamada., Isao.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*DSP 2009: 16th International Conference on Digital Signal Processing, Proceedings.*, 5201250, DSP 2009:16th International Conference on Digital Signal Processing, Santorini, Greece, 09/7/5. https://doi.org/10.1109/ICDSP.2009.5201250

}

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

SN - 9781424432981

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

ER -