Backward Gradient Interval Optimization Convergence for the Semi-Classical Signal Analysis
Maria Sara Nour Sadoun, Evangelos Piliouras, Taous‐Meriem Laleg‐Kirati · 2024
Semi-classical signal analysis (SCSA) is a signal representation algorithm based on the semi-classical Schrodinger spectral problem, that suggests the decomposition of the signal into the squared eigenfunctions of this operator. The efficiency of the SCSA decomposition directly depends on the semi-classical parameter of the operator. This implies that an optimal choice of this parameter is needed when aiming for high accuracy and low complexity reconstruction. This optimization is of paramount importance as SCSA signal processing applications; particularly denoising and characterizing biomedical signals, depend all directly on the reconstructed signal. Despite a wide and diverse use, little work has been done in the mathematical analysis of the method. In this context, this paper proposes a robust optimization scheme for the semi-classical parameter based on the Gradient-Descent (GD) algorithm, and is named Backward Gradient Interval Optimization (BGIO). This algorithm aims to yield an appropriate choice of the semi-classical parameter. The analytical expressions for the derivatives of the SCSA components are derived with respect to the semi-classical parameter, and the BGIO algorithm is implemented with a series of considerations explained rigorously. The performance of the proposed algorithm is validated on analytical signals and in-silico multi-cycle PPG data. A derivative of the present work as a denoising algorithm is presented and illustrated using real epileptic intracranial EEG signals.