Constrained Wavelet Learning via Variational Inequalities: Theory and Application to ECG Signal Analysis

We present a rigorous variational inequality framework for learning application-specific wavelets under mathematical admissibility constraints. The problem is formulated as finding a wavelet function in a convex constrained functional space that maximizes frequency localization objectives. The constraint set is characterized by zero-mean (admissibility), energy constraint (frame stability), and time-frequency localization conditions, forming a closed convex set in . Since the energy functional is convex (quadratic), we maximize it over the convex set using a variational inequality characterization and projected gradient ascent algorithm. We employ B-spline parameterization to reduce the infinite-dimensional problem to a finite-dimensional variational inequality on and establish convergence under appropriate conditions. Numerical experiments on electrocardiogram (ECG) signal analysis demonstrate the effectiveness of the theoretical framework, yielding wavelets with 68-85% energy concentration in target frequency bands while satisfying all mathematical constraints to machine precision.