A Globally Convergent Variational Framework for Mode Number Detection via Spectral Cutting Curves

Automatically determining the number of intrinsic mode functions (IMFs) and their center frequencies in Variational Mode Decomposition (VMD) remains an open mathematical challenge. Existing methods rely on heuristic settings, trial-and-error, or recursive extraction lacking theoretical convergence guarantees. We propose a variational framework that endogenously determines the number of modes. Any curve below the spectral amplitude divides the area under the spectrum into 2 parts and generate the connected intervals where spectrum locates above it, whose count defines the modal number K[g] – a topological functional induced by the cutting curve. Since K[g] is discontinuous and intractable for direct optimization, we seek the optimal cutting curve as a continuous variational surrogate: it separates distinct spectral peaks into individual regions above it while merging noise-induced fragments below. This surrogate adversarially maximizes the integral of g while penalizing its curvature, transforming the problem into iteratively solving a fourth-order boundary value problem via Lagrangian duality. We establish a rigorous proof of global convergence for the dual ascent algorithm in function space. Comprehensive numerical experiments on artificial and real-world signals including ECG data show accurate estimates of IMFs and center frequencies, avoiding redundant modes while ensuring recovery of necessary components, providing a robust, theoretically grounded initialization routine for VMD.