A Globally Convergent Variational Framework for Mode Number Detection via Spectral Cutting Curves
Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Authors: Zhong, Chenjie, Li, Zhipeng, Xu, Shangzhi, Li, Xiaohu, Zhang, Luodan, Yuan, Jianjun
Institution: Tongji University
Language: English
Subfield: Control and Systems Engineering
Field: Engineering
Domain: Physical Sciences
Open Access status: green
Awards: G1340215136, G351996751, G8505979135
OpenAlex ID: W7119181156