Decomposition for symbolic compactness: complexity-penalised pursuit of parametric mode families

Abstract

Variational mode decomposition extracts modes by penalising spectral bandwidth, yet its modes need not correspond to a single analytic family, and their quality can depend on centre-frequency initialisation. We argue that for signals whose components are \emph{a priori} describable by low-dimensional parametric families, decomposition should be posed directly as complexity-penalised approximation by a sum of symbolic components, dispensing with frequency-localisation machinery. The resulting pursuit over a library of parametric atoms (sinusoid, damped sinusoid, chirp, exponential, logarithm, low-order polynomial) uses data-driven candidate generation, BIC-governed greedy addition with a noise-floor stop, block-coordinate and joint nonlinear refinement, and compound split moves with co-removal of redundant components. On the five identifiable stationary synthetic cases, 20-seed full-structure recovery rates range from 0.65 to 1.00, with component MSE down to $10^{-6}$; the method resolves a 0.6~Hz pair below the Rayleigh limit of a one-second record. In a 20-seed paired study on all eight cases it reduces periodic-component MSE over initialisation-corrected VMD by median factors of 1653 (30~dB), 129 (5~dB) and 49 (0~dB), every stationary case at the minimum attainable significance; on semi-synthetic bearing signals (CWRU with an injected component) it recovers the injected component two orders more accurately than VMD; and on measured rotating-machine records in three operating conditions the recovered frequencies admit stable, nominally constrained shaft-harmonic lattice fits and separate recurring off-lattice lines by the same test. We analyse the penalised objective's frequency bias under unmodelled exponential content.