The Linear Series Decomposition Learner (LSDL): A Multi-Geometric Theory of Signal Structure and Representation

Signal representation underpins modern signal processing, yet many existing methods primarily transform signals into alternative domains without explicitly modelling how informative signal structure evolves during recursive localisation. This paper presents the Linear Series Decomposition Learner (LSDL), a multi-geometric theory of signal structure and representation founded on recursive support localisation. The LSDL is formulated as a recursive localisation operator acting on a fixed amplitude-reference domain, thereby establishing a mathematically rigorous framework for analysing the evolution of signal support across successive localisation levels. Theoretical analysis characterises the fundamental properties of the operator, including support evolution, monotonicity, finite recursion, perturbation stability, and admissible localisation, and it thereby provides a formal foundation for recursive signal decomposition. Building upon this operator-theoretic formulation, the proposed framework establishes that recursive support localisation induces multiple complementary geometries of signal structure. These comprise support geometry, which describes the organisation of retained signal support; discriminative geometry, which characterises class separability under recursive localisation; information geometry, which quantifies entropy redistribution and information concentration; persistence geometry, which models the emergence, evolution, and lifetime of localised signal structures across recursive filtrations; and spectral geometry, which describes recursion-induced reorganisation within the frequency domain. Collectively, these complementary geometries provide a coherent multi-geometric representation that captures structural, statistical, topological, and spectral characteristics within a common mathematical framework. The proposed theory is supported through analytical development and empirical evaluation using synthetic benchmark signals, real-world electromyographic (EMG) datasets, and comparative analyses against established signal representation approaches, including the short-time Fourier transform (STFT), wavelet transforms, empirical mode decomposition (EMD), variational mode decomposition (VMD), and sparse coding. Experimental results demonstrate that recursive support localisation produces interpretable multi-geometric representations while maintaining competitive classification performance and low online computational cost. By establishing recursive support localisation as a principled mechanism through which complementary signal geometries emerge, the LSDL provides a mathematically grounded framework for interpretable signal representation, structural analysis, and representation learning across diverse signal-processing applications.