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Data-Driven Variational Basis Learning Beyond Neural Networks: A Non-Neural Framework for Adaptive Basis Discovery

Classical representation systems such as Fourier series, wavelets, and fixed dictionaries provide analytically tractable basis expansions, but they are not intrinsically adapted to the empirical structure of modern high-dimensional data. Neural networks overcome this limitation by learning features from data, yet they do so through layered nonlinear parameterizations that often sacrifice interpretability, explicit control over basis structure, and mathematical transparency. In this manuscript we develop a non-neural alternative that learns basis functions directly from data through variational optimization. The proposed framework, termed Data Driven Variational Basis Learning (DVBL), treats basis atoms as primary optimization variables and learns them jointly with sample-specific coefficients and, when appropriate, a latent linear evolution operator. This yields a data-adaptive basis expansion that remains explicit, interpretable, and amenable to rigorous analysis. We formulate the model, establish existence of minimizers, prove blockwise descent properties for an alternating minimization algorithm, give conditions for coefficient recovery and basis identifiability, and show how manifold and dynamical regularization can be integrated without invoking neural architectures. We also discuss the conceptual novelty of the framework relative to classical dictionary learning, spectral methods, Koopman operator methods, and deep representation learning.

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