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

Classical basis expansions like Fourier series and wavelets are analytically tractable but fail to adapt to the empirical structure of modern high-dimensional data. Neural networks overcome this by learning features through layered nonlinear parameterizations, but at the cost of interpretability, mathematical transparency, and explicit control over basis structure. Andrew Kiruluta's new paper presents a non-neural alternative called Data-Driven Variational Basis Learning (DVBL), which treats basis atoms as primary optimization variables and learns them jointly with sample-specific coefficients and, when appropriate, a latent linear evolution operator. The framework remains fully explicit and interpretable while matching the adaptive capacity of deep learning.

DVBL is formulated as a variational optimization problem. Kiruluta proves existence of minimizers and establishes blockwise descent properties for an alternating minimization algorithm, along with conditions for coefficient recovery and basis identifiability. The framework also integrates manifold and dynamical regularization without invoking neural architectures. Compared to classical dictionary learning, spectral methods, Koopman operator methods, and deep representation learning, DVBL offers a unique blend of data adaptivity, mathematical rigor, and interpretability. This could open new avenues for scientific machine learning where model transparency is critical, such as in physics, biology, or finance.