Beyond the Grid: Continuous Dictionary Pursuit for Interpretable Signal Decomposition

Deep learning models achieve high accuracy in signal processing but remain difficult to interpret. Classical matching pursuit and sparse coding offer better transparency but suffer from spectral leakage and grid mismatch when signal frequencies do not align with discrete dictionary atoms. We introduce Continuous Dictionary Pursuit (CDP), a framework for interpretable signal decomposition that learns the parameters of analytical functions directly from data. Unlike traditional methods, CDP optimizes over a continuous manifold of differentiable atoms to resolve the limitations of fixed grids. The algorithm employs a greedy iterative strategy using data-driven priors and gradient-based optimization to isolate individual signal components. We provide a library of parametric atoms that covers periodic, trend, and transient structures, including discontinuous functions modeled through spectral annealing. To ensure a sparse atom set for reconstruction, the framework utilizes a stopping criterion based on a predefined atom budget and residual error convergence. Evaluations on synthetic and real-world datasets show that CDP resolves spectral leakage for the parameterized atom families considered and recovers exact mathematical components. Beyond decomposition, we demonstrate that the extracted atoms can serve as structured curriculum targets for training sequence models, leading to consistent improvements in forecasting accuracy compared to standard end-to-end training. The method offers a glass-box alternative for signal decomposition by providing high reconstruction fidelity alongside direct physical interpretability.