Spectral Basis Interpretable Unit: A Learned Orthogonal Projection Module for Interpretable Function Approximation and Unsupervised Regime Discovery

We introduce the Spectral Basis Interpretable Unit (SBIU)—a univariate function approximator that maps a scalar input to a probability distribution over a learned orthogonal set of basis vectors in a Fourier-encoded feature space. The output is a weighted sum of these probabilities, yielding a smooth, bounded function that decomposes the input domain into a set of mutually exclusive, spectrally interpretable orthogonal states. The unit inherits its structural inspiration from the measurement postulate of quantum mechanics, though it is computationally a strictly classical, deterministic module. We prove that the SBIU is a universal approximator for continuous functions on compact intervals via the Stone-Weierstrass theorem. We provide a precise characterization of when decoherence-channel augmentation is redundant: specifically, we prove that any trace-preserving channel that is diagonal in the measurement basis projects algebraically to our minimal projection-and-square form. By employing the SBIU as a spectral representation bottleneck, our architecture achieves competitive unsupervised regime segmentation (ARI = 0.6857), outperforming unconstrained Multi-Layer Perceptrons (ARI = 0.6662) on this synthetic task, while providing fully transparent, orthogonal frequency tuning curves that standard MLPs lack. Our results demonstrate that strong geometric constraints—namely, orthogonality, convex combinations, and a learned eigenbasis—can provide an interpretable alternative to traditional neural networks without sacrificing expressive power. Moving beyond synthetic benchmarks, we validate the SBIU on real-world clinical datasets spanning neuroscience (EEG) and cardiology (ECG). The architecture successfully processes physiological spectra to autonomously discover and isolate both epileptic seizures (with > 90% purity) and ventricular arrhythmias into mutually exclusive orthogonal states, entirely without supervised labels.