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 functionapproximator that maps a scalar input to a probability distribution over a learnedorthogonal set of basis vectors in a Fourier-encoded feature space. The output is aweighted sum of these probabilities, yielding a smooth, bounded function that de-composes the input domain into a set of mutually exclusive, spectrally interpretableorthogonal states. The unit inherits its structural inspiration from the measurementpostulate of quantum mechanics, though it is computationally a strictly classical,deterministic module.We prove that the SBIU is a universal approximator forcontinuous functions on compact intervals via the Stone-Weierstrass theorem. Weprovide a precise characterization of when decoherence-channel augmentation is re-dundant: specifically, we prove that any trace-preserving channel that is diagonal inthe measurement basis projects algebraically to our minimal projection-and-squareform.By employing the SBIU as a spectral representation bottleneck, our architec-ture achieves competitive unsupervised regime segmentation (ARI = 0.6857), out-performing unconstrained Multi-Layer Perceptrons (ARI = 0.6662) on this syn-thetic task, while providing fully transparent, orthogonal frequency tuning curvesthat standard MLPs lack.Our results demonstrate that strong geometric con-straints—namely, orthogonality, convex combinations, and a learned eigenba-sis—can provide an interpretable alternative to traditional neural networks withoutsacrificing expressive power.Moving beyond synthetic benchmarks, we validate the SBIU on real-world clinicaldatasets spanning neuroscience (EEG) and cardiology (ECG). The architecture suc-cessfully processes physiological spectra to autonomously discover and isolate bothepileptic seizures (with > 90% purity) and ventricular arrhythmias into mutuallyexclusive orthogonal states, entirely without supervised labels.