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

This paper introduces the Spectral Basis Interpretable Unit (SBIU), a classically inspired, quantum-mechanics-motivated neural module that uses learned orthogonal projections to achieve universal function approximation and unsupervised regime discovery with high interpretability and competitive performance on both synthetic and real-world clinical datasets.

Original authors: Kiarash Mohammadi

Published 2026-08-25

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CC BY 4.0

Original authors: Kiarash Mohammadi

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of modern artificial intelligence, machines are becoming incredibly good at finding patterns in data, yet they often remain black boxes. When a neural network makes a decision, it is usually difficult to understand why it chose one path over another, because the internal math is a tangled web of numbers that do not correspond to anything a human can easily visualize. Scientists have long sought a way to build these systems so that their inner workings are transparent, allowing researchers to see exactly how a machine breaks down a complex signal into simpler parts. This quest for clarity is particularly vital in fields like medicine, where understanding the "why" behind a diagnosis is just as important as the diagnosis itself. A recent development focuses on a new type of building block for these machines, one that forces the system to organize information into distinct, non-overlapping categories, much like sorting a mixed bag of colored marbles into separate jars based on their hue.

A researcher named Kiarash Mohammadi has introduced a new tool called the Spectral Basis Interpretable Unit, or SBIU, which acts as a specialized filter for data. Instead of using the standard, opaque mathematical functions found in most artificial intelligence, this unit takes a single number as input and transforms it into a set of probabilities. Imagine the machine looking at a sound wave or a heartbeat and asking, "How much of this signal belongs to a low-frequency rhythm, and how much belongs to a high-frequency spike?" The SBIU answers this by assigning a percentage to each possible frequency pattern it has learned. Crucially, these percentages must add up to one hundred percent, and the different patterns it identifies are mathematically forced to be completely separate from one another. This design ensures that the machine does not blur the lines between different types of signals, but rather keeps them in distinct, clean categories that a human can inspect and understand.

The power of this approach was tested first on a computer-generated simulation of a machine that switches between three different rhythmic behaviors. The machine was trained to predict the next moment in the signal without being told what the rhythms were. The SBIU successfully learned to separate the data into three distinct groups that matched the hidden rhythms perfectly, achieving a level of accuracy that outperformed standard, unstructured neural networks. More importantly, the researchers could look inside the SBIU and see exactly which frequencies it had identified as the key drivers of each rhythm. The machine had independently discovered the specific speeds of the oscillations, proving that its internal structure was not just a random guess, but a precise map of the underlying physics.

This capability was then put to the test on real-world medical data, specifically recordings of brain waves from patients with epilepsy and heartbeats from patients with arrhythmias. In the brain wave recordings, the system was able to automatically isolate the chaotic electrical storms of a seizure from the normal background activity of the brain, without ever being shown a single labeled example of a seizure. It identified a specific state that contained over ninety percent of the seizure data, separating it cleanly from healthy brain activity and other states like eyes being open or closed. Similarly, when analyzing heartbeats, the system distinguished between normal heartbeats and dangerous irregular ones by focusing on the physical shape and duration of the electrical signal. It correctly identified that abnormal heartbeats had a wider, slower shape compared to the sharp, quick pulses of a healthy heart, matching the way cardiologists diagnose these conditions in the real world.

The researchers also proved that adding complex layers of simulated noise or "decoherence" to the system, a technique sometimes used in other advanced models, was unnecessary and actually redundant for this specific type of unit. They showed mathematically that the core mechanism of the SBIU was already sufficient to handle the task, and that adding extra complexity would only confuse the training process without adding value. This finding simplifies the design, making the unit faster to train and more stable. The study concludes that by imposing strict geometric rules—forcing the machine to keep its categories separate and its outputs as valid probabilities—it is possible to create artificial intelligence that is both powerful and transparent. The result is a system that does not just predict outcomes, but reveals the distinct physical regimes hidden within the data, offering a new way to build machines that work with human understanding rather than in spite of it.

Technical Summary: Spectral Basis Interpretable Unit (SBIU)

1. Problem Statement

Traditional neural network architectures, including the recently proposed Kolmogorov–Arnold Networks (KANs), face a significant "interpretability gap." While KANs offer high flexibility and parameter efficiency by learning univariate edge functions (typically B-splines), these learned functions are local, piecewise polynomials whose coefficients lack semantic meaning. In scientific domains such as neuroscience and cardiology, where understanding the why behind a prediction is as critical as the prediction itself, this opacity limits adoption.

Existing interpretable approaches often rely on human-specified concepts (Concept Bottleneck Models), require regime labels (supervised clustering), or fail to enforce mutually exclusive partitions of the input space (e.g., standard Mixture-of-Experts with unconstrained gating). There is a need for an edge function that:

Partitions the input domain into mutually exclusive regimes without supervision.
Produces human-readable signatures for each regime.
Retains the universal approximation guarantees of spline-based methods.

2. Methodology

The Spectral Basis Interpretable Unit (SBIU)

The author introduces the SBIU, a univariate function approximator that replaces the classical B-spline with a learned orthogonal projection in a Fourier-encoded feature space. The architecture is inspired by the measurement postulate of quantum mechanics but operates as a strictly classical, deterministic module.

Core Components:

Fourier Feature Map: A scalar input $x$ is mapped to an $L_2$ -normalized $d$ -dimensional real vector $|\psi(x)\rangle$ using fixed sine and cosine features at frequencies $k\omega_0$ .
Learned Orthogonal Basis (Pointer States): A learnable special orthogonal matrix $V \in SO(d)$ rotates the input vector. The columns of $V$ define an orthonormal basis $\{|\pi_m\rangle\}$ .
Squared-Modulus Activation (Born Rule): The input is projected onto the basis vectors, and the squared moduli of these projections are computed: $p_m(x) = |\langle \pi_m | \psi(x) \rangle|^2$ . Due to the orthogonality of $V$ and normalization of $|\psi(x)\rangle$ , these values form a valid probability distribution ( $\sum p_m = 1$ ).
Convex Output: The final scalar output is a convex combination of these probabilities: $\phi(x) = \sum \lambda_m p_m(x)$ , where $\lambda_m$ are learnable coefficients.

Mathematical Formulation:
The SBIU computes a quadratic form over the Fourier-encoded input:
$\phi(x) = \langle \psi(x) | H_{\text{eff}} | \psi(x) \rangle$
where $H_{\text{eff}} = V \Lambda V^\top$ is a learned Hermitian (symmetric) operator. The pointer states are the eigenvectors of $H_{\text{eff}}$ , and the coefficients $\lambda_m$ are its eigenvalues.

Theoretical Proofs

Universal Approximation: The author proves via the Stone–Weierstrass theorem that the SBIU is a universal approximator for continuous functions on compact intervals, provided the dimension $d$ is sufficiently large.
Redundancy of Decoherence Channels: A key theoretical contribution is the proof (Theorem 1) that augmenting the SBIU with trace-preserving decoherence channels (Kraus operators) that are diagonal in the measurement basis is mathematically redundant. The measurement projection induces an algebraic collapse where the noise parameters vanish, yielding the same result as the minimal projection-and-square form. This eliminates the need for complex, unstable quantum-inspired noise simulations.

The Spectral Bottleneck Architecture

To validate the unit's clustering power, the author constructs the SBI-Bottleneck, a minimal end-to-end differentiable architecture. It takes multivariate time-series windows, applies a linear projection, normalizes the state, and passes it through a single SBIU. The unit is trained using a self-supervised next-step prediction loss (MSE) without any regime labels.

3. Key Contributions

Definition and Characterization: Rigorous mathematical definition of the SBIU, including proofs of its universal approximation capabilities and the redundancy of diagonal decoherence channels.
Spectral Bottleneck Architecture: A novel architecture leveraging SBIUs for unsupervised regime discovery in multivariate time series.
Synthetic Validation: Demonstration on a hidden-regime oscillator that the SBIU not only segments data accurately but learns pointer states corresponding exactly to known generator frequencies.
Clinical Validation: Successful application to real-world physiological data (EEG and ECG), showing the ability to autonomously isolate pathological regimes (epileptic seizures and ventricular arrhythmias) into mutually exclusive orthogonal states without supervised labels.

4. Experimental Results

Synthetic Hidden-Regime Oscillator

The model was tested on a time series switching between three regimes (2 Hz, 3 Hz, and 5 Hz oscillators).

Performance: The SBI-Bottleneck achieved an Adjusted Rand Index (ARI) of 0.6857, outperforming unconstrained Multi-Layer Perceptrons (MLP, ARI 0.6662), Spline-KANs (ARI 0.6459), and Fourier-KANs (ARI 0.1827).
Interpretability: The learned basis vectors aligned perfectly with the generator frequencies. The orthogonal constraints forced the model to decompose the signal into independent, non-overlapping spectral components, whereas unconstrained models produced entangled representations.
Robustness: The SBIU maintained stable regime boundaries under increasing Gaussian noise, while unconstrained baselines suffered representation collapse.

Real-World Clinical Validation

EEG (Epileptic Seizure Recognition):

The model autonomously isolated epileptic seizures with >90% purity (specifically 90.3% for World 7 in the primary d=10 experiment, and 93.6% for World 11 in the refined d=12 sub-regime analysis) without labels.
It successfully identified "Eyes Closed" states via Alpha wave rhythms (8–12 Hz) and distinguished tumor/healthy areas from baseline states.
The architecture naturally sparsified the representation, zeroing out unused dimensions.

ECG (MIT-BIH Arrhythmia Dataset):

The SBIU isolated Ventricular ectopic beats and Unclassiﬁed beats into a single coordinate axis (World 3) and Fusion beats into another (World 7).
Morphological Validation: The model's separation of arrhythmias aligned with clinical diagnostics. Heartbeats routed to the "arrhythmia" world exhibited a widened QRS complex (mean width 46.89 samples vs. 8.85 for normal), matching the physiological signature of ventricular ectopic beats.

5. Significance and Claims

The paper claims that the SBIU demonstrates that strong geometric constraints—specifically orthogonality, convex combinations, and a learned eigenbasis—can provide a fully interpretable alternative to traditional neural networks without sacrificing expressive power.

Interpretability without Labels: The architecture autonomously discovers and isolates pathological regimes in physiological data, providing a "transparent dashboard" of system dynamics where basis vectors correspond to physical frequencies or morphological features.
Efficiency: The SBIU acts as a parameter-sparse regularizer. It achieved peak performance with the minimum geometric capacity required ( $d=6$ for 3 frequencies), whereas unconstrained models required double the dimensions to achieve comparable segmentation.
Theoretical Clarity: By proving the redundancy of diagonal decoherence channels, the work simplifies quantum-inspired machine learning, removing optimization traps and unnecessary parameters while retaining the benefits of the Born-rule activation.

The author concludes that the SBIU offers a rigorous framework for regime discovery, bridging the gap between high-performance function approximation and human-readable scientific insight, particularly in domains governed by physical laws of wave propagation and harmonic resonance.