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Spectral Path Regression: Directional Chebyshev Harmonics for Interpretable Tabular Learning

Milo Coombs

Abstract

Classical approximation bases such as Chebyshev polynomials provide principled and interpretable representations, but their multivariate tensor-product constructions scale exponentially with dimension and impose axis-aligned structure that is poorly matched to real tabular data. We address this by replacing tensorised oscillations with directional harmonic modes of the form $\cos(\mathbf{m}^{\top}\arccos(\mathbf{x}))$, which organise multivariate structure by direction in angular space rather than by coordinate index. This representation yields a discrete spectral regression model in which complexity is controlled by selecting a small number of structured frequency vectors (spectral paths), and training reduces to a single closed-form ridge solve with no iterative optimisation. Experiments on standard continuous-feature tabular regression benchmarks show that the resulting models achieve accuracy competitive with strong nonlinear baselines while remaining compact, computationally efficient, and explicitly interpretable through analytic expressions of learned feature interactions.

Spectral Path Regression: Directional Chebyshev Harmonics for Interpretable Tabular Learning

Abstract

Classical approximation bases such as Chebyshev polynomials provide principled and interpretable representations, but their multivariate tensor-product constructions scale exponentially with dimension and impose axis-aligned structure that is poorly matched to real tabular data. We address this by replacing tensorised oscillations with directional harmonic modes of the form , which organise multivariate structure by direction in angular space rather than by coordinate index. This representation yields a discrete spectral regression model in which complexity is controlled by selecting a small number of structured frequency vectors (spectral paths), and training reduces to a single closed-form ridge solve with no iterative optimisation. Experiments on standard continuous-feature tabular regression benchmarks show that the resulting models achieve accuracy competitive with strong nonlinear baselines while remaining compact, computationally efficient, and explicitly interpretable through analytic expressions of learned feature interactions.

Paper Structure

This paper contains 58 sections, 34 equations, 7 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work & Notation
  3. Related Work
  4. Spectral feature methods and kernel approximations.
  5. Tree ensembles and tabular learning.
  6. Harmonic moments
  7. Interpretability.
  8. Symbolic Regression and Analytic Discovery
  9. Notation
  10. Background
  11. Chebyshev Polynomials
  12. Chebyshev Series and Multivariate Extensions
  13. From Tensor Products to Spectral Paths
  14. Directional Harmonics as a Change of Basis
  15. Spectral Paths, Primitive Rays, and Computational Structure
  16. ...and 43 more sections

Figures (7)

  • Figure 1: Geometric interpretation of Chebyshev polynomials. The input $x\in[-1,1]$ is lifted to an angle $\theta=\arccos(x)$ on the upper semicircle, rotated by a factor $n$, and projected onto the horizontal axis, yielding $T_n(x)=\cos(n\theta)$.
  • Figure 2: In a multivariate Chebyshev series, each variable is lifted to an angular coordinate, rotated independently by its frequency, and projected onto its coordinate axis. The resulting scalar projections combine multiplicatively, producing a rectangular “shadow” on the base plane whose area equals $\cos(m_1\theta_1)\times\cos(m_2\theta_2)$.
  • Figure 3: The new basis: directional harmonics defined by projections onto frequency vectors.
  • Figure 4: Training and validation performance of the discrete spectral model on the UCI Concrete dataset as a function of the number of paths $\mathcal{M}$. The model achieves stable generalisation after around $\mathcal{M}=9$, where validation $R^2$ plateaus and NRMSE$_\sigma$ reaches a minimum.
  • Figure 5: Feature importance on the Concrete dataset computed from normalised analytic sensitivities.
  • ...and 2 more figures