用方向性谐波构建可解释的表格数据回归模型
Spectral Path Regression: Directional Chebyshev Harmonics for Interpretable Tabular Learning
- 以方向谐波替代传统张量积多项式，按角度空间组织多维结构
- 仅需少量频向量即可控制模型复杂度，训练一步求解无需迭代
- 精度媲美非线性模型，且能解析表达特征交互关系
经典逼近基如切比雪夫多项式虽具可解释性，但其多维张量积构造随维度指数增长，且强加轴对齐结构，与真实表格数据不匹配。本文将张量化振荡替换为形式为 $\cos(\mathbf{m}^{ op}\arccos(\mathbf{x}))$ 的方向性谐波模式，按角度空间方向组织多维结构。该表示构建了离散谱回归模型，通过选择少量结构化频向量（谱路径）控制复杂度，训练退化为单步闭式岭回归求解，无需迭代优化。在标准连续特征表格回归基准上，所提模型达到与强非线性基线相当的精度，同时保持紧凑、高效，并可通过解析表达式显式解释特征交互。
原文摘要 · Abstract (English)
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.
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