Spectral Path Regression: Directional Chebyshev Harmonics for Interpretable Tabular Learning

Large-scale, high-dimensional tabular regression remains challenging: tree-based models are robust but lack end-to-end representation learning, while deep models enable flexible feature learning but often incur costly interaction modeling and sensitivity to noisy or redundant features. We propose TabNSM, a scalable regression framework that extends our earlier sparse-attention and mixer architectures. At its core, the Adaptive Sparse Interaction Module (ASIM) integrates foreground feature discovery, sparse local interaction encoding, and Feature-Token Mixing, providing near-linear complexity under fixed sparse configurations. For regression, TabNSM introduces three complementary components: a Multi-Stage Regression Head for progressive prediction refinement; GridLoss, an ordinal-aware soft-binning objective that incorporates target structure into representation learning; and RISE (Reweighted Instance Sampling by Error), a difficulty-aware sampling strategy based on loss-quantile bins. Across nine real-world regression benchmarks, TabNSM delivers strong predictive performance and practical scalability, with particularly consistent gains on high-dimensional and heterogeneous datasets. These results demonstrate that selective interaction modeling, structured regression supervision, and difficulty-aware sampling provide an effective and scalable approach to deep tabular regression.

Smooth-basis models such as Chebyshev polynomial regressors and radial basis function (RBF) networks are well established in numerical analysis. Their continuously differentiable prediction surfaces suit surrogate optimisation, sensitivity analysis, and other settings where the response varies gradually with inputs. Despite these properties, smooth models seldom appear in tabular regression, where tree ensembles dominate. We ask whether they can compete, benchmarking models across 55 regression datasets organised by application domain. We develop an anisotropic RBF network with data-driven centre placement and gradient-based width optimisation, a ridge-regularised Chebyshev polynomial regressor, and a smooth-tree hybrid (Chebyshev model tree); all three are released as scikit-learn-compatible packages. We benchmark these against tree ensembles, a pre-trained transformer, and standard baselines, evaluating accuracy alongside generalisation behaviour. The transformer ranks first on accuracy across a majority of datasets, but its GPU dependence, inference latency, and dataset-size limits constrain deployment in the CPU-based settings common across applied science and industry. Among CPU-viable models, smooth models and tree ensembles are statistically tied on accuracy, but the former tend to exhibit tighter generalisation gaps. We recommend routinely including smooth-basis models in the candidate pool, particularly when downstream use benefits from tighter generalisation and gradually varying predictions.

I show that ordinary least squares (OLS) predictions can be rewritten as the output of a restricted attention module, akin to those forming the backbone of large language models. The connection comes from viewing OLS as a similarity-based prediction rule in a learned embedding space. In this representation, least squares does not estimate coefficients per se. Instead, it selects an embedding that minimizes squared prediction error by matching training and test vectors through inner products. This maps directly onto the query-key-value structure of attention mechanisms. I then discuss extensions to dimensionality reduction, nonlinearity, and time series econometrics. Monte Carlo simulations and real-data experiments on UCI/OpenML benchmarks show that nonlinear Attention Regression performs competitively against standard machine learning baselines. In the reverse direction, I replace the attention sublayer of a transformer for tabular data with an explicit regression on polynomial features. The resulting model performs comparably to the standard transformer at a fraction of its parameter count.