Additive models buy interpretability by forbidding feature interactions, a constraint that neural instantiations enforce architecturally. We introduce the quadrilateral loss, a differentiable penalty that treats additivity as a measurable behavior instead: a second-order mixed difference on pairs of training points swapping one coordinate, which vanishes if and only if the coordinate carries no interaction, remains informative for piecewise-linear networks, and equals in expectation the per-coordinate interaction mass of the interventional Shapley-GAM. The loss turns additivity into a dial - most learned interactions prove removable almost for free, and on small datasets a moderate penalty improves accuracy and additivity simultaneously - and into an online observable: its per-feature surrender curves show, across seeds and datasets, that pre-regularization interaction magnitude barely predicts what a regularized model retains, undermining post-hoc interaction rankings. Against this instrument we compare routes to exact additivity, spanning structural masks, behavioral penalties (optionally crystallized into exact structure), weight decay, backfitting, the shared-section model, and bagged boosted stumps: constraining behavior before structure dominates weight-space constraints, rankings reverse between data regimes, and converging routes agree on the shape functions themselves. Three silent failure modes we document share one anatomy: guarantees imported into settings that quietly void their preconditions.
Deep neural networks (DNNs) exhibit attractive performance in various fields but often suffer from low interpretability. The neural additive model (NAM) and its variant called the neural basis model (NBM) use neural networks (NNs) as nonlinear shape functions in generalized additive models (GAMs). Both models are highly interpretable and exhibit good performance and flexibility for NN training. NAM and NBM can provide and visualize the contribution of each feature to the prediction owing to GAM-based architectures. However, when using two-input NNs to consider feature interactions or when applying them to high-dimensional datasets, training NAM and NBM becomes intractable due to the increase in the computational resources required. This paper proposes incorporating the feature selection mechanism into NAM and NBM to resolve computational bottlenecks. We introduce the feature selection layer in both models and update the selection weights during training. Our method is simple and can reduce computational costs and model sizes compared to vanilla NAM and NBM. In addition, it enables us to use two-input NNs even in high-dimensional datasets and capture feature interactions. We demonstrate that the proposed models are computationally efficient compared to vanilla NAM and NBM, and they exhibit better or comparable performance with state-of-the-art GAMs.
Yasutoshi Kishimoto, Kota Yamanishi, Takuya Matsuda +1
Interpretable machine learning requires models that are accurate and structurally faithful to the data. Existing explainability methods rely heavily on additive representations (e.g., Generalized Additive Models (GAMs), SHapley Additive exPlanations (SHAP), functional ANOVA), which can suffer from signal cancellation and off-support extrapolation in the presence of strong interactions. We propose Tensor Separation Learning (TSL), a regression model that learns a sum of rank-1 products of univariate per-feature functions via a stagewise greedy procedure with orthogonal refitting. By enforcing separability, TSL avoids the information loss inherent in additive projections caused by marginalizing higher-order interactions. The learned TSL model can be fully reconstructed from first-order partial dependence functions, up to constant factors. This stage-wise correspondence ensures that the resulting visualizations are faithful to the fitted components. We establish approximation-rate guarantees for functions with bounded mixed p-th order partial derivatives and demonstrate that TSL competes with black-box models on regression benchmarks.
Understanding nonlinear feature interactions is crucial in science and engineering, yet standard multilayer perceptrons (MLPs) often capture such interactions only implicitly, leading to entangled representations that can impair robustness and interpretability. We investigate product-unit residual networks (PURe) that integrate multiplicative product units with residual connections to explicitly model cross-feature couplings while stabilizing optimization. We conduct a systematic evaluation on an interaction-driven synthetic benchmark and two real-world datasets, assessing predictive accuracy, robustness to Gaussian feature noise, and performance under limited training data, and we compare real- and complex-valued variants under a matched parameter budget. Beyond accuracy, SHapley Additive exPlanations (SHAP)-based interaction analyses show that PURe learns more concentrated and structurally coherent interaction patterns than MLP baselines. Overall, PURe achieves competitive or improved performance, better robustness and sample efficiency in low-data regimes, and enhanced interaction-level interpretability.