Identifiable Convex-Concave Regression via Sub-gradient Regularised Least Squares

Abstract:We propose a novel nonparametric regression method that models complex input-output relationships as the sum of convex and concave components. The method-Identifiable Convex-Concave Nonparametric Least Squares (ICCNLS)-decomposes the target function into additive shape-constrained components, each represented via sub-gradient-constrained affine functions. To address the affine ambiguity inherent in convex-concave decompositions, we introduce global statistical orthogonality constraints, ensuring that residuals are uncorrelated with both intercept and input variables. This enforces decomposition identifiability and improves interpretability. We further incorporate L1, L2 and elastic net regularisation on sub-gradients to enhance generalisation and promote structural sparsity. The proposed method is evaluated on synthetic and real-world datasets, including healthcare pricing data, and demonstrates improved predictive accuracy and model simplicity compared to conventional CNLS and difference-of-convex (DC) regression approaches. Our results show that statistical identifiability, when paired with convex-concave structure and sub-gradient regularisation, yields interpretable models suited for forecasting, benchmarking, and policy evaluation.

Withdrawn by the author because the identifiability result in Sections 3.3-3.4 is incorrect. Residual orthogonality depends only on the fitted sum and therefore cannot identify the convex-concave decomposition; the proof also incorrectly treats the difference of two concave functions as necessarily affine. A substantially revised manuscript is in preparation

Machine Learning (stat.ML); Machine Learning (cs.LG); Statistics Theory (math.ST); Applications (stat.AP)

arXiv:2506.18078 [stat.ML]

(or arXiv:2506.18078v2 [stat.ML] for this version)