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

TL;DR

This paper introduces ICCNLS, a new nonparametric regression method that decomposes complex functions into identifiable convex and concave parts using sub-gradient regularisation, improving interpretability and predictive accuracy.

Contribution

The paper presents ICCNLS, a novel approach that ensures identifiability of convex-concave decompositions through orthogonality constraints and enhances model performance with sub-gradient regularisation.

Findings

ICCNLS outperforms traditional CNLS and DC regression in accuracy.

The method improves interpretability of complex models.

Regularisation promotes sparsity and better generalisation.

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.