Adaptively Truncated Signature-based Logistic Regression for Semi-parametric Functional Classification

Abstract

Modern sensing technologies routinely generate multi-dimensional functional data, often accompanied by scalar covariates, motivating statistical methods that can jointly accommodate nonlinear functional effects, complex cross-component dependencies, and irregular sampling patterns. Existing functional logistic regression approaches rely on basis expansions whose performance is critically sensitive to the choice of basis, smoothing parameters, and sampling design; meanwhile, current signature-based methods require heuristic selection of the signature truncation order. We propose Adaptively Truncated Signature-based Logistic Regression (ATSLR), a semi-parametric classification framework that integrates path-signature representations with a data-driven procedure for selecting the signature truncation order via penalized empirical risk minimization. Our method is supported by rigorous theoretical guarantees, including the existence of an optimal truncation order, consistency of its adaptive estimator, convergence rates for the classifier risk, a finite and computable search bound, and a general error-propagation analysis for irregular sampling. Experiments on synthetic and real-world datasets demonstrate that our framework consistently outperforms traditional functional classifiers and fixed-order signature baselines in accuracy, robustness, and interpretability. Our results highlight the practical and theoretical value of integrating rough path theory with adaptive model complexity control.

arXiv comment62 pages, 19 figures