Semi-parametric Functional Classification via Path Signatures Logistic Regression

TL;DR

This paper introduces Path Signatures Logistic Regression (PSLR), a novel semi-parametric method that uses path signatures to classify irregularly sampled functional data more accurately and robustly than traditional models.

Contribution

The paper develops PSLR, a new basis-free, nonlinear classification framework leveraging path signatures, with theoretical guarantees and superior empirical performance on real and synthetic data.

Findings

PSLR outperforms traditional classifiers in accuracy.

PSLR is robust to irregular sampling schemes.

Theoretical guarantees for estimation and risk bounds.

Abstract

We propose Path Signatures Logistic Regression (PSLR), a semi-parametric framework for classifying vector-valued functional data with scalar covariates. Classical functional logistic regression models rely on linear assumptions and fixed basis expansions, which limit flexibility and degrade performance under irregular sampling. PSLR overcomes these issues by leveraging truncated path signatures to construct a finite-dimensional, basis-free representation that captures nonlinear and cross-channel dependencies. By embedding trajectories as time-augmented paths, PSLR extracts stable, geometry-aware features that are robust to sampling irregularity without requiring a common time grid, while still preserving subject-specific timing patterns. We establish theoretical guarantees for the existence and consistent estimation of the optimal truncation order, along with non-asymptotic risk bounds. Experiments on synthetic and real-world datasets show that PSLR outperforms traditional functional classifiers in accuracy, robustness, and interpretability, particularly under non-uniform sampling schemes. Our results highlight the practical and theoretical benefits of integrating rough path theory into modern functional data analysis.