Semi-parametric Functional Classification via Path Signatures Logistic Regression

2025-07-09

AI Summary

Classification of vector-valued functional data with scalar covariates is challenging under irregular sampling, as conventional semiparametric methods rely on restrictive linear assumptions and fixed basis expansions, leading to substantial performance degradation. Method: We propose Path Signature Logistic Regression (PSLR), a geometrically principled approach grounded in rough path theory. PSLR employs truncated path signatures to construct finite-dimensional, basis-free representations that inherently capture temporal nonlinearities and inter-channel dependencies, while remaining robust to irregular sampling. It integrates time-augmented path embeddings with semiparametric logistic regression to ensure interpretability. Contribution/Results: We theoretically establish the existence of an optimal truncation level and prove consistency of the estimator. Empirical evaluations on synthetic and real-world datasets demonstrate that PSLR significantly outperforms state-of-the-art functional classification methods—particularly under non-uniform sampling—achieving superior accuracy, robustness, and interpretability.

Technology Category

• Machine Learning: Statistical Relational/Logic Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Computer Vision: Representation Learning for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Classifying vector-valued functional data with scalar covariates
• Overcoming limitations of linear assumptions and fixed basis expansions
• Handling irregular sampling and preserving subject-specific timing patterns

Innovation

• Leverages truncated path signatures for representation
• Embeds trajectories as time-augmented paths
• Estimates optimal truncation order theoretically