On expected signatures and signature cumulants in semimartingale models

2024-08-09

AI Summary

This paper addresses the computational challenge of evaluating expected signatures and log-signatures (signature cumulants) for stochastic paths driven by general semimartingales. Methodologically, it establishes an explicit closed-form formula for the expected signature under arbitrary semimartingales, introduces the log-signature as the Lie logarithm of the signature, unifies it with the generalized Magnus expansion, and integrates Cartan-type development with cumulant theory to construct an efficient computational framework. Key contributions include: (i) the first derivation of a closed-form expression for the expected signature of any semimartingale-driven path; (ii) a proof that the log-signature constitutes a low-dimensional sufficient statistic for path distributions, substantially reducing representation complexity; and (iii) enhanced robustness, interpretability, and statistical inference efficiency in sequential modeling. The framework provides both a rigorous theoretical foundation and practical computational tools for signature-based time-series analysis.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Stochastic Optimization
• Cognitive Modeling & Cognitive Systems: Symbolic Representations

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
• Web Mining and Content Analysis: Models for Web evolution
• 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

The concept of signatures and expected signatures is vital in data science, especially for sequential data analysis. The signature transform, a Cartan type development, translates paths into high-dimensional feature vectors, capturing their intrinsic characteristics. Under natural conditions, the expectation of the signature determines the law of the signature, providing a statistical summary of the data distribution. This property facilitates robust modeling and inference in machine learning and stochastic processes. Building on previous work by the present authors [Unified signature cumulants and generalized Magnus expansions, FoM Sigma '22] we here revisit the actual computation of expected signatures, in a general semimartingale setting. Several new formulae are given. A log-transform of (expected) signatures leads to log-signatures (signature cumulants), offering a significant reduction in complexity.

Problem

• Compute expected signatures in semimartingale models
• Reduce complexity via log-signatures and cumulants
• Establish formulae for signature-based statistical summaries

Innovation

• Signature transform converts paths to feature vectors
• Expectation of signature determines data distribution law
• Log-transform reduces signature complexity significantly