Probabilistic Signature Inversion: Learning Conditional Distributions from Truncated Signatures

2026-06-13 · Source: Machine Learning · Field: Technology & Digital — Artificial Intelligence & Machine Learning · Depth: Expert, quick

Summary

Probabilistic Signature Inversion (PSI) addresses the ill-posed problem of recovering a continuous-time path from its truncated signature, a feature map valued for uniqueness. The non-injective nature of the truncated signature map makes direct inversion difficult. This work reframes the challenge as learning the conditional distribution of a path given its truncated signature, employing a signature-conditioned flow matching model for estimation. The framework quantifies irreducible uncertainty using Bayes reconstruction error, deriving a closed-form Bayes-optimal error for log-GBM under linear statistics, with numerically tractable formulas for log-fBM and OU. Experiments confirm empirical reconstruction errors align with this theoretical baseline, showing further reduction when using truncated signatures. The model generates paths that faithfully recover conditioning signatures while preserving key distributional and temporal structures, demonstrating applicability to real financial data beyond theoretical parametric processes.

Key takeaway

For Machine Learning Engineers developing models for time-series reconstruction or financial data analysis, this probabilistic signature inversion framework offers a robust approach to an inherently ill-posed problem. You should consider integrating signature-conditioned flow matching models to accurately recover path dynamics from truncated signatures, especially when preserving distributional and temporal structures is critical. This method provides a quantifiable baseline for irreducible uncertainty, enabling more rigorous model validation and deployment in real-world financial applications.

Key insights

Truncated signature inversion is reframed as learning a path's conditional distribution using flow matching, quantifying irreducible uncertainty.

Principles

• Truncated signature inversion is structurally ill-posed.
• Probabilistic reframing quantifies irreducible uncertainty.
• Richer statistics reduce reconstruction error.

Method

A signature-conditioned flow matching model estimates the conditional distribution of a path given its truncated signature, providing a practical estimator for inversion.

In practice

• Apply to real financial data for path reconstruction.
• Use Bayes error for model validation baselines.
• Recover path structures from truncated signatures.

Topics

• Signature Transform
• Probabilistic Inversion
• Flow Matching Models
• Conditional Distributions
• Financial Time Series
• Bayes Reconstruction Error

Best for: AI Scientist, Research Scientist, Machine Learning Engineer

Related on AIssential

• Flow-Based Conformal Predictive Distributions — Nonconformity flows enable efficient sampling of conformal prediction set boundaries in high-dimensional spaces, leading to calibrated predictive distributions.
• Inverse Entropic Optimal Transport Solves Semi-supervised Learning via Data Likelihood Maximization — Inverse entropic optimal transport provides a unified likelihood maximization framework for semi-supervised conditional distribution learning using mixed data.
• Separating Intrinsic Ambiguity from Estimation Uncertainty in Deep Generative Models for Linear Inverse Problems — A new method separates intrinsic ambiguity from estimation uncertainty in deep generative models for inverse problems.
• Diffusion Flow Matching: Dimension-Improved KL Bounds and Wasserstein Guarantees — DFM's theoretical convergence properties are now better understood with new dimension-improved KL and Wasserstein bounds.
• Partial Causal Structure Learning for Valid Selective Conformal Inference under Interventions — Selective conformal prediction under interventions can be robustly maintained by learning partial causal structures.
• Uncertainty Quantification for Large Language Diffusion Models — Lightweight, zero-shot uncertainty quantification for LLDMs can achieve high performance with significantly reduced computational cost.

Open in AIssential →

Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning.