Probabilistic Signature Inversion: Learning Conditional Distributions from Truncated Signatures

The signature transform is a principled feature map for continuous-time paths, valued for its uniqueness and universality. Recovering a path from its truncated signature is, however, structurally ill-posed because the truncated signature map is not injective. We therefore reformulate truncated signature inversion as a probabilistic problem---learning the conditional distribution of a path given its truncated signature---and adopt a signature-conditioned flow matching model as a practical estimator. This probabilistic reformulation elucidates the fundamental difficulty of inversion: Bayes reconstruction error quantifies the irreducible uncertainty remaining after conditioning on a statistic. We derive the Bayes-optimal error under linear statistics, obtaining a closed form for log-GBM and numerically tractable formulas for log-fBM and OU---yielding a concrete theoretical baseline for model validation. This baseline upper-bounds the Bayes error under truncated-signature conditioning, since truncated signatures provide richer information than linear statistics. Experiments show that empirical reconstruction errors under linear-statistics conditioning closely match the theory-derived baseline, while errors decrease when the statistic is replaced with the truncated signature. Moreover, generated paths faithfully recover the conditioning signature while preserving key distributional and temporal structure, indicating that the estimator is well calibrated to the target conditional distribution. Together, these results establish a well-posed probabilistic framework for truncated-signature inversion, with applicability demonstrated on real financial data beyond the parametric process families covered by theory.