Probabilistic Signature Inversion Learns Path Distributions from Truncated Signatures

The signature transform is a powerful mathematical tool for representing continuous-time paths, known for its unique and universal properties. However, the inverse problem of reconstructing a path from its truncated signature is inherently ill-posed because the truncated signature map is not injective, meaning multiple paths can share the same truncated signature. This paper reformulates the challenge as a probabilistic problem: learning the conditional distribution of a path given its truncated signature. To practically estimate this, the researchers adopt a signature-conditioned flow matching model. This probabilistic framework helps to quantify the fundamental difficulty of inversion by using Bayes reconstruction error to measure the irreducible uncertainty. The study derives a closed-form Bayes-optimal error for linear statistics in specific models like log-GBM and provides numerically tractable formulas for others. Experimental results show that the empirical reconstruction errors align well with the theoretical baseline, and the model effectively recovers conditioning signatures while preserving key distributional and temporal structures, with applications demonstrated using real financial data.