# Probabilistic Signature Inversion: Learning Conditional Distributions from Truncated Signatures

arXiv: https://arxiv.org/abs/2606.15332 · HTML: https://mlsanity.com/paper/2606.15332/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 29 references that are in the map.
- First posted on arXiv 13 Jun 2026.

## Builds on (most cited first, 13 of 13)

- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation](https://mlsanity.com/paper/2107.03502/) (2021)
- [Autoregressive Denoising Diffusion Models for Multivariate Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2101.12072/) (2021)
- [Diffusion-TS: Interpretable Diffusion for General Time Series Generation](https://mlsanity.com/paper/2403.01742/) (2024)
- [A Primer on the Signature Method in Machine Learning](https://mlsanity.com/paper/1603.03788/) (2016)
- [Deep Signature Transforms](https://mlsanity.com/paper/1905.08494/) (2019)
- [Sig-Wasserstein GANs for Time Series Generation](https://mlsanity.com/paper/2111.01207/) (2021)
- [Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU](https://mlsanity.com/paper/2001.00706/) (2020)
- [A Data-driven Market Simulator for Small Data Environments](https://mlsanity.com/paper/2006.14498/) (2020)
- [Conditional Sig-Wasserstein GANs for Time Series Generation](https://mlsanity.com/paper/2006.05421/) (2020)
- [On Consistency of Signature Using Lasso](https://mlsanity.com/paper/2305.10413/) (2023)
- [The insertion method to invert the signature of a path](https://mlsanity.com/paper/2304.01862/) (2023)

## Abstract (arXiv)

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 reframe 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 formulation 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 faithfully align with the theory-derived baseline, while errors decrease when the statistic is replaced with truncated signatures. Moreover, generated paths faithfully recover the conditioning signature while preserving key distributional and temporal structures, 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.