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.

翻译：签名变换是连续时间路径的一种有原则的特征映射，以其唯一性和普适性而受到重视。然而，从截断签名中恢复路径在结构上具有不适定性，因为截断签名映射并非单射。因此，我们将截断签名反演重新构建为概率问题——学习给定截断签名的路径的条件分布——并采用签名条件的流匹配模型作为实用估计器。这种概率公式揭示了反演的基本困难：贝叶斯重建误差量化了在基于统计量进行条件化后仍存在的不可约不确定性。我们推导了线性统计量下的贝叶斯最优误差，获得了对数几何布朗运动的封闭形式以及对数分数布朗运动与奥恩斯坦-乌伦贝克过程的数值可处理公式，为模型验证提供了具体的理论基线。由于截断签名比线性统计量提供更丰富的信息，该基线是截断签名条件化下贝叶斯误差的上界。实验表明，在线性统计量条件化下，经验重建误差与理论推导的基线高度吻合，而当统计量替换为截断签名时，误差减小。此外，生成的路径在忠实恢复条件化签名的同时，保留了关键的分布与时间结构，表明该估计器能良好地标定到目标条件分布。这些结果共同为截断签名反演建立了一个适定的概率框架，并在理论涵盖的参数过程族之外的现实金融数据上展示了其适用性。