← 研踪 PaperTrailarXiv:2606.15332 · v2
Probabilistic Signature Inversion: Learning Conditional Distributions from Truncated Signatures
概率签名反演：从截断签名中学习条件分布
首次提交 2026-06-13 22:52
版本更新 2026-11-13 01:10
雷达观测 2026-11-13 14:15
摘要
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 formulation elucidates the fundamental difficulty of inversion: posterior-sampling reconstruction error quantifies the irreducible uncertainty remaining after conditioning on a statistic. For Gaussian process families, we derive this error from the tractable posterior under linear-statistics conditioning, obtaining a closed form for log-GBM and numerically tractable formulas for drifted fBM and OU. This oracle error upper-bounds the corresponding error under truncated-signature conditioning, since truncated signatures are more informative than linear statistics. The experimental results are consistent with the theoretical analysis: the learned sampler under linear-statistics conditioning closely matches the theoretically derived posterior in both reconstruction error and distribution, while reconstruction errors decrease under richer truncated-signature conditioning. Moreover, generated paths faithfully recover the conditioning signature and preserve key distributional and temporal structures. Together, these results establish a well-posed probabilistic framework for truncated-signature inversion, with applicability demonstrated on financial data beyond the process families covered by theory. Code is available at https://github.com/Junoh-Kang/ProbSigInv.
作者备注：Accepted to NeurIPS 2026. Code: https://github.com/Junoh-Kang/ProbSigInv
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v2 · 2026-11-13 01:10 · 最新观测版本
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 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 formulation elucidates the fundamental difficulty of inversion: posterior-sampling reconstruction error quantifies the irreducible uncertainty remaining after conditioning on a statistic. For Gaussian process families, we derive this error from the tractable posterior under linear-statistics conditioning, obtaining a closed form for log-GBM and numerically tractable formulas for drifted fBM and OU. This oracle error upper-bounds the corresponding error under truncated-signature conditioning, since truncated signatures are more informative than linear statistics. The experimental results are consistent with the theoretical analysis: the learned sampler under linear-statistics conditioning closely matches the theoretically derived posterior in both reconstruction error and distribution, while reconstruction errors decrease under richer truncated-signature conditioning. Moreover, generated paths faithfully recover the conditioning signature and preserve key distributional and temporal structures. Together, these results establish a well-posed probabilistic framework for truncated-signature inversion, with applicability demonstrated on financial data beyond the process families covered by theory. Code is available at https://github.com/Junoh-Kang/ProbSigInv.
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