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Probabilistic Signature Inversion: Learning Conditional Distributions from Truncated Signatures

NeurIPS 2026 · Probabilistic and scientific modeling › Variational and generative sampling · approach: Flow matching and flow sampling · 0 citations (OpenAlex)

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Reframes truncated signature inversion as learning the conditional distribution of paths given their truncated signature, using a signature-conditioned flow matching DiT, and derives Bayes reconstruction error baselines from linear statistics for log-GBM, log-fBM and OU.

Problem

Truncated path signatures are not injective, so recovering a path from its truncated signature is ill-posed; existing inversion methods return a single deterministic reconstruction that ignores this ambiguity.

Approach

Treat signature inversion as sampling from the prior-induced conditional distribution of paths given the truncated signature, learned with a signature-conditioned flow matching model, and quantify irreducible ambiguity via a Bayes reconstruction error whose closed form (log-GBM) and kernel-based numerics (log-fBM, OU) under linear statistics upper-bound richer signature conditioning.

Key results

• Empirical LS reconstruction errors closely match the analytic Bayes oracle for log-GBM (e.g., 0.173 vs 0.172 at r=6).
• TLL-conditioned flow matching at depth 6 reduces log-GBM sigma parameter error to 0.06 versus 1.69 for the TLL regression baseline.
• At depth 6, TA/TLL empirical path MSE drops below the LS oracle (e.g., log-GBM TA 0.135, TLL 0.144 vs LS oracle 0.172).

Surprising findings

• Deeper TA/TLL conditioning shows a few reversals in ordering relative to the expected LS-TA-TLL hierarchy, attributed to harder optimization.
• Conditioning consistency error grows with depth for TLL (up to 2.76% at r=6 for log-GBM), indicating weaker conditioning fidelity at high dimension.

Limitations

• Theory covers only Gaussian process families via linear statistics and does not give the exact Bayes error under truncated-signature conditioning.
• Experiments limited to one-dimensional synthetic processes and S&P 500 windows.
• Empirical error ordering LS-TA-TLL is not strict at all depths due to optimization difficulty.

Experimental setup

Datasets: synthetic log-GBM paths synthetic log-fBM paths synthetic OU paths S&P 500 cumulative log returns 2009-2022

Benchmarks: synthetic log-GBM, log-fBM, OU reconstruction error S&P 500 out-of-sample windows 2023-2025

Metrics: path MSE (Bayes reconstruction error) inter-sample spread ratio conditioning consistency relative MSE parameter estimation error signature fidelity realized volatility, skewness, kurtosis error ACF and QQ plots

Baselines: Fourier Legendre Insertion regression ablation TA regression ablation TLL

Builds on: flow matching diffusion transformer signature transform Fourier signature inversion insertion method

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