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Heavy-Tailed Flow Matching via Random Clocks

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

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Proposes HTFM, a flow-matching framework where a random clock path controls source covariance so heavy-tailed (alpha-stable, Student-t) sources arise as Gaussian scale mixtures, with truncated logsignature conditioning of the velocity field.

Problem

Standard flow matching starts from Gaussian sources, which is a poor inductive match for heavy-tailed data (rare events in imbalanced images, finance, weather); existing heavy-tailed diffusion/flow methods are tied to specific tail families.

Approach

Condition on a random clock path so the source and affine flow are Gaussian, derive the endpoint-conditioned affine target as in Gaussian CFM, marginalize over the clock to obtain alpha-stable or Student-t (including infinite-variance) sources, and feed a truncated logsignature of the clock path into the velocity network.

Key results

• On CIFAR10-LT at NFE=100, HTFM-alpha (p=1.8) reaches FID 15.89 vs 19.29 for Gaussian-clock HTFM-G and 18.18 for DLPM at the same NFE.
• On the 2D alpha-stable mixture (alpha_D=1.5), HTFM-t (nu=1.7) reaches f1-pr 0.962 vs 0.888 for HTFM-G.
• Signature order m=1 improves FID over no clock features (m=0) in all HTFM-alpha and HTFM-t settings, e.g., 16.31 vs 18.72 for HTFM-t at p=1.7.

Surprising findings

• Matched polynomial tail-decay rate does not determine the best source family: at p=1.7, HTFM-t is best on 2D mixture while HTFM-alpha is best on CIFAR10-LT and HRRR.
• Higher-order logsignature (m=2) gives no consistent gain over m=1.

Limitations

• Clock-dependent schedules yield Gaussian-mixture sources rather than exact stable or Student-t laws; asymmetric (mean-variance) extension is deferred and not evaluated.
• Clock law is hand-chosen; data-adaptive clock inference is left to future work.
• Heavy-tailed outer clock averaging relies on separate integrability assumptions; the infinite-variance pathwise equivalence is not fully validated empirically.

Experimental setup

Datasets: CIFAR10-LT 2D imbalanced alpha-stable mixture HRRR VIL

Benchmarks: CIFAR10-LT 2D imbalanced alpha-stable mixture HRRR VIL

Metrics: precision-recall f1 FID kurtosis ratio skewness ratio tail Kolmogorov-Smirnov

Baselines: LIM DLPM DLIM t-Flow HTFM-G

Builds on: conditional flow matching DLPM Levy-Itô models t-Flow rectified flow

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