Research map / Probabilistic and scientific modeling / Variational and generative sampling
Rectifying Categorical Flows on Statistical Manifolds for One-Step Generation
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Proposes ReSFlow, a reflow procedure performed on the hypersphere via the square-root map to straighten categorical flows so that one-step sampling works for discrete data.
Problem
Flow-based generative models for categorical data on statistical manifolds follow curved, non-geodesic trajectories, requiring many integration steps; Euclidean rectified flow does not respect Fisher-Rao geometry.
Approach
Apply rectified flow (reflow) on the positive orthant of the hypersphere, where Fisher-Rao geodesics are great-circle arcs, so the learned trajectories straighten toward geodesics and allow single-step categorical generation.
Key results
- On Text8 ODE-based BPC, ReSFlow reaches 1.261 vs 1.830 for SFM, 2.154 for LinearFM and 2.306 for ReDi.
- On binarized MNIST, one-step ReSFlow achieves FID 5.973 vs 35.640 for one-step SFM, at 0.452 s per 1k samples vs 112.81 s for ODE SFM (FID 4.748).
- On promoter design, one-step ReSFlow attains SP-MSE 0.028718 (vs 0.035564 for one-step SFM) with sampling time 0.409 s per 1k sequences.
Surprising findings
- Euclidean reflow (R-LFM) performs markedly worse than non-rectified LinearFM on Text8 (BPC 3.267 vs 2.154), indicating rectifying in the wrong geometry is harmful.
Limitations
- One-step FID on binarized MNIST (5.973) is still worse than the adaptive-ODE SFM result (4.748).
- Likelihood metrics require ODE integration; a single Euler/exponential step does not give reliable likelihoods.
- Theoretical guarantees are stated informally and rely on rectifiability assumptions; experiments are limited to small-scale benchmarks.
Experimental setup
Datasets: binarized MNIST Text8 promoter DNA sequences Swiss roll (synthetic)
Benchmarks: binarized MNIST Text8 promoter sequence design Swiss roll
Metrics: FID BPC SP-MSE NLL sampling time
Baselines: LinearFM SFM R-LFM ReDi
Builds on: Rectified Flow Statistical Flow Matching Flow matching ReDi
Closest papers by meaning
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- Metric Flow Matching for Smooth Interpolations on the Data Manifold (NeurIPS 2024 · similarity 0.88)
- Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes (ICML 2024 · similarity 0.88)
- Expanding Flow Maps (NeurIPS 2026 · similarity 0.88)
- Learning Distributions on Manifolds with Free-Form Flows (NeurIPS 2024 · similarity 0.88)
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