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On the Relation between Rectified Flows and Optimal Transport
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Theoretical study showing rectified flows with gradient-constrained velocity fields do not reliably compute optimal transport, via invariance properties, Gaussian/GMM closed forms, and counterexamples with disconnected supports and non-rectifiable couplings, plus a noise-injected smoothed rectification.
Problem
Whether iterative rectified flow (and its gradient-constrained variant) yields optimal transport couplings; existing claims of equivalence between rectified-flow fixed points and optimal transport.
Approach
Derives affine invariance properties and explicit Gaussian and Gaussian-mixture velocity fields for rectified flows, shows gradient-constrained rectification has weak-form solutions, and gives counterexamples (disconnected support, non-rectifiable couplings) that invalidate earlier equivalence claims; proposes noise-injected smoothed rectification that keeps iterates rectifiable.
Key results
- Gaussian coupling with jointly diagonalizable covariances: first rectification step yields the optimal coupling; in 1D any rectifiable coupling becomes optimal after one step.
- Counterexample with disconnected support: fixed point of gradient-constrained rectification has zero loss with transport cost 16 versus 4 for the optimal coupling.
- Noise-injected smoothed rectification gives min loss in O(1/K + average noise level), matching the O(1/K) rate of standard rectification when noise levels are summable.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the non-optimal coupling (tilde X0, tilde X1) as a fixed point of gradient-constrained rectification | Gradient-constrained rectification · Disconnected-support counterexample | expected squared transport cost: 4 -> 16 | Hurt |
| Use a non-rectifiable Gaussian anti-coupling with X1 = -X0 | Gradient-constrained rectification · X0, X1 ~ N(0, Id), with X1 = -X0 | rectification loss: 0 | No clear effect |
| Require the interpolated distributions to have connected support, alongside the stated regularity and rectifiability assumptions | Equivalence between gradient-constrained rectification fixed points and optimal couplings · Quadratic-cost optimal transport | optimality of fixed points | Helped |
| Inject noise into the source at each rectification step | Smoothed rectification with Gaussian source · Iterative rectification | minimum loss over K steps: O(1/K + bar c_K) | Helped |
| Use summable noise levels during rectification | Smoothed rectification with Gaussian source · Iterative rectification | minimum loss over K steps: O(1/K) | Helped |
| Use independent Gaussian coupling with jointly diagonalizable marginal covariances | One-step rectification · Gaussian marginals | coupling optimality | Helped |
Surprising findings
- Gradient-constrained rectified flow can have zero loss and straight paths yet be far from optimal, with a constant positive optimality gap under iteration.
- Vanishing loss does not imply optimality: couplings with arbitrarily small loss can be arbitrarily far from the optimal coupling.
Limitations
- Smoothed rectification guaranteed rectifiable only for Gaussian mu_0 and does not show convergence to optimal transport.
- Unclear to which fixed point iterative rectification converges from a given initial coupling.
- Analysis restricted to quadratic cost.
Experimental setup
Metrics: velocity loss transport cost optimality gap
Baselines: rectified flow (Liu et al.) minibatch OT coupling diffusion Schrödinger bridge matching
Builds on: rectified flow flow matching Benamou-Brenier formulation minibatch optimal transport
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