PaperlayerView exact revision on arXiv

Source-linked AI summary

Correcting CondOT: Exact Finite-Step Sampling in Gaussian Flow Matching

Ron Levy, Michael Elad

arXiv:2609.39488v1cs.LG

TL;DR

Finite-step flow-matching accuracy depends on the schedule, even when exact dynamics reach the same target. The paper analyzes Gaussian targets with the explicit midpoint sampler and shows that CondOT improves convergence; on uniform grids, a nearby scalar noise schedule yields exact sampling for every sufficiently large budget.

  • Problem

    Finite-step numerical approximations can have different accuracy under schedules that reach the same target with exact dynamics, raising the question of how schedule choice affects sampling error.

    Source: §1 · p. 1

  • Method

    The paper analyzes explicit midpoint sampling with exact Gaussian flow fields and uses a polynomial correction to the CondOT noise schedule while fixing α_t=t on uniform grids.

    Source: §1 · pp. 1–2

  • Results

    For α_t=t on uniform grids, scalar noise schedules within O(1/S) of CondOT yield exact Gaussian sampling for every sufficiently large S.

    Source: §7 · p. 9

  • Takeaways & Limitations

    Exact finite-budget Gaussian sampling is attainable within scalar schedules approaching CondOT on uniform grids.

    Source: §7 · p. 9

    The analysis assumes a specific solver and Gaussian targets with positive definite covariance; extensions to mixtures, other solvers, and network approximation error remain future work.

    Source: §7 · p. 9

Abstract

from arXiv · show

Flow matching generates samples by gradually transforming noise into data. In practice, using a finite number of sampling steps introduces a numerical error that depends on the chosen schedule. We study this dependence for Gaussian targets and the explicit midpoint sampling method, using the exact flow field. We measure sampling error by the squared Wasserstein distance between the target distribution and the final distribution produced by the midpoint sampler. We show that the standard conditional optimal transport (CondOT) schedule cancels the leading midpoint error and improves the general convergence bound, even when the sampling steps are unequally spaced. On a uniform grid of $S$ sampling steps, we fix the signal schedule at $\alpha_t=t$ and prove the existence of scalar noise schedules $\beta_t$ that approach the CondOT noise schedule $1-t$ at rate $1/S$ and yield exact Gaussian sampling for every sufficiently large $S$. Controlled Gaussian experiments illustrate the convergence rates and exact calibration.

1 Introduction

The paper asks how schedule choice affects finite-step sampling error when numerical samplers approximate flow matching, even though exact dynamics can reach the same target. For Gaussian targets and the explicit midpoint sampler, it analyzes the error and shows that CondOT improves convergence, while a small noise-schedule correction can yield exact sampling on uniform grids.

1 Introduction

  • The paper asks how sampling error depends on the schedule and how to choose a schedule for a given sampling budget.Different schedules can reach the same target under exact dynamics yet differ in numerical accuracy.Source: §1 · p. 1
  • The study uses Gaussian targets and the explicit midpoint sampler to isolate schedule-dependent numerical error from neural-network approximation error.Gaussian flow fields are available in closed form, and schedule changes can compensate for sampler error.Source: §1 · p. 1
  • Increasing the data contribution linearly is the unique signal schedule that preserves the terminal mean across target means, prescribed grids, and admissible noise schedules.With this choice, the remaining objective is the covariance mismatch between the target and midpoint output.Source: §1 · p. 1
  • CondOT cancels the leading midpoint error and improves convergence even when sampling steps are unequally spaced.Gaussian experiments confirm faster convergence and show calibrated terminal error reduced to numerical precision at finite budgets.Source: §1 · p. 2
  • A 1/S correction to CondOT’s noise schedule yields exact Gaussian sampling for every sufficiently large uniform-grid budget.The signal schedule remains fixed, and the correction is polynomial with one coefficient per distinct covariance eigenvalue.Source: §1 · p. 2

2 Problem Setup and Exact Sampling Error

The setup studies midpoint sampling along a Gaussian flow path, measuring its terminal discrepancy from the target with squared Wasserstein distance. It establishes a unique linear signal schedule that preserves the mean, then gives an exact finite-step objective for the remaining covariance error.

2 Problem Setup and Exact Sampling Error

  • The Gaussian path interpolates between independent standard noise and the target using signal and noise schedules with endpoint values that recover each distribution.The schedules determine the mean and covariance along the path; the reported error is squared Wasserstein distance from the target.Source: §2 · pp. 2–3

2.1 A signal schedule that preserves the mean

  • The signal schedule α_t=t is necessary and sufficient to preserve the terminal mean for every target mean, prescribed grid, and admissible noise schedule.With this choice, the sampler also preserves the exact mean at every sampling grid point and predicted midpoint.Source: §2.1 · p. 3

2.2 The exact finite-step objective

  • For α_t=t, the paper gives the midpoint output distribution and its squared Wasserstein distance from the target for every admissible noise schedule and prescribed grid.This exact objective supports schedule optimization and convergence analysis.Source: §2.2 · pp. 3–4
  • The exact objective expresses the remaining error through output standard deviations and target variances in the covariance eigendirections.The target variance in direction k is 1/λ_k; the exact path variance and sampler-produced standard deviation enter the Wasserstein expression.Source: §2.2 · pp. 3–4

3 CondOT Superconvergence

For fixed Gaussian covariance and noise schedule, the paper bounds midpoint error as the sampling grid is refined. It shows that CondOT cancels the leading error on arbitrary prescribed grids, while uniform-grid analysis characterizes its remaining error and standard-deviation discrepancy.

3 CondOT Superconvergence

  • O(h^6_max) replaces the general O(h^4_max) squared Wasserstein error bound under CondOT, without requiring equally spaced sampling times.The result assumes α_t=t and β_t=1−t as the maximum step size tends to zero.Source: §3 · p. 4
  • Θ(S^-6) is CondOT’s squared Wasserstein error rate on a uniform grid with h=S^-1.The leading coefficient is strictly positive, so CondOT is not exact at finite budgets.Source: §3 · p. 4
  • For sufficiently large uniform-grid budgets, CondOT produces a smaller standard deviation than the target in every eigendirection.This remaining discrepancy motivates changing the noise schedule to eliminate the error.Source: §3 · p. 4

4 Exact Finite-Budget Calibration Near CondOT

The section constructs a polynomial correction to CondOT’s noise schedule, with one coefficient per distinct precision eigenvalue, while keeping the signal schedule fixed. Its leading correction improves asymptotic decay, and an additional budget-specific adjustment yields exact terminal matching for every sufficiently large sampling budget.

4 Exact Finite-Budget Calibration Near CondOT

  • A single shared scalar noise schedule uses one correction coefficient per distinct precision eigenvalue and reduces to CondOT when the coefficients are zero.The schedule remains smooth and has the required endpoints for every coefficient vector.Source: §4 · p. 5
  • The unique vector K solves the linear system matching each eigendirection’s leading error coefficient to its CondOT coefficient.The resulting first-order correction is θ = K/S and has squared Wasserstein error O(S^-8).Source: §4 · p. 5
  • For every sufficiently large S, a locally unique parameter vector θS = K/S + O(S^-2) makes the midpoint output exactly match the target.The budget-specific adjustment solves r nonlinear equations; repeated eigenvalues require no additional equations, and the threshold depends on the target spectrum.Source: §4 · p. 5

5 Numerical Experiments

The experiments examine midpoint sampling on four centered Gaussian targets with fixed dimension and total variance. They show CondOT’s faster convergence, the improvement from a first-order correction, and spectrum-dependent calibrated schedules whose finite-budget terminal errors can reach numerical precision.

5 Numerical Experiments

  • The four centered Gaussian targets have dimension d = 6 and total variance tr(Σ) = 6, with two- or six-eigenvalue spectra and differing spectral spreads.The two-eigenvalue cases repeat each variance three times; the six-eigenvalue cases use geometric spacing.Source: §5 · p. 5, Table 1 · p. 6

5.1 CondOT superconvergence

  • For Target D, the four alternative fixed schedules have squared Wasserstein error Θ(S^-4), while CondOT has Θ(S^-6) on uniform grids.The curves approach the predicted rates.Source: §5.1 · p. 6, Figure 1 · p. 6

5.2 From first-order correction to exact calibration

  • For Target D, the explicit first-order correction approaches eighth-order decay, improving asymptotically on CondOT’s sixth-order decay.The comparison uses the same grids as the fixed-schedule experiment.Source: §5.2 · p. 6, Figure 2 · p. 7
  • At four budgets for Target D, the calibration iteration reduces terminal loss below 10^-170 in 90-digit arithmetic.The table compares leading and numerically calibrated schedules; separate reported results note that Targets C and D miss the matching tolerance at S = 64 and 128.Source: §5.2 · p. 7, Table 2 · p. 7

5.3 How the target spectrum changes the calibrated schedule

  • At S = 256, calibrated corrections are larger for the broader spectra in these four targets, with maximum deviations of approximately 2.42×10^-4, 8.15×10^-4, 3.15×10^-2, and 4.07×10^-2 for A–D, respectively.Two-eigenvalue corrections stay below CondOT in the interior, whereas six-eigenvalue examples cross it; these are features of the selected targets, not universal consequences of r or κ.Source: §5.3 · p. 7, Figure 3 · p. 8

6 Related Work and Scope

Prior work studies Gaussian sampling error, schedule design, and exact finite-step sampling using different models and methods. This paper’s setting combines midpoint sampling with a single scalar noise schedule shared across covariance directions.

6 Related Work and Scope

  • Gaussian schedule studies have analyzed Wasserstein and KL objectives, Lipschitz-based schedule design, and leading-error cancellation in one covariance eigendirection (Hurault et al., 2025; Benita et al., 2025; Chen et al., 2025; Hurault et al., 2026).The cited discretization work constructs schedules that cancel leading error in one eigendirection.Source: §6 · p. 8
  • For centered Gaussians with commuting covariances, Claustre et al. (2026) use direction-dependent schedules and show Euler exactly integrates the Gaussian Wasserstein geodesic on any time grid.Here, midpoint sampling uses αt = t and a single scalar noise schedule shared across covariance directions, with corrections that vanish toward CondOT as the budget grows.Source: §6 · p. 8
  • Mean preservation also connects to Levy & Elad (2026), whose Gaussian-mixture surrogate preserves posterior-component means and simplifies their Wasserstein upper bound.Fixing αt = t gives an analogous simplification here: the exact squared Wasserstein objective depends only on covariance mismatch.Source: §6 · pp. 8–9

7 Conclusion

The linear signal schedule preserves the mean, while CondOT cancels the leading numerical error and yields superconvergence even on nonuniform grids. On uniform grids, a polynomial noise-schedule correction approaching CondOT can give exact terminal sampling for sufficiently large budgets.

7 Conclusion

  • A vanishing polynomial noise-schedule correction gives zero terminal loss for every sufficiently large budget on uniform grids.The correction uses one coefficient per distinct precision eigenvalue.Source: §7 · p. 9
  • The linear signal schedule preserves the mean, and CondOT cancels the leading numerical error, yielding superconvergence even on nonuniform grids.Source: §7 · p. 9
  • The analysis assumes the explicit midpoint solver and Gaussian targets with positive definite covariance.The conclusion identifies Gaussian mixtures, other solvers, and network approximation error as future directions.Source: §7 · p. 9

A Problem Setup and Exact Sampling Error

This appendix sets up Gaussian flow matching and derives the exact flow and midpoint sampler output law. It then uses the Gaussian structure to analyze the sampler across covariance eigendirections.

A.1.1 Gaussian probability path

  • The probability path connects standard-normal base noise to target data through signal and noise schedules with specified endpoint conditions.Source: §A.1 · p. 11, §A.1.1 · p. 11

A.1.2 Marginal vector field and exact flow

  • The marginal vector field is derived from the interpolation, and its exact flow reaches the target distribution at time 1.Source: §A.1.2 · pp. 11–12

A.1.3 Spectral form of the marginal ODE

  • In the target covariance eigenbasis, the marginal ODE decouples into scalar equations that form the starting point for midpoint analysis.Source: §A.1.3 · p. 12

A.2.1 General midpoint recursion and output law

  • The explicit midpoint method uses two vector-field evaluations per step and can be applied on an arbitrary prescribed time grid.Its general-schedule output law is obtained by iterating the affine recursion.Source: §A.2 · p. 13, §A.2.1 · p. 13

A.2.2 Universal mean exactness of the linear signal schedule

The linear signal schedule is uniquely characterized by universal terminal-mean exactness under the stated assumptions. With the mean exact, the finite-step Wasserstein objective reduces to covariance mismatch, while the following analysis considers grid refinement.

A.2.2 Universal mean exactness of the linear signal schedule

  • The signal schedule α_t=t is the unique schedule that preserves the terminal mean for every target mean, prescribed grid, and admissible noise schedule.For this schedule, midpoint sampling also reproduces the exact mean at every grid point and predicted midpoint.Source: §A.2.2 · p. 14

A.3 The exact finite-step objective

  • With the mean exact, the squared Wasserstein distance reduces to covariance mismatch, which can be evaluated in the shared covariance eigenbasis.Source: §A.3 · p. 16

B CondOT Superconvergence

  • The subsequent CondOT analysis develops a general error expansion for fixed schedules as the grid is refined; on a uniform grid, hmax equals 1/S.Source: B CondOT Superconvergence · p. 16

B.1 General-grid midpoint multiplier and logarithmic scale error

The analysis relates midpoint sampling error to a logarithmic scale error and derives its leading defect on general grids. For CondOT, that complete leading defect vanishes across eigendirections.

B.1 General-grid midpoint multiplier and logarithmic scale error

  • The logarithmic relative scale error is zero when the terminal scale is correct, and its sign indicates underestimation or overestimation.Source: §B.1 · p. 17

B.2 The leading midpoint defect on a general grid

  • A stepwise expansion of the logarithmic defect is summed over the prescribed grid and eigendirections to obtain a sharper discrete error expansion.Source: §B.2 · pp. 17–18

B.3 CondOT cancellation on a general grid

  • For CondOT, the complete leading defect vanishes in every eigendirection for every positive precision eigenvalue.The stated cancellation follows from the variance being a positive quadratic, so its third derivative is zero.Source: §B.3 · p. 19

The global squared Wasserstein error therefore obeys

The section derives error rates for midpoint sampling and shows how CondOT cancels the leading error on prescribed grids. It also develops a shared polynomial noise-schedule correction for exact finite-budget calibration.

Summing over the eigendirections gives

B.4 Uniform-grid generic asymptotics

  • For fixed comparison schedules on a uniform grid, the squared Wasserstein error is Θ(S^-4) for every fixed positive precision spectrum.The result applies under the stated smoothness and positivity assumptions on the variance.Source: §B.4 · p. 21

B.5 CondOT midpoint superconvergence on the uniform grid

  • On a uniform grid, CondOT has squared Wasserstein error Θ(S^-6).The leading coefficient is strictly positive.Source: §B.5 · p. 21

C Exact finite-budget calibration near CondOT

  • With αt = t on a uniform grid, the paper constructs a polynomial correction to CondOT’s noise schedule, using one coefficient per distinct covariance precision eigenvalue to eliminate error across eigendirections.The correction is a single shared scalar schedule with the required endpoints.Source: §C.1 · p. 23

C.1 The correction family and its linear system

  • For every finite set of distinct positive precisions, the matrix governing the leading correction is invertible, so its correction system has a unique solution.The appendix proves invertibility by ruling out a nonzero left null vector.Source: §C.1 · pp. 23, 25

C.2 From the leading correction to an exact solution

This section studies schedule corrections near CondOT to reduce midpoint sampling error and calibrate exact sampling at fixed step budgets. It covers the first-order correction, iterative calibration and its tolerance across targets and budgets, and analysis of error near CondOT.

C.2 From the leading correction to an exact solution

  • The remaining O(S^-4) errors require an additional O(S^-2) coefficient adjustment; an update using inverse sensitivity L^-1 contracts distances by at least a factor of two and converges.Changing coefficient θ_j affects error direction k at leading rate L_kj/S^2.Source: §C.2 · pp. 26–28
  • Choosing θ=K/S cancels CondOT’s leading error in every eigendirection, leaving scale error O(S^-4) and squared Wasserstein loss O(S^-8).The signal schedule α_t=t already makes the mean exact.Source: §C.2 · p. 26
  • For every sufficiently large integer S, a locally unique schedule in the chosen polynomial family yields exact Gaussian sampling with zero terminal loss.The calibrated coefficients produce the target mean and covariance, so every term in the squared Wasserstein objective is zero.Source: §C.2 · pp. 26, 27, 29
  • The leading integral evaluates exactly to L_kj; with r fixed, the integrand changes by O(∥θ∥) uniformly in t, while for ∥θ∥≤R/S the remainder is O(S^-4).Source: §C.2 · pp. 32–34

C.3 Numerical calibration procedure

  • For all four targets, the first-order correction improves on CondOT at every tested S ≥ 256, and calibration reaches tolerance for all targets at S = 256, 512, and 1024.At S = 64 and 128, the iteration does not reach tolerance for Target C and stops for Target D after a midpoint factor becomes nonpositive; these failures do not establish nonexistence, since Theorem 5 guarantees calibration only for sufficiently large S, with a target-dependent threshold.Source: §C.3 · p. 35, Table 3 · p. 35
  • The first-order correction can increase loss at small budgets, including Target D at S = 64 and 128 and slightly Target C at S = 128.The Target C iteration also hits its 600-update cap at S = 64 and 128, while Target D stops after four and eight updates, respectively, when a midpoint factor becomes nonpositive.Source: §C.3 · p. 35

C.4 Convergence and calibration across the four targets

Across four Gaussian targets, observed decay exponents approach the predicted rates for fixed alternatives, CondOT, and its first-order correction. The calibrated schedules are also checked for shape and in ordinary double precision.

C.4 Convergence and calibration across the four targets

  • Observed decay exponents approach 4 for fixed alternatives, 6 for CondOT, and 8 for the first-order correction in these examples.The reported exponent is log2(J32768/J65536) for each curve and target.Source: §C.4 · p. 36, Table 4 · p. 36

C.5 Calibrated schedule shapes and coefficients

  • All four calibrated schedules decrease monotonically and have positive midpoint factors, although the calibration theorem guarantees neither property at a specified budget.The theorem gives no budget threshold for monotonicity and does not determine the number of interior crossings.Source: §C.5 · p. 37

C.6 Calibration in ordinary double precision

  • For proof iterations computed in binary64 and reevaluated at 90 digits without coefficient refinement, 16 of 20 runs converged; reevaluated scale errors were below 10^-14 and squared Wasserstein losses below 3.0 × 10^-28.The same four small-budget cases failed as in Appendix C.3.Source: §C.6 · p. 38, Table 6 · p. 38
Loading 2609.39488v1…