Difficulty-Calibrated Interpolation Paths for Conditional Flow Matching
Authors: Airin Akter Tania, Md Raihan Khan
Organizations: Dept. of Electrical and Electronic Engineering Khulna University of Engineering & Technology Khulna, Bangladesh · Dept. of Electrical and Electronic Engineering North Western University Khulna, Bangladesh
Conditional Flow Matching trains generative models by regressing a network onto the velocity of a prescribed noise-to-data interpolation path. The interpolation schedule that shapes this path is known to affect convergence and sample quality, yet it is invariably fixed in advance, independent of both the data and the model. We show that the regression difficulty of Conditional Flow Matching varies systematically along the path, and we propose Difficulty-Calibrated Flow Matching, which derives the schedule from the model itself: a short pilot run with the linear path records the per-time loss, and the schedule is set to the quantile function of this difficulty profile, so the trajectory lingers where the velocity is hardest to learn. The method has a single hyperparameter, leaves the training objective and its gradient equivalence intact, composes with classifier-free guidance, and adds about two percent training overhead. In controlled experiments on CIFAR-10, MNIST, and Fashion-MNIST with an identical compact U-Net, the calibrated path attains the best FID on CIFAR-10 at full sampling budget and clearly outperforms all fixed schedules in the large-batch, few-update regime, precisely the setting where compute is scarcest.
Figures & tables
Fig. 1: Conceptual illustration of DC-FM. A sample xt travels from noise x0 to data x1 along the interpolation path while the network vθ regresses the target velocity ut under an MSE loss. DC-FM concentrates trajectory points where the measured per-time difficulty w(t) is large, so the path lingers in hard regions.
Fig. 2: Overview of Difficulty-Calibrated Flow Matching (DC-FM). (a) Standard CFM pipeline: an interpolant mixes a data sample x1 and noise x0 , a U-Net regresses the target velocity under the flow-matching loss, and samples are drawn by integrating the ODE at inference. (b) The interpolation schedule α(t) for linear, cosine, sigmoid, and our calibrated path; only α(t) changes across the controlled comparison. (c) DC-FM measures per-time difficulty w(t) in a linear pilot and sets the schedule to the quantile function of wγ , so the trajectory slows through hard regions of the path and moves quickly through easy ones.
Fig. 3: Interpolation . schedule α(t) (left), its speed α′(t) (middle), and the pilot difficulty w(t) (right) on CIFAR-10 (top), Fashion-MNIST (middle), and MNIST (bottom). DC-FM slows down ( α′ small) exactly where w(t) is large.
Method
FID@100
FID@20
CIFAR-10
Rectified Flow (linear) [ 9 ]
5.44
5.85
GVP (cosine) [ 6 ]
5.90
5.71
Logistic (sigmoid)
5.30
6.53
DC-FM (ours)
5.13
5.15
MNIST
TABLE I: Main results: FID ( ↓ ) at NFE 100 / 20 on CIFAR-10, MNIST, and Fashion-MNIST. Best per column in bold . Schedules are named by their proposal.
γ
0.1
0.5
1
1.5
2.0
FID@100
7.43
7.90
5.81
8.85
9.52
FID@20
7.74
8.44
6.42
11.04
16.21
TABLE II: Effect of the emphasis exponent γ on DC-FM (MNIST). Best per row in bold . Linear baseline shown for reference ( ≈ 7.4 FID@100 at this budget).
B
10
20
40
80
FID@100
8.63
5.81
7.45
7.71
FID@20
9.11
6.42
8.06
8.37
TABLE III: Effect of the number of difficulty bins B on DC-FM (MNIST). Best per row in bold .
Method (batch =512 )
FID@100
FID@20
Rectified Flow (linear) [ 9 ]
10.16
10.61
GVP (cosine) [ 6 ]
10.71
22.44
Logistic (sigmoid)
11.05
13.57
DC-FM (ours)
8.13
8.39
TABLE IV: Effect of batch size on MNIST (FID ↓ , NFE 100/20). Best per column in bold . DC-FM’s advantage grows in the large-batch, few-update regime.
Fig. 4: Uncurated DC-FM samples (NFE =100 ) on CIFAR-10 (top) and Fashion-MNIST (bottom).