Random-Bridges as Stochastic Transports for Generative Models
Authors: Stefano Goria, Levent A. Mengütürk, Murat C. Mengütürk, Berkan Sesen
Organizations: Artificial Intelligence and Mathematics Research Lab James Carter Road, Mildenhall, Bury St. Edmunds IP28 7DE, UK · Department of Mathematics, University College London 25 Gordon Street, London, WC1H 0AY, UK
This paper motivates the use of random-bridges -- stochastic processes conditioned to take target distributions at fixed timepoints -- in the realm of generative modelling. Herein, random-bridges act as stochastic transports between two probability distributions when appropriately initialized, displaying either Markovian or non-Markovian, and continuous, discontinuous, or hybrid patterns depending on the choice of the driving process. We show how one can start from general probabilistic statements and branch out into specific representations for learning and simulation algorithms. Essentially, our framework formulates stochastic transports as noisy information processes governing the dynamic translation and propagation of information between arbitrary probability measures. Our theoretical and algorithmic presentation details training and simulation frameworks for both Gaussian and Lévy (specifically, Gamma, Poisson and Stable-1/2) random-bridge transports. The empirical validation focuses on the Gaussian case for demonstration and comparison purposes, where our construct delivers high-quality samples, achieving competitive Fréchet Inception Distance scores compared to widely used denoising diffusions and flow matching frameworks.
Figures & tables
Model
Network evaluations per sample
1
2
5
10
20
50
100
200
1000
Random-bridge, Euler–Maruyama
385.98
168.38
43.99
19.21
12.26
9.83
9.17
8.88
9.23
Flow Matching
372.21
173.63
41.64
19.41
13.50
10.43
9.35
8.82
8.43
Improved denoising, ancestral
364.62
364.42
46.57
25.60
11.04
6.04
5.30
5.28
5.60
Improved denoising, DDIM
364.62
363.91
70.17
33.69
–
–
–
–
–
Table 1: CIFAR-10. FID against the training split, 50000 generated samples per entry, extrapolated floor 0.67 (measured 3.15 at n=10000 ). Simulation cost is the number of network evaluations per sample. Bold marks the best entry in each column; a dash marks a configuration not run.
Model
Network evaluations per sample
1
2
5
10
20
50
100
200
1000
Random-bridge, exact transition
346.04
50.31
9.62
4.08 ± 0.06
2.14
1.37 ± 0.00
1.19
1.12 ± 0.00
–
Random-bridge, Euler–Maruyama
346.30
49.42
9.46
3.72 ± 0.02
1.89
1.27
1.17
1.12
1.06
Random-bridge, deterministic map
346.05
51.46
10.51
5.73 ± 0.08
4.19
3.42
3.19
3.05
–
Flow Matching
298.68
54.09
10.34
4.13 ± 0.06
2.05
1.23 ± 0.02
1.07
1.02 ± 0.01
1.02
Simple denoising, cosine, DDIM
418.50
380.07
17.76
6.96
4.07
2.24
1.40
2.83
16.30
Table 2: MNIST. FID against the training split, 50000 generated samples per entry; the measured floor is 0.98 at n=10000 and does not admit a defensible extrapolation to 50000 . Mean ± standard deviation over simulation seeds where repeats were run.
Target FID
Random-bridge
Flow Matching
Improved denoising
30
7
7
8
20
10
10
12
15
15
16
16
12
22
30
19
10
47
65
23
9
151
157
27
Table 3: Network evaluations per generated sample required to reach a given FID on CIFAR-10, by logarithmic interpolation between the measured entries of Table 1 .
Appendix figures & tables10 assets
Supplementary material from the paper’s appendix.
Appendix
endpoint squared error
velocity squared error
t/T
γ=0.05
γ=0.3162
velocity
γ=0.05
γ=0.3162
velocity
0.050
0.20779
0.20783
0.20801
0.230
0.230
0.230
0.400
0.04515
0.04534
0.04407
0.125
0.126
0.122
0.600
0.02017
0.02051
0.01932
0.126
0.128
0.121
0.800
0.00659
0.00698
0.00616
0.165
0.175
0.154
0.900
0.00242
0.00275
0.00216
0.242
0.275
0.216
Appendix
Table 4: CIFAR-10. Squared error of the endpoint estimate (left) and the same quantity expressed in the units of the velocity field ( 38 ) (right). The three networks agree to three decimal places over the first eighty per cent of the bridge.
Figure 1: Squared error of the velocity field along the bridge, logarithmic vertical scale, 2048 samples per point on a common σ=0 trajectory. Left: CIFAR-10. Right: MNIST. The curves are indistinguishable until t≈0.8T and separate thereafter.
Model
Simulation
Evals
FID
precision
recall
coverage
KID
Random-bridge, velocity
Euler–Maruyama
10
20.87
0.606
0.530
0.571
0.01391
Flow Matching
Euler
10
21.21
0.576
0.514
0.565
0.01873
Improved denoising
ancestral
10
27.69
0.617
0.490
0.578
0.01827
Improved denoising
DDIM
10
35.69
0.514
0.570
0.396
0.01945
Simple denoising
ancestral
10
52.97
0.472
0.286
0.340
0.05150
Random-bridge, endpoint
exact posterior
10
44.44
0.494
0.365
0.340
0.04809
Appendix
Table 5: CIFAR-10 at n=10000 . Precision is the fraction of generated samples inside the real manifold and recall the fraction of real samples inside the generated one; coverage is the outlier-robust form of recall. The Kernel Inception Distance (KID) orders the rows identically to FID at every evaluation count.
Model
Simulation
Evals
classifier FID
mode TV
least-served digit
Random-bridge
Euler–Maruyama
10
27.05
0.0464
1 at 0.819
Flow Matching
Euler
10
31.72
0.0438
0 at 0.857
Improved denoising
ancestral
10
39.64
0.0885
1 at 0.629
Random-bridge
Euler–Maruyama
100
0.85
0.0128
7 at 0.922
Flow Matching
Euler
100
1.19
0.0147
7 at 0.923
Improved denoising
ancestral
100
1.11
0.0110
2 at 0.950
Appendix
Table 6: MNIST, 50000 generated samples, scored in the feature space of a convolutional classifier trained on the MNIST training split. Mode TV is the total-variation distance between the classifier’s predicted-label histogram over generated samples and the histogram of the data. The final row compares the real test split with the training split and measures their train/test difference. It does not estimate the finite-sample FID floor.
Figure 2: MNIST FID against network evaluations for five values of γ=σT . Each value uses a separately trained model with the same architecture, training budget and evaluation code. Results use n=10000 , and the dashed line marks the measured floor of 0.98 . The heavier line corresponds to γ=0.3162 , the value used throughout the main results. The left panel uses the Euler–Maruyama scheme of Algorithm 2 . The right panel uses the deterministic map with the driving increment removed. This map does not preserve the marginals, and its bias grows with γ .
Figure 3: CIFAR-10 at fifty network evaluations with σ=I , T=0.1 , under the two regression targets at matched architecture, optimiser, sampler and budget. Both panels are Euler–Maruyama and both figures are quoted at n=2000 , the sample count at which both were measured. The velocity-trained panel resolves object boundaries and surface texture left diffuse by the endpoint-trained panel, which is the terminal region of the bridge that the uniform weight discounts.
Figure 4: CIFAR-10 samples against simulation budget, σ=I , T=0.1 , Euler–Maruyama, FID at n=50000 . Two evaluations return the conditional mean of the data; five resolve object layout; ten give recognisable objects.
Figure 5: CIFAR-10 at fifty network evaluations under one architecture and one training budget. Optimisation is AdamW throughout; the random-bridge panel uses zero weight decay and the three baselines AdamW’s default 10−2 .
Figure 7: Upper panel: TrVtx/n estimated by the regression loss of a trained network on held-out data. The dashed rule marks the mean per-pixel variance of Ψ , the value Proposition 3.10 requires at t=0 . Lower panel: the dependence of the network output on its time argument, obtained by holding the state fixed and sweeping t , normalised by the output’s own pixel standard deviation. The two curves are the same architecture and budget trained with the time argument rescaled to [0,1) and left on [0,T) respectively; they reach the same training loss at every t and differ in this sensitivity alone.