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Probability Pathq（2）μ（2)σtCond. OTMarginal OTGeneral source
Var. Exploding (Song& Ermon, 2019)q(𝑥1)T101-t××
Var.Preserving (Ho et al., 2020)q(x1)α1-t11-a²−t××
Flow Matching (Lipman et al., 2023)q(x1)tx1tσ-t+1× √××
Rectified Flow Liu (2022)q(xo)q(x1)tx1+(1-t)xo√×√
Var.Pres.Stochastic Interpolant Albergo & Vanden-Eijnden (2023)q(xo)q(x1）cos(1πt)𝑥0+sin(πt)𝑥1√×√
Independent CFMq(x0)q(x1)tc1+(1-t)𝑥√×√
(Ours) Optimal Transport CFMπ(xo,x1)tx1+(1-t)x√√√
(Ours)Schrodinger Bridge CFMπ2g²(x0,x1)tx1+(1-t)xoσ√t(1-t)√√√
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Input: Efficiently samplable q(z), Pt(x|z),and computable ut(x|z) and initial network vo.
while Trainingdo
z~q(z)；t~U(0,1)； x~pt(x|z)
LcFM(θ)←|uθ(t,x)-ut(x|z)|l²
0 ←Update(0,VθLcFm(0)) returnvθ
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Dataset→N→8gaussiansmoons-→8gaussiansN→moonsN-→scurveAvg.train time
Algorithm ↓Metric→wNPEWNPEwNPEWNPE(x10²s)
OT-CFM1.262±0.3480.018±0.0141.923±0.3910.053±0.0350.239±0.0480.087±0.0610.264±0.0930.027±0.0261.129±0.335
I-CFM1.284±0.3840.222±0.0321.977±0.2662.738±0.1810.338±0.1090.841±0.1480.333±0.0600.867±0.1170.630±0.365
2-RF (Liu, 2022)1.436±0.3440.069±0.0272.211±0.4230.149±0.1010.278±0.0260.076±0.0670.395±0.1110.112±0.0850.862±0.166
3-RF (Liu, 2022)1.337±0.3670.055±0.0432.700±0.5870.123±0.1120.305±0.0260.084±0.0510.395±0.0820.129±0.0750.954±0.116
FM (Lipman et al., 2023)1.062±0.1960.174±0.0300.246±0.0770.778±0.1440.377±0.0990.772±0.0810.708±0.370
Reg.CNF (Finlay et al., 2020)1.144±0.0750.274±0.0600.376±0.0400.620±0.0880.581±0.1950.586±0.5038.021±3.288
CNF (Chen et al., 2018)1.055±0.0590.151±0.0640.387±0.0652.937±1.9730.645±0.34310.548±8.10018.810±12.677
ICNN (Makkuva et al., 2020)1.771±0.3980.747±0.0292.193±0.1360.832±0.0040.532±0.0460.267±0.0100.753±0.0680.344±0.0452.912±0.626
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Dataset ↓Alg.→SB-CFMDSB
N-→8gaussians0.454 ± 0.1641.440 ± 0.720
moons-→8gaussians1.377 ± 0.2292.407 ± 1.025
N→moons0.283 ± 0.0480.333 ± 0.129
N-→scurve0.297 ± 0.0640.383 ± 0.134
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Algorithm↓Dataset→CiteEBMulti
TrajectoryNet (Tong et al., 2020)*0.848±-
Reg.CNF (Finlay et al., 2020)*0.825 ±-
DSB (De Bortoli et al., 2021)0.953 ± 0.1400.862 ± 0.0231.079 ± 0.117
I-CFM0.965 ± 0.1110.872 ± 0.0871.085 ± 0.099
SB-CFM1.067 ± 0.1071.221 ± 0.3801.129 ± 0.363
OT-CFM0.882 ± 0.0580.790 ± 0.0680.937 ± 0.054
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NFE / sample → Algorithm↓1001000Adaptive
FIDFIDFIDNFE
DDPM7.48274
OT-FM (reported)6.35142
VP-FM (reported)8.06183
S.I. (reported)10.27
OT-FM (reproduced)13.74212.49111.527139.83
VP-FM (ours)7.7724.0484.335525.92
OT-FM (ours)4.6403.8223.655143.00
S.I. (ours)4.4884.1324.009146.12
I-CFM (ours)4.4613.6433.659146.42
OT-CFM (ours)4.4433.7413.577133.94
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Algorithm↓g=0.1σ=0.3σ=1
Identity9.17 ± 5.689.17± 5.689.17± 5.68
I-CFM4.85 ± 5.093.44± 2.031.59 ± 0.83
OT-CFM2.81± 2.621.91 ± 1.301.04 ± 0.60
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Input:Empirical or samplable distributions qo,q1,bandwidth o,batch size b,initial network ve. while Trainingdo
/* Sample batches of size b i.i.d. from the datasets xo~qo(xo)；x1~q1(x1)
T2g2← Sinkhorn(x1,xo,2σ²)
(x0,x1）~T2g² t~u(0,1)
μt←tx1+(1-t)xo
x~N(μt,g²t(1-t)I) ut(x|z)←-(-(t∞1+(1-t)0)+(m1-00） >From (21)
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N-→8gaussiansmoons-→8gaussiansN→moonsN-→scurvemean
OT-CFM1.284 ± 0.0281.587 ± 0.2041.464 ± 0.1581.499 ± 0.1571.484 ± 0.192
CFM0.993 ± 0.0211.102 ± 0.1711.059 ± 0.1581.008 ± 0.1061.046 ± 0.132
FM0.839 ± 0.0961.076 ± 0.1261.127 ± 0.1231.014 ± 0.170
SB-CFM0.713 ± 0.3860.794 ± 0.2931.143 ± 0.3891.230 ± 0.4240.935 ± 0.397
Reg. CNF2.684 ± 0.0529.154 ± 1.5359.022 ± 3.2078.021 ± 3.288
CNF1.512 ± 0.23417.124 ± 4.39827.416 ± 13.29918.810 ± 12.677
ICNN3.712 ± 0.0913.046 ± 0.4962.558 ± 0.3902.200 ± 0.0342.912 ± 0.626
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RWISMCMC
log 2S timelog 2S time
adaptive Dormand-Prince (tolerance O.O1) integration
CFM OT-CFM-0.068±0.041 -0.076 ± 0.09826.6 ± 8.4s 13.3 ± 1.7s0.029 ± 0.037 0.009 ± 0.04534.6 ± 6.0s 12.8 ± 1.2s
FM-0.033±0.05726.5 ± 7.7s0.027 ± 0.03130.9 ± 5.8s
Euler (N=10) integration
CFM0.281 ± 0.2024.0 ± 0.8s0.336 ± 0.0303.7 ± 0.7s
OT-CFM-0.039±0.0304.2 ± 0.6s0.146±0.1074.1 ± 0.8s
FM0.176±0.0444.1 ± 0.7s0.334±0.0663.9 ± 0.6s
PIS (SDE)Euler-Maruyama (N=100) integration -0.018 ± 0.020
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