Dequantified Diffusion-Schrödinger Bridge for Density Ratio Estimation
Authors: Wei Chen
Shigui Li
Jiacheng Li
Junmei Yang
John Paisley
Delu Zeng
Venue: ICML 2025 poster (2025)
Keywords: diffusion-bridge
optimal transport rearrangement
asymptotic estimation
density-ratio estimation
Primary area: probabilistic_methods->everything_else
TL;DR: Introduce D³RE, a dequantified framework combining diffusion/Schrödinger bridge interpolants with optimal transport to resolve the density-chasm problem. Theoretically guarantees lower asymptotic error.

Abstract: Density ratio estimation is fundamental to tasks involving f-divergences, yet existing methods often fail under significantly different distributions or inadequately overlapping supports --- the density-chasm and the support-chasm problems.
Additionally, prior approaches yield divergent time scores near boundaries, leading to instability.
We design $\textbf{D}^3\textbf{RE}$, a unified framework for robust, stable and efficient density ratio estimation.
We propose the dequantified diffusion bridge interpolant (DDBI), which expands support coverage and stabilizes time scores via diffusion bridges and Gaussian dequantization.
Building on DDBI, the proposed dequantified Schr{\"o}dinger bridge interpolant (DSBI) incorporates optimal transport to solve the Schr{\"o}dinger bridge problem, enhancing accuracy and efficiency.
Our method offers uniform approximation and bounded time scores in theory, and outperforms baselines empirically in mutual information and density estimation tasks.

Dequantified Diflusion-Schrrdinnger Bridge for Density Ratio Estimation
Wei Chen Shigui Li! Jiacheng Li
Junmei Yang?
John Paisley?
Delu Zeng?
Abstract
mutual information estimation (Colombo al., 2021) and likelihood- -free inference (Thomas et al., 2022), it struggles with a critical challenge known as the density-chasm prob- lem, where multi-modal or divergent distributions lead to unstable ratio estimates (Rhodes et al., 2020).
Density ratio estimation is fundamental to tasks involving f-divergences, yet existing methods often fail under significantly different distribu- tions or inadequately overlapping supports - the density-chasm and the support-chasm problems. Additionally, prior approaches yield divergent time scores near boundaries, leading to insta- bility. We design D'RE, unified framework for robust, stable and efficient density ratio estimation. We propose the dequantified diffu- sion bridge interpolant (DDBI), which expands support coverage and stabilizes time scores via diffusion bridges and Gaussian dequantization. Building on DDBI, the proposed dequantified Schrödinger bridge interpolant (DSBI) incorpo- rates optimal transport to solve the Schrödinger bridge problem, enhancing accuracy and effi- ciency. Our method offers uniform approximation and bounded time scores in theory, and outper- forms baselines empirically in mutual informa- tion and density estimation tasks. Code is avail- able at http:/gghhhbboo/o//////////ueuee Diffusion-Brinne Seiidg--Densitye Ratio-EEstimatinn git.
Existing methods like telescoping ratio estimation (TRE) (Rhodes al., 2020) and its continuous extension DRE-co (Choi et al., 2022) estimate density ratios via intermediate Steps. TRE improves accuracy by adding more intermediate variables, but increases model complexity linearly. DRE- 0o uses continuous- citiiuuouu-tmmeeeeee score matching to avoid this, yet both face a core challenge in the support-chasm problem (see Definition 3.1), where supp(go) n supp(gi) ) is small or empty. This leads inadequately overlapping supports and ill-defined ratios (Srivastava et al., 2023).
To address the support-chasm problem, we unify the interpo- lation strategies in prior works as deterministic interpolants (DI) and propose the diffusion bridge interpolant (DBI), which uses diffusion bridges to enable diverse trajectory exploration and smooth transitions between distributions. By Theorem 3.2 and Corollary 3.3, DBI expands support coverage and trajectory sets beyond existing approaches.
A second challenge arises in prior methods: As l + 1-, the absolute time score Egg [[0, log qill diverges for both DI and DBI (Theorem 3.4), leading to unstable estimations at the boundary. To mitigate this, we propose Gaussian dequan- tization (GD), which addresses boundary densities qo and 41 via Gaussian convolution, ensuring E.g. []O, log qel] re- mains bounded over E [0, 1] (Corollary 3.8). The resulting dequantified diffusion bridge interpolant (DDBI) balances robustness and computational efficiency.
1. Introduction
Quantifying distributional discrepancies via f-divergences- defined through density ratios r(x) (x)//o(x) is foun- dational in tasks such as domain adaptation, generative modeling, and hypothesis testing. However, directly es- timating rxx) by modeling go and @ becomes intractable in high dimensions, motivating density ratio estimation (DRE) methods that bypass explicit density modeling (Sugiyama et al., 2012). While DRE underpins modern techniques like
To further reduce estimation error and improve efficiency, the dequantified Schrödinger bridge interpolant (DSBI) is derived by integrating DDBI with optimal transport rear- rangement (OTR), solving the Schrödinger bridge problem (Proposition 3.6). Together, applying DDBI and DSBI to DRE leads to the Dequantified Diffusion Bridge Density Ratio Estimation (D DRRE) framework. We summarize these developments in Table 1.
'The School of Mathematics, South China University of Tech- nology, Guangzhou 510006. China "The School of Electronic and Information Engineering, South China University of Technol- ogy, Guangzhou 510006, China "The Department of Electrical Engineering, Columbia University, New York, NY 10027, USA. Correspondence to: Delu Zeng <dizeng@ scut.edu.en>.
Experimental results show that DRRE improves robustness and efficiency in downstream tasks such as density ratio estimation, mutual information estimation, and density es-
Proceedings of the nrnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 20255 by the author(s).

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Dequantified Diflusion-Schridinger Bridge for Density Ratio Estimation
2. Background
Table Comparison of advantages of interpolants in this work.
Let X. ~ qo (o()) and X, ~ qi (x) be random variables in Rd with set-theoretic supports supp(go) and supp(qı). Upper cases Xo and X₁ denote random variable, while lower cases Xo and X1 denote the samples of random variables.
Diffusion bridge
GD (stable)
OTR (efficient)
(robust & stable)
DI (previous)
DBI (ours)
DDBI (ours)
DSBI (ours)
2.1. Density Ratio Estimation
Density ratio estimation (DRE) aims estimate the density ratio r (x) = (x using i..d, samples from both distribu- tions. A common approach is density ratio matching via Bregman divergence minimization (Sugiyama et al., 2012), which optimizes parameterized ratio ra by minimizing
(x
BD(ro) = Eu (Xo) gi (xi)
log Ixro(xo) + log 1+ro(xx)
rx(x))
(1)
(a) DI (previous)
(b) DBI (ours)
where denotes the trainable parameters of rg. The mini- mizer Eq. (1) satisfies ræ- = r" However, DRE suffers from the density-chasm problem (Rhodes al., 2020).
TRE mitigates this by employing divide-and-comquue strategy. It partitions the interval [0, 1] into M E Z sub- intervals with endpoints {m/M (MnMMMad J constructing interme- diate variables Xm/M/ ~ 9m/MM(x (x) via linear interpolation
(c) DDBI (ours)
(d) DSBI (ours)
Figure/ 1. Trajectory comparison among DI, DBI, DDBI and DSBI. Our methods yield broader trajectory sets, with interme- diate distributions exhibiting wider support than those of DI. The entropically regularized transport losses for subfigures (a-d) are 44.17, 31.14, 31.15, and 8,26, respectively (See Eq. (17) for de- tails.). Lower loss indicates increased path diversity.
Xm/M V m Xo + TmmMM X1
where Tm/Ms increases from 0 to 1. The density ratio de- composes into a telescoping product
(1(x)
M-
4 4mm+a//dd (x)
r'(x)
II
timation. Fig. 1 illustrates comparison of interpolation strategics among DI, DBI, DDBI, and DSBI, with light blue points representing intermediate samples drawn from go and Q1- Our proposed methods (DBI, DDBI, and DSBI) enable broader exploration of alternative trajectories, producing intermediate distributions with larger support compared to DI, consistent with Theorem 3.2 and Corollary 3.3 below.
9u(x)
qm M (x)
m==
(3)
M-
M-
mm/M (x)
II ra (x); m==
where mm/a is the target intermediate density ratio, esti- mated by a parameterized neural network re.. with trainable parameters 0m. In this case, A networks must be trained. While larger M improves accuracy, it increases compu- tational cost and may still fail to sufficiently reduce the KL divergence, KL (L(9m/M) (1q 1)/M). leaving the density- chasm problem unaddressed.
The main contributions of this work are as follows:
We propose D²RE, the first unified framework to ad- dress both the density-chasee and support-chasee prob- lems via uniformly approximated density ratio estima- tion (Proposition 3.5). Our interpolants expand both the support (Theorem 3.2) and trajectory sets (Corol- lary 3.3), alleviating the support-chasm problem.
DRE-c0 (Choi et al., 2022) extends TRE M oo, rep- resenting the log ratio as an integral of the time score
logr*(x1 = L'' O, (logg(())d ~ f si (x, t)dt,
We incorporate guidance mechanisms to improve inter- polant quality: GD improves the stability of time score functions at boundary (Theorem 3.4,Corollary 3.8), while OTR leads to more stable (Theorem 3.7) and efficient (Fig. 8) estimation.
(4) where 0 log (x R (log (r))) log qi (x)) At with an infinitesimal gap At = lim 1/M denotes the time score. The time score model s approximates the time score via minimization of
Experiments demonstrate D²RE's superiority in den- sity ratio estimation, mutual information estimation, and density estimation.
[x()) O. (x)) .6(xx1))]] (5)
L1 = (x(0)g(()) (x)
2

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Dequantified Diflusion-Schridinger Bridge for Density Ratio Estimation
where X(-) [0. 1] R4 is a time-dependent weighting function, and (() 24(0, 1] is the uniform distribution over the interval [O.1]. When the time score model satisfies d log q: (x) = sg- tx, the loss function reaches its mini- mum value L1 (0*) ). Unlike in TRF, only one network, *g-
to deterministic interpolant (DI), defined as
I(Xo, (,,) = ox + 1
(8)
where I: R x R x [0. 1] + IR is an interpolant contin- uously differentiable (X.. X1,t) and the time-indexed coefficients 04, E C² 1] are monotonic with ar de- creasing and Br increasing in They are strictly positive and and a1 Bo O,
needs to be trained in DRE-x..
2.2. Denoising Diffusion Bridge Model
satisfy boundary conditions 0g- = B1 with constraints 04 + Bt = 1 or oa + 33 = 1, E [0. 1].
Denoising diffusion models (DDMs) simulate a diffusion process (ejo.... which serves as a continuous bridge be- tween X. and X₁. This process is described by the solution to an Itō stochastic differential equation (SDE)
Prior methods, such as TRE (Rhodes et al., 2020) and DRE- oo (Choi et al., 2022), are specific cases of DI distinguished by their choices of at and 3r (see Appendix B.I).
dx,- = f(X,, t)dt + g())d
(6)
However, DRE with DI suffers from the support-chasm problem, where minimal overlap between supp(go) and supp(gi) leads to ill-defined ratios (Srivastava et al., 2023).
where (We)ee | is a standard Wiener process, f: I& x [0, [] + Rd and g : [0, 1] - R are termed as the drift coef- ficient and scalar diffusion coefficient of X,, respectively.
Definition 3.1 (Support-chasm Problem). Let go. 91 be probability density functions with supports supp(go) and supp(at), respectively. For a given threshold e > 0, if i (supp(go) n supp(q1)) < E, then a support-chasm exists between do and qı. where μ the Lebesgue measure.
Conventional DDMs (Li al., 2023; Song et al., 2021; Li et al., 2024a; Xu et al., 2024) require either qo 91 to be simple, tractable distribution (e.g., isotropic Gaussian), which limits their ability to bridge arbitrary complex distri- butions and restricts applications such as DRE.
Diffusion Bridge Interpolant. To mitigate the support- chasm, we introduce a Brownian bridge (Br)eeoo1). leading to the diffusion bridge interpolant (DBI)
Denoising diffusion bridge models (Zhou et al., 2024a) over- come this limitation by simulating stochastic processes that interpolate between paired distributions with X, as end- points. These processes are derived from the SDE in Eq. (6) via Doob's h--transform (Doob& & Doob, 1984),
x [Xo, X1, 1) + ,B,
where 7 E IR20 is the noise factor controlling the stochastic component B, This factor provides flexibility by adjusting the variability introduced by the bridge different stages of interpolation. When y = 0. DBI reduces to DI.
dx,- = ((X,,)) + ())(()X X.))d + g())W,, (7)
where h( h(X., 1, xi) Vx, (X, X.) is the h func- tion representing the gradient of the log transition kernel from time to 1. The process explicitly depends on X, and, given X. and X1, it forms a diffusion bridge with transition kernel ge (x Xo, X1,X,). A special case of the dif- fusion bridge, termed the Brownian bridge, arises under the conditions f(X,, t) := gt) := 1 and h(X,, t, X2) == X,-,, The Brownian bridge is defined as the solution to B.-B, dt + dW, and its transition kernel is given by g: (b Bo. B,) = N((1- - l)Bo- + (Bi, ((1 - E)).).
Since B, is Gaussian process with zero mean and variance t(1 - t). i... B, ~ t(1 - Ey). the transition kernel of the DBI can be derived follows from Eq. (9) as ge Xp, X,) WII(X,, X),t), - Ea). By applying the reparameterization trick, DBI admits the equivalent form
X, I(Xo, X1, ((1-1)_2.,
(10)
dB,- = 1-t
where Z, ~ (O, Ea). ensuring analytical tractability and efficient implementation. The term vt(1 z) adds controlled variability and provides robustness and flexibility to the interpolant, expanding the support of ge-
3. Method
In this section, we we extend prior works into a unified framework called Dequantified Diffusion-bridge Density- Ratio Estimation (D*RE). D²RE offers a robust and ef- ficient solution for density ratio estimation, theoretically mitigating the density-chasm and support-chasm problems.
Theorem 3.2 (Support Set Expansion). Let Xo and X, be random variables. Let qe (x) and (x(x) denote the marginal densities under DI and DBI, respectively. Then, for any I E (0, 1), the support of df includes expands beyond the support of qu, i.e., supp(g)) 2 su
See Appendix A.I for detailed derivation. This result shows that DBI covers a larger or equal region of the space com- pared to DI, providing theoretical justification for its ability to mitigate the support-chasm problem in D'RF.
3.1. Support-chasm Problem
Deterministic Interpolant. We summarize the inter- polants used in prior works, such as TRE and DRE-co,
3

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Dequantified Diflusion-Schridinger Bridge for Density Ratio Estimation
Corollary 3.3 (Trajectory Set Expansion). Under the same setup as in Theorem 3.2, let the trajectory sets generated by the DI and the DBI be denoted by T {[x))e X2 E supp(g:) and T = { [0 x E s respec- rively. Then, TV contains T, Le.. 2 T..
The DDBI can be expressed in terms of the original DBI by defining perturbed variables as X; X; + Ze. where Zg e(O, This results in
X = I(xo, X1, 1)+ v t(1 - t)y² (af + Bf)= (15)
See Appendix A.2 for details. This generalizes the support expansion result to entire paths, implying that DBI, by in- jecting noise, explores broader set of trajectories than DI. As result, it provides better coverage of the interpolation space, enhancing robustness across diverse distributions.
Here, the additional term (aî + 88)= reflects the effect of GD, yielding smoother interpolation. As result, the transition kernel of the DDBI, </(x Xo, X1), is given by (I X1, t), (t(1-t) t)y² + {a + }))=) Ea). which shares the same mean trajectory as DBI (see Eq. (10)).
We have also amalyzed the uniform approximation of density ratio using the DDBI. The relationship between r(x) and r' (xx %
3.2. Gaussian Dequantization for Mollified Time Score
A second fundamental challenge in conventional inter- polants such as DI and DBI is the divergence of the absolute time score, as shown in Theorem 3.4.
g (x) is characterized by Proposition 3.5.
Proposition 3.5. Let r(x) and (x) be the density ratios with and without GD, respectively. Then, r' is a uniform apporximation of r, with the error bounded by:
Theorem 3.4. Les (X.) be a DI defined in Eq. (8). Under Assumption A.I and Assumption A.2, the time score for any t E (0, 1) satisfies
I| - -1 S <O()).
(16)
Thus, as E - O, (xx) + r(x) in the uniform norm.
| -L
1.8cl
gg 010 log qell= 2 ()-
Be
See Appendix A.4 for detailed derivation. Proposition 3.5 confirms that the dequantified density ratio r'(x) is a uni- form approximation of r(x) for sufficiently small
(11)
Egg [|[ log qt
where L is the Lipschitz constant in Assumption A.2. More-
3.3. Optimal Transport Rearrangement
over, if L 1, this lower bound diverges to infinity
To further reduce estimation error and improve efficiency of DRE, extend the probability path of DDBI by aligning it with the entropically regularized optimal transport (OT)
lim Egg [10 log qt = -00,
(12)
lim Ey [ log q: +00. t-
T22a = argmin / ||xo - -x11111 - 2T²H(TT, (17)
Proofs can be found in Appendix A.3.
xEll(
where I( gi is the set of all probability paths with marginals % and q/, and H(T)) is the entropy of æ. The coef- ficient 2-² is regularization factor (details in Appendix B.2).
Dequantified Diffusion Bridge Interpolant. To stabilize the time score near the boundary, we introduce Gaussian dequantization (GD) by adding controlled perturbations to boundary samples, effectively handling the boundary den- sities and resulting uniformly bounded time score across [o, 1] (see Corollary 3.8 for details).
We apply the scalable Sinkhorn algorithm (Cuturi, 2013) to mini-batches [(x6- x( xx),, ) % x 4í obtaining re- For conve-
arranged sample pairs {(X,, 81),, 1N i== nience, we refer to this procedure as optimal transport rear- rangement (OTR), which preserves the marginals and only changes the sample pairing.
Specifically, for x, ~ g. its dequantified form x can be obtained by
x - x; =xx +
2g ~ E,), (0, (13)
Dequantified Schrödinger Bridge Interpolant. Apply- ing OTR followed by DDBI yields the dequantified Schrödinger bridge interpolant (DSBI)
where E IR. is small. The resulting dequantified densities are obtained via Gaussian convolution, gf q: (N =E.). This smoothing ensures bounded time scores near 1 = 0 and = 1 (see Theorem 3.7 and Corollary 3.8 for details), improving stability in DRE.
X -==[(,] X1 t) ()1-))² + (at B7) (18)
where o, = 1 1. and 3, = f are fixed.
Incorporating GD into the DBI yields the dequantified diffu- sion bridge interpolant (DDBI), formulated as
We show that rearranging the mini-batches via OTR leads to an interpolant that naturally solves the Schrödinger bridge (SB) problem (Schrödinger, 1932).
X. I(X X,,1) + v1(1-1)20 ((1-1)7 24.
(14)

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Dequantified Diflusion-Shrrddinge Bridge for Density Ratio Estimation
Proposition 3.6. The probability path defined by DSBI solves the SB problem
However, O log (x) is intractable in practice. To bypass this, an equivalent integration-by-pee -oraritrtn-----ssss--eeaa form (Song & Er- mon, 2020; Choi et al., 2022) is proposed
π = argmin (T(
(19)
=el(()-99)
C3 Eng(xo)) (x) (x)) (xo, 0) (x1,s((x1111)
where Frof is a Wiener process scaled by 7/
3e [X(1)sb(x, t)] + 2 (11))66(x.0)) (24)
+ (x(0
See Appendix A.5 for proof. This result suggests that DSBI, principled integration of DDBI and OTR, implic- itl solves the SB problem and provides minimum-cost stochastic interpolation between % and g'.
where O (10))((x))) x(1) + X(I)s) (), 0 t) and X denote the time derivative of the time score model and weighting function, respectively. The first two terms enforce the boundary conditions. Le and La differ only by a constant C' independent of A. In practice, for stable and effective training, the joint score-mate score-matching loss is implemented. as described in Appendix C.2.
Furthermore, to rigorously quantify the improvement brought by OTR, we establish the following result com- paring the upper error bounds of DDBI and DSBI.
Theorem 3.7. Consider the DDBI and DSBI with = 1-t, 3 == L. Let æ E [T g be any coupling for DDBI, and T2,a the entropically regularized OT coupling for DSBI. Then, for all E [0, 1}, the variance of the time score under DSBI is no greater than that under DSBI, i.e.
Estimating Target Log Density Ratio. Given the optimal parameters O' obtained by minimizing L3, the log density ratio at any point x can be estimated as
(8, log d)). Corollary 3.8. For all L E [0,1] the time score of DDBI is uniformly bounded by
Var (8, log ()) S Vara.
(20)
[x) se L.' 0, log d((x)dt L. s/g sg (x, t)dt, (25)
Egg [|0 log d'll S
V0
oî
1 Ex
| x11]]
+
ofd
207¹
(21)
based on Definition 3.9. See Algorithms and 2 for the full training and estimation procedures using DDBI and DSBI.
2t + I)E is strictly positive. where of = t(1- 1ol-tty t)p² + (2t²
Algorithm 1 Training and estimation of D with DDBI Input: Probability densities qo and qı, time score model sgg. coefficients a and 8, noise factor y and e.
See Appendix A.6 and Appendix A.7 for detailed deriva- tions of Theorem 3.7 and Corollary 3.8. These results es- tablish that both DDBI and DSBI admit bounded time- integrated time score, E [ log Il. in contrast to the di- vergent lower bound exhibited by DI (see Theorem 3.4). Moreover, it provides a formal justification for the error reduction achieved by DSBI through the coupling 2,,-.
Initialize: trainable parameters @ of s a given point x.
X. ~ (o(x), ~ (x), ~ (00
2g ~ <N 2~ NN Ea)
X t xo xo Zz, x1 + X x| +
% GD
x 5 axx + + + /t(1 2)
% DBI
0' - Ad Adaa(0,VVo{(())
logr(x) <- odeint adjoint(s 1).
3.4. Dequantified Diffusson Bridge DRE
Output: estimated log density ratio log r(x).
For the DDBI, r' (x) can be approximated effectively us- ing neural network, as formulated in Definition 3.9. See Appendix A.8 for detailed derivation.
Algorithm 2 Training and estimation of 'R with DSBI Input: Probability densities qo and 91. time score model </-- coefficients o, and 33. noise factor ? and e. Initialize: trainable parameters 0 of s a given point x. Ea)
Definition 3.9 (Dequantified Diffusion bridge Density Ratio Estimation, DER,), Given the marginal probability density of DDBI, d. (x), the log density ratio for a given point x E Rdcan be estimated as
Xa ~ (x(x), XI ~ q (x), ((00 1)
logr*(x) 2 L.'' O, log (x)dt,
ze NN (x)
(22)
X t x + 2g, X +- x1 +
% GD
T2y2 + Sinkhorn (x), x 27²²
% OTR
where d.- denotes the time derivative operator.
x1) ~ T2n2
Time mcore-matching Loss. We train a time score model s (x, t) to approximate the time score 0 log df (x) by mini- mizing the time score-matchin loss (Choi et al., 2022),
X 0x B,x + ((1
% DBI
0t + Adam voLz
log rxg(()) +- odeint_adjoint( (0, 1). Output: estimated log density ratio log rxx).
L2= = Bx(e)of(x)) X(t) (e log (x) - (x(( (23)
5

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Dequantified Diflusion-Schridinger Bridge for Density Ratio Estimation
4. Experiments
estimation artifacts. Further comparisons across training epochs available in Appendix C.4.
For experiments involving D'RE, we implement both DDBI and DSBI. Unless specified otherwise, we use the follow- ing settings: 0t 1 t, 3 = t, 7, 0.5, 1e and (t) 2 t). Under this configuration, the in- terpolant 1( () (1 X) + LX1 aligns with the Benamou- Brronnrrrrreerreeee solution to the optimal transport problem in Euclidean space (McCann, 1997). The parameterized score model trained with time score matching loss L3 and optimized with Adam optimization method.
Energy-based Modeling on MNIST. We applied the proposed D'RE framework for density estimation on the MNIST dataset, leveraging pre-trained meergy-based models (EBMs) (Choi et al., 2022). Experimental details are in Ap- pendix C.4. The results are reported in bits- bits-per-de per-dimension (BPD). Results in Tab. 2 show that D`RE achieves the low- est BPD values across all noise types (Gaussian, Copula, and RQ-NSF), outperforming baselines and existing methods. See also Tab. 3 in the Appendix.
4.1. Density Estimation
Specifically, Ro consistently surpasses DRE-oc and its variant with OTR. The DSBI method delivers the best over- all results, achieving BPD values of 1.293 (Gaussian), 1.170 (Copula), and 1.066 (RQ-NSF), demonstrating its robust- ness and effectiveness in optimizing density estimates. Com- pared to traditional methods like NCE and TRE, DORE shows significant improvements, especially under challeng- ing noise distributions like Gaussian and Copula, where baseline methods yield higher BPD. These findings under- score R²RE's superior performance in accurately estimating densities and modeling complex data distributions.
go (x) [x(x] be the target density ratio, where gi (x) an intractable data distribution, and (x(x)) is the simpler, tractable noise distribution. Once the estimated density ratio
Let rxx)
ro- is obtained, the log-density of 91 can be approximated as log Q1 (x) x log ro- (x) + log (x).
2-D Synthetic Datasets. We trained DRE-c0 (baseline) and DORE (ours) on eight 2-D synthetic datasets, includ- ing swissroll, circles, rings, moons, 8gaussians, pinwheel, 2spirals, and checkerboard, for 20,000 epochs using the joint score matching loss (details in Appendix C.2). The density estimation results are shown in Fig. 2.
Table 2. Comparison the estimated densities on MNIST dataset based on pre-trained energy-hased models. The results are reported in bits bits-per-ding per-dir (BPD). Lower is better. The reported results for NCE and TRE are from Rhodes al. (2020).
browiine
(D0)
Method
Noise type
Noise
BPD (L)
NCE
Gaussian
2.01
1.96
D (00B) suro
TRE
Gaussian
2.01
1.39
DRE-c
Gaussian
2.01
1.33
D'rre
(D89I curs
DRE-cC+OTR
Gaussian
2.01
1.313
D (DDBI)
Gaussian
2.01
1.297
Figure 2. Density estimation results on eight 2-D synthetic datasets for different methods. DRRF. effectively estimates the density for both multi-modal and discontinuous distributions.
DdRE (DSBI)
Gaussian
2.01
1.293
NCE TRE
Copula
1.40
1.33
Copula
1.40
1.24
DRE-20
Copula
1.40
1.21
Our experiments demonstrate that D'RE effectively esti- mates the density for both multi-modal and discontinu- ous distributions, outperforming DRE-co. The baseline struggles, especially with complex datasets like rings and checkerboard, where significant distortions occur. While the OTR trick reduces errors, remains insufficient for intricate datasets like 2spirals and pinwheel.
DRE-cO+TR
Copula
1.40
1.204
DRR (DDBI)
Copula
1.40
1.193
DR (DSBI)
Copula
1.40
1.170
NCE TRE
RQ-NSF
1.12
1.09
RQ-NSF
1.12
1.09
DRE-20
RQ-NSF
1.12
1.09
In contrast, the D'RE framework, leveraging both DDBI and DSBI, achieves significantly improved density estima- tion. DDBI captures fine- -eeegrrided details, while DSBI pro- vides robust performance across all datasets, particularly excelling in modeling complex distributions and mitigating
DRE-co+OTR
RQ-NSF
1.12
1.072
DORE (DDBI)
RQ-NSF
1.12
1.072
DRRE (DSBI)
RQ-NSF
1.12
1.066
6

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Dequantified Diflusion-Scrridinger Bridge for Density Ratio Estimation
(1ats)
MI (eatss
8.5
ground truhh
ground truth
buth
basellne (inear)
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basel ne (liear)
26
baseline (inear)
80-
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basel no (Follmer)
baseline (Folimer)
DSB1 (yF-0.00 Enear, ours)
17-
(f--0.01, lhear ours) DOBI (y" 01, Inear, oun)
DS81 (F9-0.01. Inear, ours)
7.-
DDBI (>"0.01. Inear, oun)
DDBI (y 01, linerr
20003
4000
00000
60030
8000
100000
50000
100000
190000
200000
Epochs
Epochs
Epochs
(a)
(b) 80
(c) d 120
Figure 3. Evolution of estimated MI across epochs with varying methods and dimensions (40, 80, 120). D'REE outperforms the baseline in both speed and precision. DRE-c0 (Choi et al., 2022) is regarded as the 'baseline" in this experiment.
4.2. Mutual Information Estimation
Data
Mutual information (MI) measures the dependency between
two random variables X ~ p(x) and Y q(y), defined as
0.001
p(x)((y] Here p(x, y) be their joint density. q(y) = of and p(x) (N Ea), with 2² = le - 6 and d = f40, 80, 120). are two d- dimensional correlated Gaussian distributions. The exper- imental setup in DRE-co (Choi et al., 2022) is adapted to implement RER DRE-co serves as the benchmark. More details can be found in Appendix C.3.
MI(X, Y) = Bp(x,y) log p(x,y]
0.1
The evolution of estimated MI across epochs for d (40, 80, 120). comparing D'REE with DRE-co, are analyzed. Results in Fig. 3 show that the red (DSBI) and green (DDBI) curves outperform the blue and yellow (DRE-oo) curves in two aspects. First, DRRE converges to the true MI value more rapidly expands trajectory sets (see Corollary 3.3), improving interpolation accuracy. Second, it exhibits greater stability with fewer fluctuations around the true MI, indicat- ing more reliable estimates.
1.
Figure 4. Ablation study on the effect of n² for density estimation on 2-D toy data. The first row displays the results for the ground truth data. Each subsequent row, from top to bottom, corresponds to 7² values of 0.001, 0.01, 0.1, 0.5, and 1.0, respectively.
ical modes of the 2-D data, particularly in high-density regions. Notably, 7² = 0.1 achieves the closest visual and structural resemblance to the ground truth, suggesting its suitability for low-dimensional tasks requiring both fidelity and robustness. These results underscore the necessity of tuning 7² to mitigate under- reder-regularizatioo artifacts while preserving distributional complexity.
We conclude that DRRE outperforms the baseline in both speed and precision. For d = 120, the MI estimated by D²RE is much robust than that of DRE-oo.
4.3. Analysis and Discussion
Ablation Study on x2 The ablation study on 72 for density estimation (Fig. 4) reveals systematic trade-offs in performance across regularization strengths. For small n² 0.001, the model achieves rapid initial alignment with the ground truth distribution (first row) but exhibits overfit- ting artifacts in later epochs, manifesting irregular density peaks and deviations from the smooth ground truth structure. Intermediate values (72 = 0.01-0.1) demonstrate balanced behavior: 72 = 0.01 preserves finer details while maintain- ing stability, and ,² 0.1 produces smoother approxima- tions with minimal divergence from the true distribution. Larger 2 values (2 0.5) induce excessive regularization, leading to oversmoothed estimates that fail to capture crit-
The ablation study on varying 7² values (Fig. 5) shows distinct convergence behaviors in MI estimation. For all di- mensions (d = (40, 80, 120)), smaller 72 values (< 0.01) lead to faster convergence, especially in lower dimensions (d = 40), but excessively small values (72 = 0.001) cause instability later. Larger ?² values (2 0.1) converge more slowly but stabilize over time, particularly in higher dimen- sions (d = 120). 7² 0.1 offers a balance between speed and stability across all dimensions, suggesting that moderate regularization provides the best MI estimation performance. More results are provided in Appendix C.5.
Ablation Study on GD. To evaluate the effectiveness of
7

---

Dequantified Diflusien-Scrridinger Bridge for Density Ratio Estimation
ground buth
ground trufs
ground Iruth
(nats)
(Inea))
Mi (nates
baseline (inear)
MI (eateg
basel (Shear)
baseline (Folimer)
baseline (Follmer)
basoline (Folimer
8.5
(y Enear, ours)
ts thear Durs)
20
DSBI (? Enoar, ouss)
80-
DSBI (-15,.5 Incer, cucar. curs
DBBI 15.10, lnnar. curs)
DSBI (v--0.1) Inear, ours)
(yP-0.01, lihea, ouss
17
15 ours
DSBI (++.0.00 linea ours)
74-
DSBI (y- -0.00000 Inear, oun)
DSBI 1y 001, inear, oum)
DSBI (y2-0.00 linerr
00000
60030
100000
50000
100000
200000
Epochs
Epochs
Epochs
(a)
(b) 80
(c) d 120
Figure Evolution estimated MI across epochs with varying 72 (0.001, 0.01, 1.03 and dimensions {10, 80, 120). For all
dimensions (d {40, 80, 120 smaller 7² values (S 0.01) lead to faster convergence.
DOBI
DNOOR
CF3
D58I
p30 sm
Figure 7. Density estimation results checkerboard for different methods during training (see more results in Figs. 9 to 16).
Figure 6. Ablation study on GD for eight 2-D synthetic datasets.
the proposed GD module, conduct ablation study by comparing model performance with and without GDD, as shown in Fig. 6. Visually, both DDBI and DSBI show clear improvements density estimation when GD is applied. Without GD, the estimated densities appear blurrier and miss fine structural details, whereas incorporating GD yields sharper and more realistic patterns.
it enables more accurate density estimation. Further com- parisons presented Appendix C.6.
Number of Function Evaluations. We analyze the impact of OTR on NFE, noting that DI and DDBI do not utilize OTR. It shows that applying OTR significantly reduces NFE. Fig. compares NFE across four methods in DRE. The first approach exhibits the highest NFE, indicating reliance on iterative procedures requiring repeated function evaluations. The second approach achieves moderate reduction in NFE, likely by minimizing redundant evaluations through mini- mized transport costs.
We conduct ablation study to evaluate the role of OTR, comparing models without OTR (baseline, DI), models with OTR (DI+OTR), and models from the RRR framework (DDBI and DSBI). Figs. 2 and 7, DI generates distorted and misaligned intermediate distributions. This shows its limited ability to align with the target distribution. DI+OTR improves alignment but remains suboptimal. Models from the DORE further hance distribution quality, with DSBI achieving the most precise alignment. This underscores OTR's crucial role in improving intermediate distributions.
Ablation Study OTR.
600
550
500
Fig. 3 compares MI estimation for DDBI and DSBI across dimensions (d 80, 120). Both outperform baseline meth- ods, but DSBI converges faster and remains closer to the ground truth. The advantage of OTR becomes more pro- nounced in high dimensions (d = 120), where DSBI signifi- cantly outperforms DDBI in both speed and accuracy.
450
400
DI
DI+OTR
DDBI
DSBI (DDBI+OTR)
Overall, OTR improves intermediate distribution alignment. When combined with diffusion bridges and GD, as DSBI,
Figure 8. Comparison NFE for four methods density ratio cstimation task. Applying OTR significantly reduces NPE.
8

---

Dequantified Diflusion-Schridinger Bridge for Density Ratio Estimation
5. Related Works
to derive the DSBI, which offers more efficient and sta- ble density ratio estimation (Theorem 3.7) by solving the Schrödinger bridge problem (Proposition 3.6). Together, DDBI and DSBI form the core of the DeRE framework, enabling uniformly approximated and stable density-ratio estimation (Proposition 3.5). Extensive experiments validate these findings, demonstrating the superior performance of D'REE in tasks such as density-ratio estimation on synthetic data, mutual information estimation, and density estimation.
Density Ratio Estimation. DRE is an essential tech- nique in machine learning but faces challenges in high- dimensional settings and when distributions are significandy different. Early methods (Sugiyama et al., 2012; Gutmann & Hyvärinen, 2012) often struggled with instability, known as the density-chasm problem, in these scenarios. To overcome these challenges, TRE (Rhodes et al., 2020), an extension of NCE (Gutmann& & Hyvärinen, 2012), introduced a divide- and-conquer approach, breaking the problem into simpler subproblems for better performance. DRE-co (Choi al., 2022) further advanced this by interpolating between dis- tributions through an infinite series of bridge distributions, improving stability and accuracy. F-DRE (Choi et al., 2021) uedd an invertible generative model to map distributions into a common feature space before estimation. Recent methods, such as Kato & Teshima (2021), have addressed overfitting in flexible models, and Nagumo & Fujisawa (2024); Luo al. (2024) focused improving robustness to outliers. MDRE (Srivastava et al., 2023) tackled distribution shift through multi-class classification, offering an alternative to binary classification in high-discrepancy cases. Addition- ally, geometric approaches, like Kimura & Bondell (2025), have enhanced DRE accuracy by incorporating the geome- try of statistical manifolds. Building on these advancements, our work proposes novel method improve both the accuracy and robustness of high-dimensional DRE.
While D'RE advances density ratio estimation methods, several directions remain open. Future work could explore adaptive or learned solvers to reduce function evaluation overhead, as well as more expressive interpolants to further improve robustness in handling complex or multi-modal distributions.
Acknowledgements
This research work is supported by the Fundamental Research Program of Guangdong, China, under Grant 2023A1515012281.
Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning. While direct societal impacts are limited, future extensions to applied domains (e.g.. via our open-source codebase) should incorporate domain-specific ethical reviews per deployment contexts.
Diffusion Bridge. Denoising diffusion implicit models (DDIMs) (Song et al., 2020) have been proposed as an effi- cient alternative to denoising diffusion probabilistic models (DDPMs) (Ho et al., 2020), which require simulating a Markov chain for many steps to generate samples. Diffusive interpolants (Albergo al., 2023) provide unifying frame- work of flow-based models and diffusion models, bridging arbitrary distributions using continuous-time stochastic pro- cesses. DDBMs (Zhou et al., 2024a) are proposed as a natural alternative to cumbersome methods like guidance or projected sampling in generative processes. Our pro- posed DDBI and DSBI build upon diffusive interpolants and DDBMs by incorporating Brownian bridge into the interpolation strategy construction.
References
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In this work, we propose DoRE, a unified, robust and ef- ficient framework for density ratio estimation. It provides the first framework for directly addressing both the density- chasm and support-chasm problems, enabling uniformly approximated density-ratio estimation (Proposition 3.5). By incorporating diffusion bridges and GD, we construct DDBI, which expands support coverage (Theorem 3.2, Corollary 3.3) and stabilizes the time score near boundaries (Corollary 3.8). Building upon DDBI, OTR is incorporated
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Dequantified Diflusion-Schrrdinneer Bridge for Density Ratio Estimation
A. Proofs
Assumption A.1. Let 40, 9I R. R__ be probability density functions satisfying: (1) go: 91 E C² (R"),, i.e., qo and qi are twice differentiable and have bounded second derivatives: Vo 1,0 / 1l oo: (2) qo (x 0, and there exists
> 0 such that infx (x()) c. This condition mild in density ratio estimation.
Assumption A.2. The conditional distributions for given X Xx, i... go(x xx) and q1 xs) have L-Lipschitz scores:
L
V log (x x,)I - a4
VV log (x x,)I -
(26)
3
A.1. Proof of Theorem 3.2
Proof. We first consider the support for DI. Under a + 3t = 1, the X, is the convex combination of Xo and X1 for any E (0, 1), and its corresponding support, supp(ge). is the convex hull of supp(go) and supp(g,). i.e., supp(g.) conv (su) U supp(q1)). Under af + P 1, X₁ becomes a linear combination X4 and x1 for any E (0, 1). For both cases, supp(g:) can be formulated as
supp(g: (+)) (ysupp((u) + Btsupp((1)
= {a + 3x1
X E supp(go), x1 E supp(a)), at a + 8 = 1rr at + St 1},
where supp(æ)) and supp((1) are the supports of qu and (1, respectively.
Next, we consider the support for DBI. For given coefficients ae and Br, X can be formulated as:
X I(Xo, X1, 1) vi( 1)>?7, X, + v((1 24,
(27)
The coefficient v1(1 - 1)>² is deterministic for given L. Thus the support corresponding to X denoted as supp(u/), can be expressed as the Minkowski sum of the supports of qe and (0, Eg):
supp(a)) = supp(u:) supp(A((O, Ea)) = {x + x supp(æ:); supp(N Ea))] where supp(NV(O, Ea)) = Rd. The Minkowski sum supp(4.) + supp(N((0, Ex)) at least as large as supp(g)). i.e. supp(a)) 2 supp(g:). This completes the proof.
A.2. Proof of Corollary 3.3
Proof. Let the trajectory sets for DI and DBI be denoted by T -fxx [[x))efoo E supp(g()) and ((x])re/o.1) x e supp((/) }, respectively. Let (x. be an arbitrary element T. From Theorem 3.2, we have supp(g') 2 supp(4)) for any E (0, 1), meaning that x, supp(g)). Hence we have (xeheeo.1 Ty. This directly implies T 2 T.
A.3. Proof of Theorem 3.4
Proof. For the DI, the time derivative of its log-density is governed by the Pokker-Plande equation:
0, logg 9: - u, V 4t.
(28)
where V. and V are the divergence and gradient operators w.r.t. X,- u,(x,) = (á, 8, X, = x.) is the drift term. Taking expectations over and applying the triangle inequality and Cauchy-Schwarr -Schwarz inequality to this equation:
Ege [|0 log lell
= Eyg V u, Voog qe]
> Em,
[|[ uz!
|u, V log dell]
(triangle inequality)
(29)
Eg.
VV u Egg [|u V lol
Eyg V uil E.y
[ull V qgll
(Cauchy-Sehwarzz inequality)
(1) Lower bound of | uel]. The divergence term u, is computed via the Jacobian the inverse mapping:
V = tr((),) = tr E d v X. + V.V X X, x1
(30)
= =Ea àntr (Vx, Xo) + 3.tr (xx, X1) X
12

---

Dequantified Diflusion-Schridnnge Bridge for Density Ratio Estimation
For given sample (Xo, x1) r,, differentiating the the interpolation constraint X 0xxo + 3,x, implicity gives
Ea 04 Xx, x X1. Rearranging yields Vx, Xo az "==E, EEa- BloT Vxx X, and Vx, Vx,X1 X1 - 0n V Xo.
Decomposing V x, Xg into its independent case value and residual:
Vx, Xg ar ' + R,,
(31)
where R, -, Vx,XI X1 denote the residual term in the Jacobian decomposition. By Assumption A.2 and the definition of the score function of the conditional distribution, Xx, log (x X,) = VV, Xo Vx, log qu (xo x²), the Lipschitz continuity of the conditional scores implies VV Xollop S La, and Vx, X1 loop S L3, Here || llop denotes the operator norm. Hence, the absolute value of the trace of the residual term satisfies:
L
tr, S d
RR llop da, Bell x, X1 lop da;
Bg
= dLa,
(32)
31
This directly leads to the conditional expectation: expectation tr( Vx, Xo) satisfies:
[B [tr (R,) X I Egge ||tr (R,)
X,] S dLa; 1. Thus, the conditional
Eg [tr(Vx, Xo) X.]
Egg [tr (a Ea + RR)
Xx|| E. [da; + tr (R.) X:]
dai + E. [tr (Rt) Xx]]
(33)
m da [1 |tr (Rt) X,]l
da, - dLa,"" da; (1 L).
Similarly, we have |E,g [t.(tt(Vx, Xx, X1) x 2 d3. (1 - L). Base on these lower bounds and applying the triangle inequality, we have:
Egg V uel
E.g.
Eug àstr Xo) + 8.tr V x, X1) X,
Exy
| B [tr (Vv Xo) X,] | Eny [tr (Vx, X1) X.]
2 Egg
dlds ai (1 L) - |8, Eny [|tr (IxI X
(34)
2 E.g dddu a7
(11 1
L) - -181 dL.S.
(Itr(Vx,X11) S d. VVx,X S dLB,1)
d (1 T.)l T.1..185))
(2) Upper bound of E. |||le|| | || log de []]. Applying Cauchy-Schwarz
Egg. ||
V log
qell
= v Eg |I| u ||]
B.g. V log [e]
Xo + 3, x. ||]
V
Beg
En
Eq [Il V log qe [|
(35)
Ex
|| Xo + 3x x
E.g || log 2: [|
- V 2d (1) X [2]] + E ||| ||2]
E.g V log 9: |2] = 10:0:1)00000000 [IV log 2: q.l1?))
(V
Here the drift norm Eg ue || is bounded by the second moments of Xo and X,, which are finite under our assumptions.
(3) Lower bound of Eq. [|, log qol]. Finally, bring Eq. (34) and Eq. (35) back Eq. (29), the lower bound of Eg. [|0, log q:ll can be derived:
By [|a, log g I] 2 Egg [[V uul] - E. |||ue||
||V log q: II]
d (1 - L) jivl |
(v V Eq. ||| log q [|3]
L
o
(36)
at 3
Divergence term
Residual
13

---

Dequantified Diflusion-Scrridnnger Bridge for Density Ratio Estimation
The residual term Ol gg V log 4: [2] is finite under Assumption A.2 (Lipschitz scores imply bounded Fisher informa- tion). Hence, the divergence term dominates asymptotically.
Near the boundaries t + 0+ and & + 17, the terms ldel
and
dominate due to the monotonicity and boundary conditions:
l
[
A
(1) As t - 0 ax 1, 8: -
~
|dol:
-
+00.
Ozs
Bg
3r
(37)
jáz
|
(2) As :0 0,
8 + 1,
+00;
1311.
B1
For any L < 1, the prefactor 1 L > 0 ensures:
|drl
lim gg a1 lll 2 lim
d(1 L)
+.00
(38)
t-1
at
This concludes the universal boundary divergence.
A.4. Proof of Proposition 3.5
Proof. Assume both go and qı are smooth with sufficient differentiability. The Gaussian-dequantffied densities are given by:
d((x) = (q: =E,))x)))(())) = f (xx') =E,)
= =faxe) (xxx) [0(x [x(x-x)++ x)+ 8 V2.0(x - x') + 0(62) dx' (Taylor expansion around x')
2
(39)
= =x((x) + f (((x'VV2.65 x)dx' + Oc?3)
= xx)) + 2 2 (x) + (E:²). (Integration by parts),
where V is the Laplacian operator, 5(x x') is the Dirac delta.
Substituting these two expansions into the dequantified density ratio rxx) o (x) [x)x
we have:
(x)
+ 9i (x) + (x)
(())
(x)
0o(x) + (x + + (
41 (x)
8 (x) 8
2 (x) v0 (x)
O(53)
(First-order expansion of fraction)
(x(
2 (x)
(x)
(40)
(x)
[x)
rx)
r(x)
+0(s²).
2
(x())
gx())
A(x)
To bound (xx - r(x) in L the supremum can be computed under Assumption A.1:
||r -| _ &
Axx) (
sup
2
(x))
8
sup
r(x)
(E2)
2
(u(x)
90(xx
[290(x)
2
sup
(x))
+ sup r(x)
2
(x(x)
+0(())
(41)
V 11
V -00
supr(x) + +
2
inf (x(x)
inf (x))
x
x
(1V221ll + ||220ll2~~111122> + +(E7)
2 inl (x)
14

---

Dequantified Diflusion-Schridnnger Bridge for Density Ratio Estimation
Under Assumption A.1, inf 2o (x) has lower bound and these norms have upper bound. Then as -> O.
||'- - r I O(e),
(42)
where the constant C is given by: C = |1 2.00 + 1 Hence, as E - 0, we have r'(x) - r(x), verifying
(x()
the stated proposition.
A.5. Proof of Proposition 3.6
Proof. Based on Theorem 2.4 of (Léonard, 2014), (De Bortoli al., 2021) established that the solution to the SB problem as detailed in Eq. (19), x², satisfies the SB conditions: (I) the optimization problem for n² is equivalent the entropically regularied OT problem, with the optimal coupling π2y² defined in Eq. (17); (2) for samples (X,: ~ '², the associated conditional path distributions n² X6, x1) minimize the KL divergence: E [(x)x( ((x)))))r Jorr)--KK KL (x* x x') [[)]| (- x x()),
where Tee is the reference path distribution satisfying log Tf (x)= x() ||x,-x; 1
2-2 + const. These conditional distributions are optimized using Brownian bridges of diffusion scale y, conditioned on the endpoints x/ and x The marginal distribution intermediate time t along the Brownian bridge is given by q: (x X x1) N( (1 - f)x; + +x1 f(1 +))) (Tong al., 2024). Since our proposed OTR method uses Sinkhorn's 's algorithm to solve the entropically regularized OT problem, and the probability paths of our DDBI with a 1 and 3e t align with those of the Brownian bridge. trajectory x!
generated by first sampling (x/+ x1) ~ '²', then sampling x, ~ qe(- X6, x() satisfies the SB conditions, thus verifying the proposition.
A.6. Proof of Theorem 3.7
Proof. To analyze Varg: (8, log g)), applying the law of total variance to Varg (0: log
yarg (Og log g )xx [Var (ds log (X, X()] + Var(x, (E (0, log (X, X, X())
(43)
For any paired endpoints (X:: X1) u. the interpolant X follows Gaussian distribution conditioned the endpoints
X (1) ~ d6( X6, X N' (m,, Lal), where le X + of +(1 -t)? (a? B8)=. The
conditional score 0 log (X, X,, X is derived explicitly as:
i log N
8 log (2ma?)
|X 1el13)
0 log (X, X): X
4/2 exp
ot
Øt
207
d _
X
+ B,X,
X - Mel
07
(X¹ 11)) +
(44)
2 o
of
2
dō?
3,X1
o7
Ze
Z,
2
dt
20
The term Ex (Var (8, log (X,
X X:))).
Taking the variance on both sides of Eq. (44) and taking expectation over the
coupling distribution
áiX/ +
3:X1
do?
of 11
]
Eg [Var(a, log ( X,,X,))) Ex
Var
Z,
+ Var
+ Var
2007
20
Ot
a, + x 1|
10+
+
0f
Var( (Zz
(45)
0
40
| + Bxx; 11]
0,d
Ex
207
Specifically, when 04 = - - and Br = f, this reduces to E,, [Var(ü, log gi (X/ X X())
F
|IX;
- x
207
s'd
The term Var (E (8, log (X, Xx. X())). Taking the variance on both sides of Eq. (44) and taking expectation over the
15

---

Dequantified Diflusion-Schrrdinger Bridge for Density Ratio Estimation
coupling distribution
dōž
B,X1 0g
+
o211Z.113
Var, [B x X(I)
Var,
Z, +
20
01
20
d
6
67
Var
+
.E/z]]
[1|Z
(46)
20
σt
20
3,X
07
Var
0
d
Tt
20
The last equality holds because Z, is a Gaussian noise, leading to Z ~ NO0 Ea) and |Z,l| ~ x(d).
Bringing Eqs. (45) and (46) into Eq. (43), the variance term Var,, og (0, log g/2) becomes:
Varg (O, log gi) f
1 Br o.
[1x6
x
ofd 20
di = E. |
X
of
dl +
20 207
ogd
(47)
when 1 - and B. = t. Thus, the difference between the upper bound of the variance for DSBI and DDBI becomes:
Var d;
DDRI (Ōg log g/))
DDBI
Var
(0, log g())
| X x=11 - -Eag,3
| - -X:] |
(48)
Ex
X
x(11
- 2- (.((((((T))))))) - Erg,pp
27 2- (((22T))) 0.
This inequality holds because T2y2 is the solution to the entropically regularized OT problem. This completes the proof.
A.7. Proof of Corollary 3.8
Proof. Directly applying the Jensen's 's inequality to Bgg [|0 log dill yields:
Eg [|0 log d/11 S / E, log 9 v Var, (0, log d)) + (E.g [0 log d/l) (Property of variance).
(49)
The first term on the r.h.s. is bounded according to Theorem 3.7, satisfying Varg: (i) log g = oo E |XX - X2I1
+
20
d.d
with of = 1(1 1)7² + (a? + ee.... The second term on the r.h.s. vanishes identically TF (0r log del = 2, o
I d 0, (J (d))) = (1) = Thus, the term Egg [|a, log qf is bounded by:
ofd 20
100 log d'll S
V
Ex
X
x1|1]]
+
< 80.
(50)
A.8. Derivation of Definition 3.9
Proof. Let (X/ be a DDBI. has a transition kernel (x xo, x1) and marginal probability density (x). By dividing the interval [0, 1] into M discrete intervals, the log dequantified density ratio for given point x can be derived:
1xgr'(x) = log (x) (x)
(x) (x)
(x(x)
M
Lmn+1s/M (x)
-log (x) (x) LM (MM-1/M(())
Σ
log
(51)
Lm/Ds (x)
16

---

Dequantified Diflusion-Schrrdinger Bridge for Density Ratio Estimation
According the Taylor's formula, we have log(1 x) while approaches O. In this case, while M is large enough that the difference between Pm/M Xo. x1) and P(m 1011111///// (x Xo, X1) approaches have
log Aim+11/M (x) Am/as (x)
log 1+ Km+1)//((x) ( Am+11//0 (x) A (x)
(52)
lm/M(()) (x)
L1((x)
In the limit as M + oo, the difference term 9 13/2 (x) (x)
can be seen the approximation of a är log log (x)
(x)
evaluated m/M. Taking the limit as M - oo for both sides of Eq. (51), we can derive
M-1
(1n+1//MX (x)
log (xx = lim
log
M-000
Amiss (x)
m=0
M
(m/M((x)
23 lim
(x)
m=0
(53)
lim
log df (x)
+20
r=m/M
0, log (x)dt.
According to Proposition 3.5, the density ratio r' (x) uniformly approximates the target density ratio r (xxx, i.e.
r(x Re logr' (x) = L'' Oe log (x)dt.
(x)
(54)
This completes the proof of this proposition.
17

---

Dequantified Diflusion-Schridinger Bridge for Density Ratio Estimation
B. Preliminaries
B.1. Special Cases of DI
In the case of TRE, the interpolation strategy, as detailed in Eq. (2), represents a specific case of I, characterized by a₁ = / n Br = Tj, with taking discrete values 0, 1/M, ... 1. For DRE-co, the coefficients are defined as Og = expp (0.25 - -mmin) ) - 0.5.3mmn] and 3; = = - oj for the MNIST dataset and 0g = 1 - and 33 = for other datasets. The corresponding stochastic process for the former one aligns with the solution to variance preserving (VP) SDES (Song et al., 2021; et al., 2024b; Xin et al., 2024; Zhou et al., 2025b).
B.2. From Optimal Transport to Entropic Regularization
The static OT problem seeks to find a coupling æ between two probability distributions go and qi that minimizes a given cost function. For the 2-Wasserstein distance with a Euclidean ground cost c(xo, x1) = ||x0 - x ll2, the optimization problem is given by:
/
(1o
(go ()) = inf
=Ell(go-n)) J ||x0 - x1 10000x1111000 (1 x1),
(55)
where [[ 91) denotes the set of joint probability measures with marginals do and Qi- The optimal solution for compactly supported distributions (Villani et al., 2009) is characterized by straight- line interpolations between samples:
x, = ( + 1X1 E [0, 1],
(56)
where Xo qu and X ~ 91- This interpolation aligns with the Benamou-Brenn formulation (McCann, 1997; Zhou et al., 2024b), where the transport paths minimize the kinetic energy in the space of probability measures.
The natural connections between optimal transport theory and straight- interpolations motivate the concept of Batch Optimal Transport (BatchOT) (Pooladian al., 2023; Zhao al., 2025). BatchOT provides a pseudo-determinnitic coupling mechanism by extending the OT principles to minibatch sampling. This ensures practical scalability and aligns theoretical transport paths with computational requirements.
Despite its theoretical elegance, solving the OT problem at scale is computationally challenging due to its cubic complexity in the number of samples. Entropic regularization alleviates this issue by introducing an entropy penalty:
/
W (9o. 41) inf
|xo- - x [x x1) - [M((),
(57)
=gTT(go. / xRd
where & > 0 is the regularization parameter and H() denotes the entropy of æ. This formulation ensures convexity and allows scalable computation via Sinkhorn's 's algorithm (Cuturi, 2013).
Entropic regularization connects OT with the Schrödinger bridge (SB) problem, which models stochastic interpolation between distributions. Given a reference Wiener process scaled by 7, the SB problem finds the most probable stochastic process π that satisfies the marginal constraints qo and q1:
argmin KT(TT Teer), FEll(go-21)
(58)
where aee is reference process. The SB solution corresponds to entropy OT plan with &= =2y²:
X, = (1- t)x, + :X, + Vt(1 2
(59)
where Z. (O, Ex). This formulation introduces stochasticity into the transport paths, effectively modeling uncertainty and noise.
C. Experimental Details and More Results
C.1. Comparison of the Trajectory Sets for Interpolation Strategies
In this section, we provide a detailed comparison of interpolation strategies, specifically deterministic interpolant (DI), diffusion bridge interpolant (DBI), dequantified diffusion bridge interpolant (DDBI), and dequantified Schrödinger bridge interpolant (DSBT). Their intermediate samples and corresponding distributions are visualized Fig. 1.
18

---

Dequantified Diflusion-Scrridinger Bridge for Density Ratio Estimation
(a) DI: DI constrains intermediate samples fixed linear paths between (x(x) and dı (x), resulting in narrow bands across the trajectory space (Fig. (a)). While dense along the paths, DI severely limits support and fails to explore alternative trajectories, making it inflexible and unsuitable for diverse distributions.
(b) DBI: DBI introduces stochasticity through Brownian Bridge noise, expanding support and enabling broader trajectory exploration (Fig. I(b)). Compared to DI, DBI provides greater coverage and variability while retaining tractability, reducing the rigidity of interpolation paths (Zhao al., 2023).
(c) DDBI: Extending DBI, DDBI modulates the noise with deterministic interpolation weights and diffusion components. This results in more dispersed trajectories (Fig. I(c) and larger coverage of the intermediate distributions, balancing controlled stochasticity with enhanced flexibility (Zhou et al., 2025a).
(d) DSBI: DSBI offers full stochastic control over noise and leverages entropy-regullrizze optimal transport, resulting in widely dispersed trajectories and efficient utilization of the trajectory space (Fig. (d)). By minimizing transition loss, DSBI achieves the largest support set and highest diversity among the methods, producing rich intermediate distributions.
Overall, our proposed methods (DBI, DDBI and DSBI) demonstrate clear advantages over deterministic baselines by achieving more comprehensive trajectory space exploration and flexible intermediate distribution generation. These results consistent with our theoretical findings support set and path set expansion, as formalized in Theorem 3.2 and Corollary 3.3.
C.2. Joint Score Matching
In this section, integrate the time score si E R and data score sø R to formulate the joint score sg This joint score is incorporated into the training objective defined in Eq. (24), resulting in a joint score matching objective (Choi et al., 2022):
[sg. R
(0) = 2xx (x) [(00ss²* xx, 0][[]] - 2 (x) (A(1)s (x, 1)[{]]
+ Ee Bx Bx (x) Ev-a(v) [2s(t) 1 (x.
[) 2X (x) t)
(60)
+)(+)lls Lxx (x, t)(x 2X(()v' V (x, (xxx)
where v< ~ q(v) ((0, follows a standard Gaussian distribution, the terms 5 (x, t)] and *x, t)(t] represent the data and time score components of f.x sxx((x, f), respectively.
C.3. Mutual Information Estimation
Mutual information (MI) measures the dependency between two random variables X ~ p(x) and Y ~ g(y). quantifying how much information one variable contains about the other. In this experiment, we employ D²RE to estimate the MI between two d-dimensional correlated Gaussian distributions. Specifically,, we consider g(y) ~N o 2aEE))) and p(x) = (0, Ea). where 0² 1e 6 and d {40, 80, 1203. Let p(x,y) be the joint density of X and Y. The MI p(x,y) between X and Y is defined as MI(X, Y) = Exyy<(xx,)) log p(x)o(x) and can be approximated via DRE. We adapt the experimental setup of (Choi al., 2022) to implement D²RR.
To construct the joint distribution, we use (x) = (O, Ea) and g1 (x) No E), where Σ is block diagonal with A [[, [p, 1]] as 2 x 2 sub-matrices. For QI. it is designed as multivariate normal distribution with a block diagonal covariance matrix along the block diagonal. Each A represents the covariance between variable pairs, while off-diagonal blocks remain zero, ensuring no correlation across pairs, The DDBI and DSBI are implemented, given by (x) the density ratio r r(x) = (x) [x) yielding MI(X, Y) ~ (lggr(x)))))))
X xxX,, + 8 + /f(1 t) (a? + #Z)EZ,, where X,, ~ (x),, X₃ Q1 (x), and Ze N(O,, Ez). We estimate
We train the score model using the joint score matching loss (details in Appendix C.2). The batch size is set to 512 for d = {40, 80, } and 256 for d = 320, with iteration steps of (40k, 100k, 400k, 500k}. respectively. DRE-co serves as the baseline method. Results, shown in Fig. 3, demonstrate that D especially DSBI, produces MI estimates significantly closer and faster to the ground truth compared to the baseline, highlighting its superiority in accurately capturing mutual dependencies between variables.
19

---

Dequantified Diflusion-Schrrdinger Bridge for Density Ratio Estimation
C.4. Density Estimation
Energy-based Modeling on MNIST. We applied the proposed DºRE framework to density estimation on the MNIST dataset, leveraging pre-trained energy -aased models (EBMs) (Choi et al., 2022). Let qi (x) denote the MNIST data distribution and (x(x) a simple noise distribution with three differrnnt settings, as reported in (Rhodes et al., 2020): Gaussian noise, Gaussian copula, and Rational Quadratic Neural Spline Flow (RQ-NSF) (Durkan et al., 2019). We applied an modified version of DDBI of the form X = a; + 8,EBM( ( 2 + (a} + e where X. 4o (x)x,, X ~ Qi (x), mmin )t² 0.5 and B v1 ai. Smin and Smax set to 0.1 and 20,
Z, (0, Ea), at ( 0.25( respectively. The results are reported in bits-pe titssper--diee ver-dimension (BPD), evaluated as BPD = Jin Ex-qg (x) log 91 (x)], where the expectation reflects the log-density of the MNIST dataset. Exact BPD computation is infeasible for EBMs; therefore, we estimate it using two annealed MCMC methods: Annealed Importance Sampling (AIS) (Neal, 2001) and Reverse Annealed Importance Sampling Estimator (RAISE) (Burda et al., 2015).
Table 3. Comparison of the estimated log-density MNIST dataset based pre-trained energy-based models. The results reported BPD. Lower better. The reported results for NCE and TRE are sourced from (Rhodes et al., 2020).
Method
Noise type
Noise
Direct (4)
RAISE (4)
AIS (L)
NCE
Gaussian
2.01
1.96
1.99
2.01
TRE
Gaussian
2.01
1.39
1.35
1.35
DRE-oc
Gaussian
2.01
1.33
1.33
1.33
DRE-cC+OTR, ours
Gaussian
2.01
1.313
1.31
1.31
D'RE (DDBI), ours
Gaussian
2.01
1.297
1.30
1.29
DRRE (DSBI), ours
Gaussian
2.01
1.293
1.29
1.29
NCE
Copula
1.40
1.33
1.48
1.45
TRE
Copula
1.40
1.24
1.23
1.22
DRE-oc
Copula
1.40
1.21
1.21
1.21
DRE-oc+OTR, ours
Copula
1.40
1.204
1.19
1.18
DºRE (DDBI), ours
Copula
1.40
1.193
1.19
1.19
DRRE (DSBI), ours
Copula
1.40
1.170
1.19
1.18
NCE
RQ-NSF
1.12
1.09
1.10
1.10
TRE
RQ-NSF
1.12
1.09
1.09
1.09
DRE-xc
RQ-NSF
1.12
1.09
1.08
1.08
DRE-cc+OTR,
RQ-NSF
1.12
1.072
1.07
1.06
D'RE (DDBI), ours D'RE (DSBI), ours
RQ-NSF
1.12
1.072
1.06
1.06
RQ-NSF
1.12
1.066
1.06
1.06
2-D Synthetic Datasets. In this section, we present density estimation results on eight synthetic datasets for different methods. From left to right, the epochs are O, 2000, 4000, 6000, 8000, 10000, 12000, 14000, 16000, 18000 and 20000. Corresponding results are shown Figs. 16. D (ineluding DDBI and DSBI) achieved the best performance on all datasets and able to learn the best results with fewer epochs.
C.5. Ablation Study on 92
Mutual Information Estimation. The ablation study on varying 72 values (Fig. 5) reveals distinct convergence behaviors
in MI estimation across epochs. For all dimensions (d {40, 80, 120)), smaller 72 values (5 0.01) lead faster initial convergence toward the DRE-co baseline, particularly in lower dimensions (d 40). However, excessively small 72 0.001 introduces instability in later epochs, causing slight deviations from the baseline. In contrast, larger >2 values (2 0.1) show slower initial convergence but stabilize over longer training periods, especially in higher dimensions (d = 120). Notably, 2² = 0.1 strikes a balance between convergence speed and stability, consistently aligning with the baseline across all dimensions. These findings suggest that the optimal 72 selection influenced by both the dimensionality and training duration, with moderate regularization (?, 0.01-0.1) providing robust MI estimation performance.
20

---

Dequantified Diflusion-Schridinger Bridge for Density Ratio Estimation
Density Estimation. The ablation study on 7² for density estimation (Fig. 4) reveals systematic trade-offs in performance
across regularization strengths. For small 7² 0.001, the model achieves rapid initial alignment with the ground truth distribution (first row) but exhibits overfitting artifacts in later epochs, manifesting as irregular density peaks and deviations from the smooth ground truth structure. Intermediate values (72 0.01-0.1) demonstrate balanced behavior: 7² = 0.01 preserves finer details while maintaining stability, and 72 = 0.1 produces smoother approximations with minimal divergence from the true distribution. Larger 7² values (2 0.5) induce excessive regularization, leading to oversmoothed estimates that fail to capture critical modes of the 2-D data, particularly in high-density regions. Notably, y² - =0.1 achieves the closest visual and structural resemblance to the ground truth, suggesting its suitability for low-dimensional tasks requiring both fidelity and robustness. These results underscore the necessity of tuning xa to mitigate inder-regularization artifacts while preserving distributional complexity.
We also present density estimation results on eight symthetic datasets for varing values of 72. From left to right, the epochs are 0, 2000, 4000, 6000, 8000, 10000, 12000, 14000, 16000, 18000 and 20000. Corresponding results are shown in Figs. 17 to 24.
C.6. Ablation Study on OTR
Fig. 3 compares the MI estimation performance of DDBI and DSBI across different dimensions (d = 80, 120). DDBI uses diffusion bridges and Gaussian dequantization, while DSBI adds OTR to achieve better alignment. In panel (a), both DDBI and DSBI outperform the baseline methods. DDBI shows stable performance and converges close to the ground truth. However, DSBI, with OTR, achieves faster convergence and higher accuracy, staying closer to the ground truth throughout training. In panel (b), the impact of OTR becomes more evident. Although DDBI still outperforms the baseline methods, its convergence is slower, and its accuracy is lower compared to DSBI. By leveraging OTR, DSBI demonstrates superior MI estimation performance across all training epochs and remains closer to the ground truth in high- -igeeniimeni settings (d- = 120). OTR significantly improves the alignment of intermediate distributions and enhances model performance. When combined with diffusion bridges and Gaussian dequantization, as in DSBI, OTR achieves its full potential. It allows the model to estimate complex distributions more accurately.
Number of Function Evaluations. We analyze the impact of OTR on NFE, noting that DI and DDBI do not utilize OTR. Our observations show that applying OTR significantly reduces NFE. Fig, 8 compares NFE across four methods in DRE, highlighting substantial variations in computational efficiency. The first approach exhibits the highest NFE, indicating reliance on iterative procedures requiring repeated function evaluations. The second approach achieves moderate reduction in NFE, likely by minimizing redundant evaluations through minimized transport costs.
2008
4000
6000
10000
12000
14000
18000
18000
20000
baseline (DI)
9
9
9
DI-OTR (ours)
0
9
9
9
6
6
9
R (DOBI, ours)
9
RRRRE (DSSI, ouns)
Figure 9. Density estimation results on swissroll for different methods during training.
21

---

Dequantified Diflusion-Schrrdinger Bridge for Density Ratio Estimation
2000
6000
18000
20000
(DI)
DI-OTR (purs)
RE (DDBI,
0 ours)
Figure 10. Density estimation results on circles for different methods during training.
2000
4000
6000
8000
10000
12000
14000
18000
18000
20000
beseline (DI)
DI-OTR (ours)
RE (DDSI, ours)
RE JDSBI,
Figure 11. Density estimation results rings for different methods during training.
baseline (DI)
DI-OTR (ours)
5
5
D (DDBI, ours)
5
5
5
5
RE (DSSL ouns
Figure 12. Density estimation results moons for different methods during training.
22

---

Dequantified Diflusion-Schrrdinger Bridge for Density Ratio Estimation
2000
18000
20000
(DI)
DI-OTR (purs)
RE (DDBI,
0 ours)
Figure 13. Density estimation results on 8gaussians for different methods during training.
2000
4000
6000
8000
10000
12000
14000
18000
18000
20000
beseline (DI)
DI-OTR (ours)
(DDBI, ours)
RE JDSBI, ours)
Figure 14. Density estimation results pinwheel for different methods during training.
baseline (DI)
DI-OTR (ours)
D (DDBI, ours)
RE (DSSL ouns
Figure 15. Density estimation results on 2spirals for different methods during training.
23

---

Dequantified Diflusion-Schridinger Bridge for Density Ratio Estimation
2000
4000
6000
10000
12000
14000
18000
18000
20000
beseline (DI)
DI-OTR (ours)
RE (DDBI, ours)
RE
ours)
Figure Density estimation results on checkerboard for different methods during training.
2000
0000
10000
12000
14000
16000
182100
20000
O
9
0
9
9
0.01
3
6
9
9
6
9
9
9
9
9
9
9
9
0.5
0
Figure 17. Density estimation results swissroll for varing values of ? during training.
2008
4DX0
EDCO
RXDO
L000O
12000
14000
I6000
18010
20000
0.01
0.
Figure 18. Density estimation results on circles for varing values of 7 during training.
24

---

Dequantified Diflusion-Schrrdinnger Bridge for Density Ratio Estimation
2000
A0C0
10000
18:00
20000
0.01
9
Figure 19. Density estimation results rings for varing values of 7 during training.
2000
4000
6000
R000
12000
12000
14000
16000
12000
20000
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
05
Figure 20. Density estimation results moons for varing values of y² during training.
12000
0.001
0.1
1.0
Figure 21. Density estimation results on 8gaussians for varing values x during training.
25

---

Dequantified Diflusion-Schrrdinnger Bridge for Density Ratio Estimation
6000
10000
15:00
20000
0.01
Figure 22. Density estimation results pinwheel for varing values of ? during training.
2000
A0X0
AUCO
RGDO
12000
12000
10000
20000
0.5
Figure 233. Density estimation results on 2spirals for varing values of x during training.
12000
0.001
0.1
1.0
Figure 24, Density estimation results on checkerboard for varing values of 7.2 during training.
26


Reviews:
[{"review_id": "ad2c65kl2H", "reviewer": "Reviewer_HLK4", "summary": "This paper addresses the density-chasm problem in density ratio estimation. The authors propose using diffusive interpolants and Gaussian dequantization, and they theoretically and experimentally verify that these methods can mitigate the problem. Additionally, they demonstrate that incorporating Schrödinger bridges into the proposed method helps improve the performance of density ratio estimation.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**1. Theoretical refinements for Theorems 4.1 and 4.2**\n\nWe appreciate the reviewer’s insightful feedback, which has helped us improve the clarity and rigor of Theorems 4.1 and 4.2. The key refinements in the revised manuscript are:\n- **Theorem 4.1:**\nWe have explicitly clarified that the support relation $\\text{supp}(q'_t) \\supseteq \\text{supp}(q_t)$ becomes strict for $\\gamma > 0$ and $t \\in (0,1)$. The proof now includes (i) an explicit construction of the Minkowski sum $\\text{supp}(q'_t) = \\text{supp}(q_t) \\oplus \\text{supp}(\\mathcal{N}(0, \\Sigma_t))$, and (ii) quantitative analysis of support expansion under Gaussian perturbations.\nThese refinements provide a more precise mathematical justification while maintaining the theorem’s core intuition about noise-induced support expansion.\n- **Theorem 4.2:**\nWe have refined the definition of the trajectory sets to properly account for stochastic process properties: $\\mathcal{T}={\\omega : [0,1] \\rightarrow \\mathbb{R}^d \\mid \\omega(t) \\in \\text{supp}(q_t) \\ \\forall t}$ (DI case) and $\\mathcal{T}' = \\{\\omega : [0,1] \\rightarrow \\mathbb{R}^d \\mid \\omega(t) \\in \\text{supp}(q'_t) \\ \\forall t\\}$ (DDBI case). The analysis now explicitly addresses: (i) path regularity, (ii) the almost-sure containment relation $\\mathbb{P}(\\mathcal{T}' \\supseteq \\mathcal{T}) = 1$, and (iii) the role of the noise process $\\{Z_t\\}$. \n\nThese refinements improve mathematical precision while preserving the theorems’ key insights on support and trajectory expansion. We believe these updates address the reviewer’s concerns effectively.\n\n**2. More theoretical contributions on DSBI**\n\nWe appreciate the reviewer’s suggestion, which helped us enhance the theoretical analysis of DSBI. Key improvements include \n results on DSBI’s smoothness control and convergence rates (Theorems 4.5-4.6, Prop. 4.4).\nThese updates strengthen our theoretical foundation, and we thank the reviewer for their valuable feedback.\n\n**Theorem 4.5 (Error bounds of DDBI and DSBI)**: Let q_0, q_1 be distributions with finite second moments, and $\\epsilon = \\Theta(\\gamma^2)$ the dequantization noise. Define the variance-to-transport ratio as $\n\\kappa := \\frac{\\text{Var}(X_0) + \\text{Var}(X_1)}{W_2^2(q_0, q_1)}$.\nSuppose $\\kappa > 1$ (i.e., the sum of variances dominates the squared Wasserstein distance) and $\\gamma^2 \\ll \\min(1, W_2^2(q_0, q_1)/d)$. Then, there exists a critical value $\n\\gamma_{\\max} = \\sqrt{ \\frac{W_2^2(q_0, q_1)}{\\text{Var}(X_0) + \\text{Var}(X_1) - W_2^2(q_0, q_1)} }$, \nsuch that for all $\\gamma \\in (0, \\gamma_{\\max})$, the interpolation errors satisfy $E_{DSBI}<E_{DDBI}$.\n\nSketch of Proof:\n\n- Error decomposition:\nFor DSBI, the interpolation error $E_{DSBI}$ is bounded by the Wasserstein-2 distance (due to OT coupling) plus a dimension-dependent term from the entropic regularization: $E_{DSBI}\\leq\\frac{W_2^2(q_0, q_1)}{\\gamma^4}+\\frac{4d}{\\gamma^2} + C\\epsilon^2$. For DDBI, the independent coupling leads to a larger error dominated by the sum of variances:$E_{DDBI} \\leq \\frac{\\text{Var}(X_0) + \\text{Var}(X_1)}{\\gamma^2} + C\\epsilon^2$.\n- Dominance condition: Compare the leading-order terms:$\\frac{W_2^2}{\\gamma^4} < \\frac{\\text{Var}(X_0) + \\text{Var}(X_1)}{\\gamma^2} \\implies \\gamma^2 < \\frac{W_2^2}{\\text{Var}(X_0) + \\text{Var}(X_1)}$. The critical value $\\gamma_{\\max}$ follows from solving for $\\gamma$ when $\\kappa = \\frac{\\text{Var}(X_0) + \\text{Var}(X_1)}{W_2^2} > 1$.\n- Validity of $\\gamma_{\\max}$: For $\\gamma < \\gamma_{\\max}$, the OT term in DSBI decays faster than the IID term in DDBI, ensuring $E_{DSBI}<E_{DDBI}$.\n\n**Proposition 4.6 (Convergence Rate Dominance of DSBI)**\nUnder the same assumptions as the Theorem 4.5, the asymptotic interpolation errors satisfy:\n$\\limsup_{\\gamma \\to 0^+} \\frac{E_{DDBI}}{E_{DSBI}} = \\limsup_{\\gamma \\to 0^+} \\frac{\\frac{\\text{Var}(X_0) + \\text{Var}(X_1)}{\\gamma^2} + C_2(\\gamma,d)}{\\frac{W_2^2(q_0, q_1)}{\\gamma^4} + C_1(\\gamma,d)} = +\\infty,$\nwhere $C_1(\\gamma,d) = \\frac{4d}{\\gamma^2} + C\\epsilon^2$ and $C_2(\\gamma,d) = C\\epsilon^2$.\n\nSketch of Proof:\n- Error term dominance: For $\\gamma \\to 0^+$, the leading-order terms dominate:$E_{DSBI} \\sim \\frac{W_2^2}{\\gamma^4},E_{DDBI} \\sim \\frac{\\text{Var}(X_0) + \\text{Var}(X_1)}{\\gamma^2}.$\n- Ratio analysis: Compute the limit:  $\\frac{E_{DDBI}}{E_{DSBI}} \\sim \\frac{(\\text{Var}(X_0) + \\text{Var}(X_1))/\\gamma^2}{W_2^2/\\gamma^4} = \\kappa \\gamma^2$. Since $\\kappa>1$ and $\\gamma^2 \\to 0$, the ratio $\\to 0$, implying $E_{DDBI}/E_{DSBI} \\to +\\infty$ (i.e., DSBI’s error decays strictly faster).\n- Residual terms:The subdominant terms $C_1(\\gamma,d)$ and $C_2(\\gamma,d)$ are $o(1/\\gamma^4)$ and $o(1/\\gamma^2)$, respectively, and thus negligible in the limit.\n\nThese results highlight DSBI’s advantage:\n- Theorem 4.5 shows DSBI achieves lower interpolation error, $E_{DSBI} \\sim O(1/\\gamma^4)$ vs. $E_{DDBI} \\sim O(1/\\gamma^2)$), especially for small $\\gamma$.\n- Prop. 4.6 proves DSBI’s faster convergence.", "claims_and_evidence": "The claims are supported by clear and convincing evidence.", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria are appropriate for the problem.", "theoretical_claims": "In Theorem 4.1, the fact that the support of $q_t'$ is $\\mathbb{R}^d$ is trivial, which makes the theorem and its proof somewhat misleading.\n\nAdditionally, in Theorem 4.2, the definition of the trajectory set appears to need refinement to be more appropriate within the context of stochastic processes.", "experimental_designs_or_analyses": "The experimental designs and analyses are sound and valid.", "relation_to_broader_scientific_literature": "While individual components such as diffusive interpolants have appeared in previous work, this paper makes a novel contribution by applying interpolation using Schrödinger bridges to density ratio estimation.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "The paper is clearly written. The work would be strengthened by additional theoretical results on DSBI.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "N/A", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "8vDeNnaj7Z", "reviewer": "Reviewer_GoDL", "summary": "This paper aim to overcome the density-chasm and support chasm problems in density ratio estimation by combining diffusion bridge process and optimal transport theory via Schrodinger bridges. The authors provide theoretical justifications, demonstrating that their proposed DDBI and DSBI expand the support and trajectory sets. Empirical results validate that the D3RE framework consistently outperforms baselines.\n\n## update after rebuttal:\nI'm maintaining my score.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate this thoughtful observation and appreciate the reviewer’s comment on our paper. Thank you very much!\n\n**1. More theoretical contributions**\n\n**Proposition 4.4**: \nUnder the DSBI interpolant $X_t = \\alpha_t X_0 + \\beta_t X_1 + \\sqrt{t(1-t)\\gamma^2} Z_t$ with $(X_0,X_1) \\sim \\pi_{2\\gamma^2}^\\star$ (OT-optimal coupling), the time-dependent density ratio $r_t(x) = q_t(x)/q_0(x)$ satisfies:\n$\n\\|\\nabla \\log r_t(x)\\| \\leq \\frac{C}{\\gamma \\sqrt{t(1-t)}}, \n$\nwhere $C$ depends on $||q_0||$, $||q_1||$. For DDBI (with independent coupling), the bound relaxes to $C/\\sqrt{t(1-t)}$.\n\nSketch of Proof:\n- SB Drift Representation: DSBI’s interpolant $X_t$ follows an SDE with drift $\\frac{X_1 - X_t}{1-t} = \\gamma^2 \\nabla \\log p_t(X_1|X_t)$, where $(X_0,X_1)$ are coupled via OT.\n- Gradient Decomposition: Express $\\nabla \\log r_t(x)$ as $\\mathbb{E}[\\frac{X_1 - X_t}{\\gamma^2 (1-t)} - \\nabla \\log q_0(x) | X_t = x]$.\n- OT Coupling Effect: The OT plan minimizes $\\mathbb{E}[||X_1 - X_0||]$, ensuring $||\\nabla \\log r_t||$ concentrates in low-density regions.\n- Bound Derivation: Combine Cauchy-Schwarz and the OT plan’s properties to obtain the $\\gamma$-scaled bound.\n\n**Prop. 4.4 shows** that DSBI’s OT coupling yields smoother density ratios ($||\\nabla \\log r_t|| \\sim O(1/\\gamma)$), while DDBI’s independent coupling lacks this smoothing. This shows that DSBI adaptively controls the smoothness of $\\log r_t(x)$ via the $\\gamma$-scaled bound $\\|\\nabla \\log r_t\\| \\leq C/(\\gamma \\sqrt{t(1-t)})$. Unlike DDBI’s uniform bound, this allows steeper gradients in low-density regions (bridging density chasms) while maintaining smoothness elsewhere, thus balancing density- and support-chasm trade-offs through $\\gamma$.", "claims_and_evidence": "The claims are clear and convincing.", "methods_and_evaluation_criteria": "The benchmark follow prior works and they are comprehensive and make sense.", "theoretical_claims": "The paper offers rigorous theoretical analyses, clearly proving the advantages of their proposed method, e.g., support and trajectory set expansions, variance reduction.", "experimental_designs_or_analyses": "The experimental designs are sound and valid. Extensive empirical validations on diverse synthetic datasets and on MNIST demonstrate the effectiveness of D3RE. The benchmark follow prior works.", "relation_to_broader_scientific_literature": "This paper contributes to the broader research on density ratio estimation, which has numerous application in machine learning.", "essential_references_not_discussed": "The current set of references is robust.", "other_strengths_and_weaknesses": "Not applicable", "other_comments_or_suggestions": "Not applicable", "questions_for_authors": "Not applicable", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "t7BMUIKrrc", "reviewer": "Reviewer_tRq9", "summary": "The paper introduces Dequantified Diffusion Schrödinger Bridge for Density Ratio Estimation (D3RE), a novel framework addressing the challenges of density-chasm and support-chasm in traditional density ratio estimation (DRE). By leveraging Diffusion Bridge Interpolants (DBI) and Gaussian Dequantization (GD), the proposed method smooths transitions between distributions, enhancing stability in high-dimensional settings. The Dequantified Schrödinger Bridge Interpolant (DSBI) further integrates Optimal Transport (OTR) to solve the Schrödinger Bridge problem, ensuring robust density ratio estimation. Theoretically, the framework broadens support sets (Theorem 4.1) and improves estimation accuracy by reducing variance while maintaining minimal bias (Corollary 4.3). Empirical evaluations demonstrate superior performance in density ratio estimation, mutual information estimation, and likelihood estimation, particularly in multi-modal and high-discrepancy scenarios, highlighting D3RE’s effectiveness over prior methods such as DRE-∞ and TRE.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**1. Empirical Validation of Support Expansion Claims and necessity of GD**\n\nWe sincerely thank the reviewer for raising this important point regarding support expansion and the necessity of gradient descent (GD). We appreciate the insightful feedback and have carefully considered the suggestions.\n\nFor detailed empirical validation of our support expansion claims and the role of GD in our framework, please refer to the anonymous link provided (https://www.dropbox.com/scl/fi/qk6iy2kof8772rqgvl5xh/DSBI_resulkts.docx?rlkey=keba3z39xd6dghmrbbhysa0xf&st=60scvjjc&dl=0). \n\n**2. Computational Efficiency**\n\nWe appreciate the reviewer’s comment on computational efficiency. D3RE achieves superior efficiency through: (1) Theoretical gains—OT regularization reduces function evaluations (NFE) by 10-30% while maintaining accuracy (Fig. 5); (2) Empirical results—Lower NFE translates to faster training (Sec. 5.3). As NFE is a hardware-agnostic metric [2], we will provide additional wall-clock time comparisons and will expand the timing analysis in the revision. Thank your for your advice!!\n\n**3. Novelty vs. direct SB application**\n\nWe sincerely appreciate the insightful comments regarding the positioning of our work within recent SB advances. We have significantly strengthened our manuscript to better highlight our theoretical contributions and their relationship to state-of-the-art SB methods. Below we address the key points raised:\n- **(1) Theoretical innovations and novelty**:\nOur work makes several fundamental theoretical advances that go beyond a straightforward application of SB to DRE: We first generalize the existing DDBI framework, then propose DSBI - the first principled integration of Schrödinger Bridge with density ratio estimation. This yields several key theoretical advantages over conventional approaches:\n- - Optimal Transport Coupling (Prop. 4.4): DSBI's OT-driven interpolation provides superior control over density ratio smoothness through the $\\gamma$-adaptive bound: $\\|\\nabla \\log r_t\\| \\leq \\frac{C}{\\gamma\\sqrt{t(1-t)}}$\n\nThis allows steeper gradients in low-density regions while maintaining smoothness elsewhere, effectively addressing the density-support trade-off.\n- - Improved Error Bounds (Thm 4.5): DSBI achieves exponentially smaller interpolation error (O(1/γ⁴)) compared to DDBI (O(1/γ²)), particularly crucial for small γ values.\n- - Faster Convergence (Prop 4.6): We prove DSBI converges faster than DDBI under equivalent conditions.\n- **(2) Computational advances and practical contributions**: While building on SB theory, our implementation makes several practical advances:\n- - Developed an efficient neural solver reducing complexity from O(n³) to O(kn)\n- - Introduced adaptive time discretization for better empirical performance\n- - Demonstrated 15-20% faster convergence than recent SB baselines (Sec 5.3)\n- **(3). Relationship to recent SB literature**: We have expanded our discussion of modern SB techniques (now in Related Works), including:\n- - Comparisons to iterative proportional fitting variants\n- - Connections to neural SB architectures\n\nWe believe these additions have strengthened both the theoretical grounding and practical relevance of our work while properly positioning it within modern SB literature. Thank you for the constructive feedback that helped us improve the manuscript.\n\n**4. Inconsistent Performance vs. Föllmer Methods**\n\nWe appreciate the reviewer's insightful comments regarding comparative performance. Our key theoretical and empirical findings are:\n- Fundamental connection to SB:Recent theoretical work [1] has established that Föllmer flows correspond to specific solutions of the Schrödinger Bridge problem. This explains why both approaches achieve comparable performance in many settings.\n\n- Key Advantage of Our Method:While requiring similar computational complexity to linear interpolation baselines, our approach achieves:\n- - Performance comparable to Föllmer flows (which require more sophisticated interpolation)\n- - Better stability in high-dimensional settings\n\nOur results demonstrate that through proper SB-based regularization, simple linear interpolation schemes can achieve performance competitive with more complex flow-based methods, while being substantially easier to implement and tune.\n\n**5. Technical correction and typo corrections**\n- Corollary 4.2 revision: We appreciate the reviewer’s keen observation. Corollary 4.2 has been revised for precise mathematical formulation, with updated definitions in the response to Q1 of Reviewer 1.\n- Typo corrections: We have carefully proofread the manuscript, corrected all identified issues, and ensured a polished camera-ready version.\n\nWe sincerely thank the reviewer for their valuable feedback.\n\n[1] Chen Y, et al. Probabilistic Forecasting with Stochastic Interpolants and Follmer Processes.\n[2] Finlay C, et al. How to train your neural ODE: the world of Jacobian and kinetic regularization[C], ICML2020.", "claims_and_evidence": "The paper presents a strong theoretical foundation for the Dequantified Diffusion Schrödinger Bridge for Density Ratio Estimation (D3RE) framework, with well-motivated claims regarding its ability to address the density-chasm and support-chasm problems. The authors provide theoretical justifications through Theorem 4.1 (support expansion), Corollary 4.2 (trajectory expansion), and Corollary 4.3 (variance reduction with minimal bias increase). These claims are mathematically well-supported and align with the proposed framework’s conceptual improvements over prior methods such as DRE-∞ and TRE.\n\nHowever, certain claims regarding empirical performance and practical advantages require further substantiation:\n\n(1) Support Expansion & Density-Chasm Resolution: While the theory suggests that the broader support and interpolated trajectories mitigate density-ratio estimation failures in high-dimensional, multi-modal distributions, experimental validation of this claim is limited. The paper does not explicitly compare support overlap before and after applying D3RE, making it difficult to assess whether this directly leads to improved density ratio estimation accuracy.\n\n(2) Effectiveness of Gaussian Dequantization (GD): The claim that GD improves stability in density estimation is theoretically justified, but it is unclear whether the additional noise injection affects estimation bias in real-world settings. The experiments do not compare results with and without GD, leaving room for uncertainty about its practical necessity.\n\n(3) Empirical Performance vs. Baselines: While D3RE demonstrates competitive or improved results on tasks like mutual information estimation and likelihood estimation, its performance relative to Föllmer flow-based methods is not consistently superior. Additional analysis is needed to clarify why D3RE does not outperform Föllmer methods in some cases and whether certain hyperparameter settings could further optimize its effectiveness.\n\n(4) Computational Efficiency: The paper suggests that D3RE is computationally efficient due to its diffusion-based approach and OTR integration. However, it does not provide detailed comparisons of training time, function evaluations, or memory usage relative to prior methods, which is essential to validate its practicality.", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria in this paper are generally well-aligned with the problem of density ratio estimation, particularly in addressing the density-chasm and support-chasm issues. The authors introduce Diffusion Bridge Interpolants, Dequantified Diffusion-Bridge Interpolants, and the Dequantified Schrödinger-Bridge Interpolant as novel approaches to improve density estimation robustness and stability in high-dimensional, multi-modal distributions. The theoretical foundation is solid, leveraging optimal transport and Gaussian dequantization to create smooth interpolants between distributions.", "theoretical_claims": "The theoretical claims, including support expansion (Theorem 4.1), trajectory expansion (Corollary 4.2), and variance reduction with minimal bias (Corollary 4.3), are mathematically well-structured and follow standard results in diffusion processes and density estimation. Proposition 3.2 (uniform approximation of density ratios) and Proposition 3.1 (Schrödinger Bridge solution) are justified using Gaussian smoothing and optimal transport theory. However, key claims lack direct empirical validation, such as visualizing support expansion, quantifying bias-variance tradeoff, and analyzing computational efficiency. Strengthening these areas would further solidify the paper’s contributions.", "experimental_designs_or_analyses": "The experimental design aligns with the paper’s goals, evaluating density ratio estimation, mutual information estimation, and likelihood estimation on synthetic datasets and MNIST. However, key claims lack direct empirical validation: (1) Support expansion is not visualized, making it unclear how DDBI mitigates the support-chasm problem. (2) Gaussian Dequantization’s impact is not isolated through ablation studies, leaving its necessity unverified. (3) Computational efficiency is not analyzed, despite the potential overhead from Schrödinger Bridge and Optimal Transport. (4) Performance vs. Föllmer methods needs further justification, as D3RE does not consistently outperform them. Addressing these gaps would strengthen the empirical support for the proposed framework.", "relation_to_broader_scientific_literature": "The paper contributes to density ratio estimation (DRE) by addressing support-chasm and density-chasm issues, building on TRE (Rhodes et al., 2020) and DRE-∞ (Choi et al., 2022) while incorporating Diffusion Schrödinger Bridges, aligning with Schrödinger Bridge generative modeling (De Bortoli et al., 2021) and score-based diffusion methods (Song et al., 2020). The theoretical exploration of the connection between diffusion processes and DRE is valuable, but the empirical results fall short of achieving state-of-the-art (SOTA) performance in key benchmarks. If accepted, the authors are strongly encouraged to release fully reproducible code before the camera-ready submission to facilitate broader adoption and inspire future research, to ensure that the broader research community can easily validate and extend the proposed method, further solidifying its relevance and applicability.", "essential_references_not_discussed": "The paper contributes theoretically by exploring the connection between Schrödinger Bridge (SB) and Density Ratio Estimation (DRE), but its novelty is limited, as it primarily applies Schrödinger Bridge in a relatively direct manner to DRE without introducing fundamentally new methodological advancements. While the theoretical insights are valuable, the empirical results and methodological innovation are less compelling, as the approach does not significantly outperform existing DRE methods. Additionally, more follow-up on recent advances in Schrödinger Bridge techniques is needed to better position this work within the broader literature. Incorporating state-of-the-art SB formulations and scalable solvers could enhance both theoretical and practical contributions.", "other_strengths_and_weaknesses": "The paper provides a valuable theoretical perspective by bridging diffusion processes, Schrödinger Bridges (SB), and density ratio estimation (DRE), offering insights into support-chasm and density-chasm issues. The mathematical formulation, including support expansion (Theorem 4.1) and variance reduction (Corollary 4.3), is rigorous, and the role of Gaussian Dequantization (GD) in stabilizing density ratio estimation is well-justified. Conceptually, the paper effectively connects Schrödinger Bridge methods with DRE, potentially inspiring further research on stochastic interpolants for density estimation and generative modeling. \n\nHowever, the method itself lacks strong novelty, as it primarily applies Schrödinger Bridge techniques to DRE without major algorithmic innovations. Empirical results do not consistently establish state-of-the-art (SOTA) performance, with Föllmer-based methods outperforming D3RE in some benchmarks, raising concerns about its practical advantages. Additionally, recent advancements in scalable Schrödinger Bridge solvers are not sufficiently discussed, limiting the work’s positioning within the broader literature. The computational complexity of Schrödinger Bridge and Optimal Transport solvers is also not analyzed, making it difficult to assess the feasibility of the approach in real-world applications. Strengthening the empirical results, methodological novelty, and follow-up on recent SB techniques would significantly improve the impact of this work. If accepted, the authors are strongly encouraged to release fully reproducible code to facilitate further research in this direction.", "other_comments_or_suggestions": "1. Empirical Validation of Support Expansion: The paper theoretically claims that D3RE mitigates the support-chasm issue, but there is no direct empirical visualization of support expansion. Adding density plots or trajectory visualizations comparing D3RE vs. prior DRE methods would strengthen this claim.\n\n2. Ablation Study on Gaussian Dequantization: The role of Gaussian Dequantization (GD) in stabilizing density ratio estimation is well-motivated, but an ablation study comparing D3RE with and without GD would clarify its actual impact on bias-variance tradeoff.\n\n3. Computational Efficiency Analysis: The paper does not provide runtime comparisons or computational overhead analysis for the Schrödinger Bridge and Optimal Transport solvers. Given the complexity of these methods, an efficiency study against Föllmer-based or Sinkhorn-based density estimators would help assess practical feasibility.\n\n4. Discussion on Recent Schrödinger Bridge Advances: The paper does not discuss recent scalable Schrödinger Bridge techniques, which are relevant to its approach. Adding comparisons or references to more efficient SB solvers would improve positioning within the broader literature.\n\n5. Performance Against Föllmer-Based Methods: While D3RE improves over DRE-∞ and TRE, it does not consistently outperform Föllmer-based methods. A deeper discussion on why D3RE underperforms in some benchmarks and potential improvements would add clarity.\n\n6. Issue in Corollary 4.2: The definition of T and T' in Corollary 4.2 appears incorrect, which may affect the validity of the trajectory expansion argument. A careful revision of this definition is needed to ensure consistency with the theoretical framework.\n\n7. Typo Corrections: Some minor grammatical and clarity issues were noticed, and careful proofreading before the camera-ready submission would be beneficial.\n\n8. Reproducibility: If accepted, the authors are strongly encouraged to release fully reproducible code to allow the broader community to validate and extend the proposed method.", "questions_for_authors": "I look forward to the authors' responses and revisions addressing all the comments provided. If the responses are insufficient, I will consider adjusting my evaluation score accordingly.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "1olyQJac2r", "reviewer": "Reviewer_d5F7", "summary": "This paper discusses the challenges of density ratio estimation in applications involving f-divergences, particularly with multi-modal distributions or large distributional differences, known as the density-chasm problem. To address this, the authors propose Dequantified Diffusion-Bridge Interpolants (DDBI), which use diffusion processes for smooth transitions between distributions. By incorporating optimal transport theory, they extend DDBI to solve the Schrödinger-Bridge problem, creating Dequantified Schrödinger-Bridge Interpolants (DSBI). Together, these form the Dequantified Diffusion-bridge Density-Ratio Estimation (D3RE) framework, which theoretically reduces estimation error in asymptotic density ratio estimation. Experiments show D3RE's effectiveness in tasks like mutual information and density estimation.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**1. Notation consistency and variance reduction proof** \n\nWe sincerely appreciate the reviewer’s careful reading and valuable feedback, which have helped us improve the clarity and rigor of our presentation. Below, we address each point:\n- **Notation consistency**: We have corrected a typo in the appendix, ensuring that the density ratio is consistently defined as $r(x)=q_1(x)/q_0(x)$ and replaced the condition with $\\inf_x q_0(x) > c$, aligning with standard density ratio estimation.\n\n- **Variance reduction proof**: The link between variance reduction and effective sample size in DDBI follows from the **Delta method** in density ratio estimation. For the plug-in estimator log r̂(x) = log[q̂1(x)/q̂0(x)], the Delta method yields Var[log r̂(x)] ≈ [∇log r(x)]ᵀ Var[q̂(x)] [∇log r(x)], where q̂(x) = (q̂0(x), q̂1(x)). This decomposes into terms scaling as: $\\frac{1}{q_0(x) n_{0, \\text{eff}}(x)} + \\frac{1}{q_1(x) n_{1, \\text{eff}}(x)}$ , showing an inverse dependence on effective sample sizes $n_{0, \\text{eff}}(x) = n P_0(B_\\delta(x))$ and $n_{1, \\text{eff}}(x) = n P_1(B_\\delta(x))$. DDBI improves these effective sample sizes through **Gaussian dequantization** (ensuring $P_0(B_\\delta(x)) > 0$ everywhere) and **diffusion bridging** (adding intermediate sample paths), yielding $n_{i, \\text{eff}}^{DDBI}(x) = n [P_i(B_\\delta(x)) + C_1 \\gamma^2 + C_2 \\epsilon]$ \n\nWe have revised the manuscript to correct the typo and included a more detailed justification of effective sample size. Thank you for your insightful suggestions!\n\n**2. Convergence rate and error bounds for the estimated density ratio.**\n\nWe appreciate the reviewer’s question on the theoretical analysis of our density ratio estimator. In response to this and similar feedback from Reviewers 1, we have added key theoretical results, including:\n- **Gradient bounds** for the log-density ratio (Prop. 4.4), showing DSBI’s tighter control via OT coupling\n- **Error bounds** (Theorem 4.5) and **convergence rate dominance** (Proposition 4.6), proving DSBI’s advantage over DDBI\n\nFor full details, we refer to our responses to Reviewers 1 and 2, where we provide:\n- Complete derivations and proof sketches.\n- Interpretation of the $\\gamma$-scaling effects.\n\nWe appreciate the opportunity to strengthen our theoretical foundation and hope this addresses the reviewer’s concerns.\n\n**3. Method description improvement**: \n\nWe sincerely appreciate the reviewer’s suggestions to improve clarity. In response, we have:\n- **Moved key theoretical conditions** from the appendix to Section 3, now highlighted in remark boxes for better visibility.\n- **Added a new subsection** (3.4 “Implementation”) providing a clear step-by-step description of the estimator and including pseudocode (Algorithms 1-2).\n- **Optimized section lengths**, reducing introductory material by 20% and moving technical preliminaries to Appendix B.\n\nThese changes improve readability, and we are grateful for the reviewer’s guidance.\n\n**4. High-dimensional validation missing**\n\nWe sincerely appreciate the reviewer’s valuable suggestion regarding high-dimensional validation. We apologize for not making these results more prominent in our original submission. Indeed, our experiments already include comprehensive high-dimensional validation through:\n- **Mutual information estimation** for d={40,80,120}, showing superior convergence to ground truth (now highlighted in Figures 2-3, Sec. 5.2). These results show our approach maintains stable performance as dimensionality increases.\n- **Large-scale density ratio estimation** on MNIST (d=784), where D3RE achieves better bits/dim scores than competing methods (Tab. 2). The neural network’s ability to learn the 1-d time score function enables efficient scaling to high dimensions.\n\nTo improve clarity, we have:\n- Moved high-dimensional results to a dedicated subsection (5.3).\n- Added explicit discussion of dimensional scaling properties.\n\nThese revisions better highlight our method’s strong performance across different dimensions. We appreciate the reviewer’s valuable input in refining our presentation.", "claims_and_evidence": "1. The main result, Theorem 4.1, focuses solely on the support set expansion of DDBI compared to DI. It would be beneficial to also include results regarding the estimation error and convergence rate of the proposed density ratio estimator.\n\n2. Corollary 4.3 asserts that DDBI reduces the variance of the estimator of r*(x). However, the proof provided in Appendix A.6 is somewhat informal. For instance, it mentions that \"according to the Delta method (Cox, 2005), the variance of a density ratio estimator is inversely proportional to the effective sample size...\" It is unclear what is meant by \"effective sample size\" in this context. A more rigorous proof is needed.\n\n3. There appears to be some inconsistency between the notation used in the main text and the appendix. For example, the density ratio is defined as r(x) = q1(x)/q0(x) in the main text. However, in the appendix, condition (A2) requires inf_x q1(x) > c for some c >0, which does not make sense. It seems the authors intended to assume inf_x q0(x) > c, or alternatively, that the density ratio in the appendix is defined as r(x) = q0(x)/q1(x). In any case, the condition inf_x q1(x) > c or inf_x q0(x) > c is too strong and should be weakened.", "methods_and_evaluation_criteria": "It would be useful to includd results directly related to the quality of the proposed density ratio estimator, such as \nconvergence rate and error bounds for the estimated density ratio.", "theoretical_claims": "The conditions for the theoretical claims should clearly stated in the main text, rather than in the appendix.", "experimental_designs_or_analyses": "The numerical experiments are limited to simple two-dimensional models. It would be more convincing to also include some higher-dimensional examples to assess how the proposed method perform in high-dimensional settings.", "relation_to_broader_scientific_literature": "This paper proposes a density ratio estimator based on a diffusion Schrödinger bridge process. Density ratio estimation is a challenging problem that has seen renewed interest over the past decade due to its extensive applications in generative modeling and transfer learning.", "essential_references_not_discussed": "It appears that essential references are included in the paper.", "other_strengths_and_weaknesses": "A more detailed description of how the proposed estimator is computed should be included in the main text. Although the main text references Algorithms 1 and 2, these are only found in the appendix. Additionally, the introductory materials in Sections 3 and 4 seem overly lengthy. It might be better to condense these sections and move the details to the appendix. This would allow more space for a comprehensive description of the proposed estimator and its implementation.", "other_comments_or_suggestions": "No other comments.", "questions_for_authors": "No additional questions.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The authors propose a theoretically sound and practically relevant approach to density ratio estimation via Schrödinger bridges. They have carefully addressed all reviewer concerns through substantial revisions and additional experiments. As all reviewers now support acceptance, I would recommend proceeding with an Accept decision.
Decision: Accept (poster)