Summary
Flow matching and alpha blending have achieved tremendous attention in the matching problems. Although the straight trajectories based on $X_t = (1-\alpha)X_0+\alpha X_1$ yield **fast inference**, it doesn't necessarily mean they are efficient in score estimations in training. To make the training theoretically more efficient, rectified flow leverages the optimal transport ideas to minimize the transport cost. However, the stochastic alternative is still lacking (stochastic interpolant may not be optimal in terms of optimal transport). To tackle this issue, the authors proposed the stochastic version of flow matching/ alpha blending with entropic optimized transport. A novel scheme IMF that resembles the IPF iteration is proposed.
Strengths
1. Despite the advances of stochastic interpolant, the transport from marginals may not be optimal in general. The **optimal-transport**-based stochastic version of flow matching and alpha blending is **still missing** in the community.
2. In my understanding, the proposal of the IMF algorithm is the key/ most important contribution of this paper.
3. solid experiments and comparisons are studied, which verify the promising potential of DSBM.
4. Connections to flow matching, rectified flow, and stochastic interpolant are studied, which makes the authors easier to understand the benefit of this paper.
Weaknesses
1. Clearer clarification between IPF and IMF would be required to facilitate the understanding. IPF is based on forward K; IMF is based on backward KL, this is good in math, but more detailed information is lacking. for example, if we solely check proposition 2, we can derive almost the same loss based on forward KL (which leads to the IPF algorithm) except that the underlying measure is not based on $\Pi_{0, t}$. So **what is the real benefit/ advantages of simulating $\Pi_{0, t}$ to train the loss in proposition 2 would clarify our concerns**
2. Algorithm 1 is not clear enough. The authors may consider providing us more details in the learning of $\upsilon_{\phi^{\star}}$ in Alg 1. For example, in the training of alpha-blending (line 60-64 in https://github.com/tchambon/IADB/blob/main/iadb.py), given some samples of $x_1$ and $x_0$, very simple code would be used to train the models. However, when it comes to DSBM, how is it implemented? pseudocode would work if it clarifies our concerns.
3. My concern is that this algorithm may be quite expensive.
Minor
1. missing one related work:
Discussions on the accumulations of IPF errors based on general optimal-transport cost functions studied in [1] would be appreciated.
[1] Provably Convergent Schrödinger Bridge with Applications to Probabilistic Time Series Imputation. ICML'23.
2. writing of Equation between line 637 and line 638 is weird, because given X_t=x_t and x_T, what is the randomness there and why do we need expectation w.r.t. $\Pi_{T|t}$?
3. derivation of eq after line 113 is not clear to me. why $E_{\Pi_{0, T}}[||X_0-X_T||]^2$? is there any reference for that?
4. first part of section 4 in the main paper, the need the alternating schemes is still not well motivated
5. what is IMF-b?
Questions
1. DDMs can be seen as the first iteration of DSM, which implies that we can use DDMs model weights to initialize the model weights of DSM. Does it mean that empirically we can also use weights from stochastic interpolant / the stochastic version of flow matching to train the first step of DSBM?
2. In line 993 in the appendix, the authors mentioned that "The advantage of this DSM loss is that it does **not rely on any divergence computation**". I believe this is quite an important part of scalability, but why it is only emphasized in the joint training instead of stating it in the main paper?
Rating
7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
Confidence
3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.