Summary
The paper derives theoretical guarantees for a flow matching procedure which constructs a diffusion-like process which allows for samples obtained from a source distribution $\mu$ to a target distribution $\nu^*$. The goal is to build a suitable simple sampling procedure where a source sample is updated through a series of relatively simple Markovian procedure to an approximate sample from $\nu^*$. In the typical setting of diffusion models, this takes the form of a score function augmented with additional Gaussian noise. However, this suffers from two main drawbacks -- firstly, the choice of base distribution is fixed (the base function is always Gaussian) and secondly, the diffusion is truncated after a finite number of timesteps which incurs error in the approximation of the intermediate distribution by a Gaussian. Flow matching addresses both of these by allowing for transformations of arbitrary distributions to each other.
To define a flow-matching process as in the paper, one obtains samples $(X, Y)$ generated from a coupling (joint distribution) $\pi (X, Y)$ where the marginals of $X$ and $Y$ are $\mu$ and $\nu^*$ (the source and target distribution respectively). Given these samples, an interpolating distribution, $\{X_t\}_{t \in [0, 1]}$, is constructed with $X_0 \coloneqq X$ and $X_1 \coloneqq Y$ and the intermediate values are distributed based on the choice of an appropriate bridge distribution. This paper utilizes the Brownian bridge which is the conditional distribution of Brownian motion conditioned on fixing its endpoints. The main issue with this approach is that such an interpolating distribution is not \emph{Markovian} which makes sampling from it in a diffusion-like process challenging. This may be addressed through the notion of
a Markovian projection which constructs another \emph{Markovian} process with the same marginals as the interpolating process for which a score-function-like analogue which enables efficient Markovian sampling. The paper then shows that under some relatively mild assumptions on the approximation of the score function and the moments of the coupled distributions, this procedure guarantees closeness to the target distribution in Total Variation distance. All previous approaches either require additional assumptions on the estimated score function, and/or do not account for the discretization error. One of the main challenges of the proof is in analyzing the discretization error of the scheme where the continuous process is approximated with one with discrete time steps. The paper observes that the terms controlling this error may be rather large at the end points of the interval. Therefore, the paper considers a truncated process on $[\delta, 1 - \delta]$ and show that on the truncated set, the gradients of the densities with control the discretization error are bounded in ($L^8$) norm.
Overall, the results of the paper are novel and interesting. The observation that the truncated distribution is well-behaved with respect to the discretization error is valuable for future theoretical and practical work on flow models. This also illustrates the drawbacks of prior work which require additional Lipschitz assumptions on the scores/estimated scores and presents a natural approach towards circumventing them. This is a nice contribution and I recommend acceptance.