Flow models parameterized as time-dependent velocity fields can generate data with noise by integrating the ODE. These models are often trained using flow simulation, i.e. by randomly sampling noise and target points. and to ensure that the velocity field corresponds, on average, to when tested with segment connectivity to . While these pairs are sampled independently by default, they can also be selected more carefully by cluster matching sound in target points using an optimal transport (OT) solver. Although promising in theory, the OT flow matching (OT-FM) method is not widely used in practice. Zhang et al. (2025) showed recently that OT-FM really starts to pay off when the batch size increases significantly, which can only be done by the multi-GPU implementation of the Sinkhorn algorithm. Unfortunately, the cost of using Sinkhorn can balloon quickly, which it needs everyone's jobs pairs are used to fit the velocity field, where the normalization parameter should be smaller in general to produce better results. To fulfill the theoretical promises of OT-FM, we propose to move away from batch-OT and instead rely on a semidiscrete architecture that exploits the fact that the target dataset distribution is often of finite size. . The SD-OT problem is solved by estimating a two-dimensional vector using SGD; using that vector, the newly generated audio samples during the train can be matched to the data points at the cost of internal product search (MIPS). Semidiscrete FM (SD-FM) removes the quadratic dependence that shuts down OT-FM. SD-FM outperforms both FM and OT-FM on all training metrics and target budget constraints, on all multiple data sets, in unconditional/conditional generation, or when using slow-flow models.
- ** Work done while at Apple
