The free lunch: conditional training induces the marginal velocity field

these arbitrarily looking decisions come from. At a high level, flow matching proposes a divide and conquer approach to training. The big problem to be conquered is transporting P0 into a single monolithic P1. Flow matching breaks it down into multiple simpler tasks. Each training step selects a real image X1, then it transports P0 into a Gaussian centered around it. So, a single training iteration teaches the model how to transport P0 into a density P1 conditioned on X1. To simulate these simpler transports, we need the corresponding conditional velocity fields. Instead of VT, we'll simulate VT conditioned on X1, and we'll discuss soon how to do that. Now, a model that only learns how to perform simple P0 to Gaussian transports is not very useful. It needs to know the destination X1 in order to produce the velocity field that leads to it. And of course, at inference time, we don't have access to X1. The whole point is to generate it. However, here's the magic behind flow matching. When a model is trained on these basic tasks, the skill to perform the challenging P0 to P1 transport can emerge automatically. In other words, a model that observes conditional velocity fields can implicitly learn an unconditional velocity field, also called marginal. The flow matching algorithm verifiably has this property. It was really not obvious that such a free lunch existed, and that's probably why flow-based methods took a while to arrive at this formulation. While this realization is incredible,