Conflicting velocity labels are the mechanism, not a bug

still time-dependent. There's one more issue with these velocity labels. Consider two training steps. The first one samples end points X0, X1, and time T. The second one samples two different end points, but the same time T. Their interpolations intersect at point XT. Now, here's the conundrum. For the same input, our model will now see two different velocity labels. They can't both be correct, so are we training the model on garbage labels? The saving grace is the specific choice for the loss function. As we saw before, flow matching uses the mean squared error loss to minimize the distance between a model prediction V theta and the velocity label X1 - X0. Given conflicting labels for the same input, it can be shown that a model trained with the MSE loss will learn their weighted average. So, in this case, the learned model prediction V theta will be somewhere in between the two V stars. Okay, so the model has a well-defined behavior on conflicting labels. But is learning their average useful? Well, it turns out that this weighted average is the marginal velocity field. So, the MSE loss drives the model towards the correct marginal because of the conflicting labels, not despite them.