Let the distribution of the population be denoted with a sample from the population. Note in the problem of finding , we have no access to directly, only samples from denoted .
The flow matching objective is to find s.t. can be sampled from a known distribution (e.g. a known distribution such as ) while approximates . Let such a target distribution be denoted .
Note that is completely determined by an invertible transformation called “flow” as demonstrated in Eq. . But , as shown in and is merely the unique solution admitted by a first order differential equation determined completely by the velocity field . Let the corresponding velocity field for be written as .
Thus given a target probability density path and it’s corresponding velocity field we want to find a vector field with learnable parameters such that the loss parameter defined by the mean square error (MSE)
In other words, we want to regress to . For sufficiently small , it is clear the probability path generated by will closely approximate . The problem with is how to find and to calculate . It is clear there is more than one possible (that admits the normal distribution at and the target distribution and ). Even more importantly, it is unknown how to find .
The authors aim to show and can be approximated by samples from through an appropriate method of aggregation. The construction is shown in Constructing a Velocity Field.