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Road to Flow Matching

· 5 min read
Yuxuan (Jerry) Lin
M.S. in Computer Engineering @ Columbia University

Introduction

Generative models aim to transform a simple base distribution, such as a Gaussian, into a complex target data distribution. The core idea is to construct a generator that maps samples from the base distribution into data space. To compute likelihoods, this generator must be invertible. If a generator transforms ZZ into X=G(Z)X = G(Z), then Z=G1(X)Z = G^{-1}(X) must exist, and the density is computed using the change of variables:

PX(X)=PZ(G1(X))det(G1(X)/X)P_X(X) = P_Z(G^{-1}(X)) \left|\det(\partial G^{-1}(X)/\partial X)\right|