Whitening Convergence Rate of Coupling-based Normalizing Flows
This provides foundational insights for practitioners in machine learning using normalizing flows, addressing a key theoretical gap in convergence analysis.
The paper tackles the theoretical understanding of coupling-based normalizing flows by proving that they perform whitening of the data distribution and deriving linear convergence bounds in flow depth, with numerical experiments validating the theory.
Coupling-based normalizing flows (e.g. RealNVP) are a popular family of normalizing flow architectures that work surprisingly well in practice. This calls for theoretical understanding. Existing work shows that such flows weakly converge to arbitrary data distributions. However, they make no statement about the stricter convergence criterion used in practice, the maximum likelihood loss. For the first time, we make a quantitative statement about this kind of convergence: We prove that all coupling-based normalizing flows perform whitening of the data distribution (i.e. diagonalize the covariance matrix) and derive corresponding convergence bounds that show a linear convergence rate in the depth of the flow. Numerical experiments demonstrate the implications of our theory and point at open questions.