Woodbury Transformations for Deep Generative Flows
This work addresses a bottleneck in deep generative models for researchers and practitioners by providing a more efficient and effective flow operation, though it is incremental as it builds on existing normalizing flow architectures.
The paper tackles the problem of enabling rich interactions among variables in normalizing flows without high computational costs by introducing Woodbury transformations, which achieve higher likelihood models on image datasets while maintaining efficiency in sampling and likelihood evaluation.
Normalizing flows are deep generative models that allow efficient likelihood calculation and sampling. The core requirement for this advantage is that they are constructed using functions that can be efficiently inverted and for which the determinant of the function's Jacobian can be efficiently computed. Researchers have introduced various such flow operations, but few of these allow rich interactions among variables without incurring significant computational costs. In this paper, we introduce Woodbury transformations, which achieve efficient invertibility via the Woodbury matrix identity and efficient determinant calculation via Sylvester's determinant identity. In contrast with other operations used in state-of-the-art normalizing flows, Woodbury transformations enable (1) high-dimensional interactions, (2) efficient sampling, and (3) efficient likelihood evaluation. Other similar operations, such as 1x1 convolutions, emerging convolutions, or periodic convolutions allow at most two of these three advantages. In our experiments on multiple image datasets, we find that Woodbury transformations allow learning of higher-likelihood models than other flow architectures while still enjoying their efficiency advantages.