Copula & Marginal Flows: Disentangling the Marginal from its Joint
This work tackles a foundational limitation in deep generative modeling for high-dimensional tasks, offering a novel approach to control distributional properties, though it appears incremental in its specific focus on tail modeling.
The paper addresses the inability of deep generative networks to exactly model or extrapolate distributional properties like tail asymptotics, deriving upper bounds for tail expressions and introducing copula and marginal generative flows (CM flows) that enable exact tail modeling and prior CDF assumptions.
Deep generative networks such as GANs and normalizing flows flourish in the context of high-dimensional tasks such as image generation. However, so far exact modeling or extrapolation of distributional properties such as the tail asymptotics generated by a generative network is not available. In this paper, we address this issue for the first time in the deep learning literature by making two novel contributions. First, we derive upper bounds for the tails that can be expressed by a generative network and demonstrate Lp-space related properties. There we show specifically that in various situations an optimal generative network does not exist. Second, we introduce and propose copula and marginal generative flows (CM flows) which allow for an exact modeling of the tail and any prior assumption on the CDF up to an approximation of the uniform distribution. Our numerical results support the use of CM flows.