Localised Generative Flows

A BSTRACT We argue that flow-based density models based on continuous bijections are limited in their ability to learn target distributions with complicated topologies, and propose localised generative flows (LGFs) to address this problem. LGFs are composed of stacked continuous mixtures of bijections, which enables each bijection to learn a local region of the target rather than its entirety. Our method is a generalisation of existing flow-based methods, which can be used without modification as the basis for an LGF model. Unlike normalising flows, LGFs do not permit exact computation of log likelihoods, but we propose a simple variational scheme that performs well in practice. We show empirically that LGFs yield improved performance across a variety of density estimation tasks. 1 I NTRODUCTION Flow-based generative models, often referred to as normalising flows, have become popular methods for density estimation because of their flexibility, expressiveness, and tractable likelihoods. Given the problem of learning an unknown target density p null X on a data space X, normalising flows model p null X as the marginal of X obtained by the generative process Z p Z, X: g 1 ( Z), (1) where p Z is a prior density on a space Z, and g: X Z is a bijection. The parameters of g can be learned via maximum likelihood given i.i.d. To be effective, a normalising flow model must specify an expressive family of bijections with tractable Jacobians. Affine coupling layers (Dinh et al., 2014; 2016), autoregressive transformations (Germain et al., 2015; Papamakarios et al., 2017), ODEbased transformations (Grathwohl et al., 2018), and invertible ResNet blocks (Behrmann et al., 2019) are all examples of such bijections that can be composed to produce complicated flows. These models have demonstrated significant promise in their ability to model complex datasets (Papamakarios et al., 2017) and to synthesise novel data points (Kingma & Dhariwal, 2018). However, in all these cases, g is continuous in x .