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Recent work shows that path gradient estimators for normalizing flows have lower
variance compared to standard estimators for variational inference, resulting in
improved training. However, they are often prohibitively more expensive from a
computational point of view and cannot be applied to maximum likelihood train-
ing in a scalable manner, which severely hinders their widespread adoption. In
this work, we overcome these crucial limitations. Specifically, we propose a fast
path gradient estimator which improves computational efficiency significantly and
works for all normalizing flow architectures of practical relevance. We then show
that this estimator can also be applied to maximum likelihood training for which
it has a regularizing effect as it can take the form of a given target energy func-
tion into account. We empirically establish its superior performance and reduced
variance for several natural sciences applications.
Generative modeling via stochastic processes has led to remarkable empirical results as well as to recent advances in their theoretical understanding. In principle, both space and time of the processes can be discrete or continuous. In this work, we study time-continuous Markov jump processes on discrete state spaces and investigate their correspondence to state-continuous diffusion processes given by SDEs. In particular, we revisit the Ehrenfest process, which converges to an Ornstein-Uhlenbeck process in the infinite state space limit. Likewise, we can show that the time-reversal of the Ehrenfest process converges to the time-reversed Ornstein-Uhlenbeck process. This observation bridges discrete and continuous state spaces and allows to carry over methods from one to the respective other setting. Additionally, we suggest an algorithm for training the time-reversal of Markov jump processes which relies on conditional expectations and can thus be directly related to denoising score matching. We demonstrate our methods in multiple convincing numerical experiments.