We propose an approach for sequence modeling based on autoregressive normalizing flows. Each autoregressive transform, acting across time, serves as a moving reference frame for modeling higher-level dynamics. This technique provides a simple, general-purpose method for improving sequence modeling, with connections to existing and classical techniques. We demonstrate the proposed approach both with standalone models, as well as a part of larger sequential latent variable models. Results are presented on three benchmark video datasets, where flow-based dynamics improve log-likelihood performance over baseline models. Data often contain sequential structure, providing a rich signal for learning models of the world. Such models are useful for learning self-supervised representations of sequences (Li & Mandt, 2018; Ha & Schmidhuber, 2018) and planning sequences of actions (Chua et al., 2018; Hafner et al., 2019) . While sequential models have a longstanding tradition in probabilistic modeling (Kalman et al., 1960) , it is only recently that improved computational techniques, primarily deep networks, have facilitated learning such models from high-dimensional data (Graves, 2013) , particularly video and audio. Dynamics in these models typically contain a combination of stochastic and deterministic variables (Bayer & Osendorfer, 2014; Chung et al., 2015; Gan et al., 2015; Fraccaro et al., 2016) , using simple distributions (e.g. Gaussian) to directly model the likelihood of data observations. However, attempting to capture all sequential dependencies with relatively unstructured dynamics may make it more difficult to learn such models. Intuitively, the model should use its dynamical components to track changes in the input instead of simultaneously modeling the entire signal. Rather than expanding the computational capacity of the model, we seek a method for altering the representation of the data to provide a more structured form of dynamics. To incorporate more structured dynamics, we propose an approach for sequence modeling based on autoregressive normalizing flows (Kingma et al., 2016; Papamakarios et al., 2017) , consisting of one or more autoregressive transforms in time. A single transform is equivalent to a Gaussian autoregressive model. However, by stacking additional transforms or latent variables on top, we can arrive at more expressive models. Each autoregressive transform serves as a moving reference frame in which higher-level structure is modeled. This provides a general mechanism for separating different forms of dynamics, with higher-level stochastic dynamics modeled in the simplified space provided by lower-level deterministic transforms. In fact, as we discuss, this approach generalizes the technique of modeling temporal derivatives to simplify dynamics estimation (Friston, 2008 ). We empirically demonstrate this approach, both with standalone autoregressive normalizing flows, as well as by incorporating these flows within more flexible sequential latent variable models. While normalizing flows have been applied in a few sequential contexts previously, we emphasize the use of these models in conjunction with sequential latent variable models. We present experimental results on three benchmark video datasets, showing improved quantitative performance in terms of log-likelihood. In formulating this general technique for improving dynamics estimation in the framework of normalizing flows, we also help to contextualize previous work. Figure 1: Affine Autoregressive Transforms. Computational diagrams for forward and inverse affine autoregressive transforms (Papamakarios et al., 2017) . Each y t is an affine transform of x t , with the affine parameters potentially non-linear functions of x <t . The inverse transform is capable of converting a correlated input, x 1:T , into a less correlated variable, y 1:T . We have presented a technique for improving sequence modeling based on autoregressive normalizing flows. This technique uses affine transforms to temporally decorrelate sequential data, thereby simplifying the estimation of dynamics. We have drawn connections to classical approaches, which involve modeling temporal derivatives. Finally, we have empirically shown how this technique can improve sequential latent variable models.
We show how autoregressive flows can be used to improve sequential latent variable models.
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