The two key characteristics of a normalizing flow is that it is invertible (in particular, dimension preserving) and that it monitors the amount by which it changes the likelihood of data points as samples are propagated along the network. Recently, multiple generalizations of normalizing flows have been introduced that relax these two conditions \citep{nielsen2020survae,huang2020augmented}. On the other hand, neural networks only perform a forward pass on the input, there is neither a notion of an inverse of a neural network nor is there one of its likelihood contribution. In this paper we argue that certain neural network architectures can be enriched with a stochastic inverse pass and that their likelihood contribution can be monitored in a way that they fall under the generalized notion of a normalizing flow mentioned above.

The last choice especially makes sense when p( z) is symmetric.

The two key characteristics of a normalizing flow is that it is invertible (in particular, dimension preserving) and that it monitors the amount by which it changes the likelihood of data points as samples are propagated along the network. Recently, multiple generalizations of normalizing flows have been introduced that relax these two conditions \citep{nielsen2020survae,huang2020augmented}. On the other hand, neural networks only perform a forward pass on the input, there is neither a notion of an inverse of a neural network nor is there one of its likelihood contribution. In this paper we argue that certain neural network architectures can be enriched with a stochastic inverse pass and that their likelihood contribution can be monitored in a way that they fall under the generalized notion of a normalizing flow mentioned above. We prove that neural networks only containing linear and convolutional layers and invertible activations such as LeakyReLU can be flowified and evaluate them in the generative setting on image datasets.

The two key characteristics of a normalizing flow is that it is invertible (in particular, dimension preserving) and that it monitors the amount by which it changes the likelihood of data points as samples are propagated along the network. Recently, multiple generalizations of normalizing flows have been introduced that relax these two conditions. On the other hand, neural networks only perform a forward pass on the input, there is neither a notion of an inverse of a neural network nor is there one of its likelihood contribution. In this paper we argue that certain neural network architectures can be enriched with a stochastic inverse pass and that their likelihood contribution can be monitored in a way that they fall under the generalized notion of a normalizing flow mentioned above. We term this enrichment flowification. We prove that neural networks only containing linear layers, convolutional layers and invertible activations such as LeakyReLU can be flowified and evaluate them in the generative setting on image datasets.

Normalizing flows are diffeomorphic, typically dimension-preserving, models trained using the likelihood of the model. We use the SurVAE framework to construct dimension reducing surjective flows via a new layer, known as the funnel. We demonstrate its efficacy on a variety of datasets, and show it improves upon or matches the performance of existing flows while having a reduced latent space size. The funnel layer can be constructed from a wide range of transformations including restricted convolution and feed forward layers.

Normalizing flows and variational autoencoders are powerful generative models that can represent complicated density functions. However, they both impose constraints on the models: Normalizing flows use bijective transformations to model densities whereas VAEs learn stochastic transformations that are non-invertible and thus typically do not provide tractable estimates of the marginal likelihood. In this paper, we introduce SurVAE Flows: A modular framework of composable transformations that encompasses VAEs and normalizing flows. SurVAE Flows bridge the gap between normalizing flows and VAEs with surjective transformations, wherein the transformations are deterministic in one direction -- thereby allowing exact likelihood computation, and stochastic in the reverse direction -- hence providing a lower bound on the corresponding likelihood. We show that several recently proposed methods, including dequantization and augmented normalizing flows, can be expressed as SurVAE Flows. Finally, we introduce common operations such as the max value, the absolute value, sorting and stochastic permutation as composable layers in SurVAE Flows.

Deep generative modelling using flows has gained popularity owing to the tractable exact log-likelihood estimation with efficient training and synthesis process. Trained flow models carry rich information about the structure and local variance in input data. However, a bottleneck for flow models to scale with increasing dimensions is that the latent space has same size as the high-dimensional input space. In this paper, we propose methods to reduce the latent space dimension of flow models. Our first approach includes replacing standard high dimensional prior with a learned prior from a low dimensional noise space. Further improving to achieve exact log-likelihood with reduced dimensionality, our second approach presents an improved multi-scale architecture (Dinh et al., 2016) via likelihood contribution based factorization of dimensions. Using our method over state-of-the-art flow models, we demonstrate improvements in log-likelihood score on standard image benchmarks. Our work ventures a data dependent factorization scheme which is more efficient than static counterparts in prior works.