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dirmeier committed Feb 19, 2024
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# Summary

Normalizing flows [NFs, @papamakarios2021normalizing] are tractable neural density estimators which have in the recent past been applied successfully for, e.g.,
generative modelling [@kingma2018glow,@ping20wave], Bayesian inference [@rezende15flow,@hoffman2019neutra] or simulation-based inference [@papamakarios2019sequential,@dirmeier2023simulation]. `Surjectors` is a Python library in particular
for *surjective*, i.e., dimensionality-reducing normalizing flows (SNFs, @klein2021funnels). `Surjectors` is based on the libraries JAX, Haiku and Distrax [@jax2018github, @deepmind2020jax] and is fully compatible with them.
generative modelling (@kingma2018glow, @ping20wave), Bayesian inference (@rezende15flow, @hoffman2019neutra) or simulation-based inference (@papamakarios2019sequential, @dirmeier2023simulation). `Surjectors` is a Python library in particular
for *surjective*, i.e., dimensionality-reducing normalizing flows (SNFs, @klein2021funnels). `Surjectors` is based on the libraries JAX, Haiku and Distrax (@jax2018github, @deepmind2020jax) and is fully compatible with them.
By virtue of being entirely written in JAX [@jax2018github], `Surjectors` naturally supports usage on either CPU, GPU or TPU.

# Statement of Need

Real-world data are often lying in a high-dimensional ambient space embedded in a lower-dimensional manifold [@fefferman2016testing] which can complicate estimation of probability densities (@dai2020sliced, @klein2021funnels,@nalisnick2018deep).
Real-world data are often lying in a high-dimensional ambient space embedded in a lower-dimensional manifold [@fefferman2016testing] which can complicate estimation of probability densities (@dai2020sliced, @klein2021funnels, @nalisnick2018deep).
As a remedy, recently neural density estimators using surjective normalizing flows (SNFs) have been proposed which reduce the dimensionality of the data while still allowing for exact computation of data likelihoods [@klein2021funnels].
While several computational libraries exist that implement *bijective* normalizing flows, i.e., flows that are dimensionality-preserving, currently none exist that efficiently implement dimensionality-reducing flows.

`Surjectors` is a normalizing flow library that implements both bijective and surjective normalizing flows. `Surjectors` is light-weight, conceptually simple to understand if familiar with the JAX ecosystem, and
computationally efficient due to leveraging the XLA compilation and vectorization from JAX.
We additionally make use of several well-established packages within the JAX ecosystem [@jax2018github] and probabilistic deep learning community.
For composing the conditioning networks that NFs facilitate, `Surjectors` uses the deep learning library Haiku [@haiku2020github]. For training and optimisation, we utilize the gradient transformation library
For composing the conditioning networks that NFs facilitate, `Surjectors` uses the deep learning library Haiku [@haiku2020github]. For training and optimization, we utilize the gradient transformation library
Optax [@deepmind2020jax]. `Surjectors` leverages Distrax [@deepmind2020jax] and TensorFlow probability [@dillon2017tensorflow] for probability distributions and several base bijector implementations.

# Adoption

@dirmeier2023simulation have proposed a novel method for simulation-based inference where they make use autoregressive inference surjections for density estimation and where they
are using `Surjectors` for their implementations.
are using `Surjectors` for their implementations. @dirmeier2023uncertainty used `Surjectors` for uncertainty quantification and out-of-distribution detection in deep neural network models.

# References

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