# Machine Learning Trivializing Maps: A First Step Towards Understanding How Flow-Based Samplers Scale Up

arXiv: https://arxiv.org/abs/2112.15532 · HTML: https://mlsanity.com/paper/2112.15532/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 15 references that are in the map.
- First posted on arXiv 31 Dec 2021.

## Builds on (most cited first, 8 of 8)

- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Flow-based generative models for Markov chain Monte Carlo in lattice field theory](https://mlsanity.com/paper/1904.12072/) (2019)
- [Equivariant flow-based sampling for lattice gauge theory](https://mlsanity.com/paper/2003.06413/) (2020)
- [Sampling using SU(N) gauge equivariant flows](https://mlsanity.com/paper/2008.05456/) (2020)
- [Flow-based sampling for fermionic lattice field theories](https://mlsanity.com/paper/2106.05934/) (2021)
- [Flow-based sampling for multimodal and extended-mode distributions in lattice field theory](https://mlsanity.com/paper/2107.00734/) (2021)
- [Estimation of Thermodynamic Observables in Lattice Field Theories with Deep Generative Models](https://mlsanity.com/paper/2007.07115/) (2020)

## Built on by (most cited first, 5 of 5)

- [Detecting and Mitigating Mode-Collapse for Flow-based Sampling of Lattice Field Theories](https://mlsanity.com/paper/2302.14082/) (2023)
- [Path-Gradient Estimators for Continuous Normalizing Flows](https://mlsanity.com/paper/2206.09016/) (2022)
- [Simulating first-order phase transition with hierarchical autoregressive networks](https://mlsanity.com/paper/2212.04955/) (2022)
- [Comparison of Affine and Rational Quadratic Spline Coupling and Autoregressive Flows through Robust Statistical Tests](https://mlsanity.com/paper/2302.12024/) (2023)
- [Empirical evaluation of normalizing flows in Markov Chain Monte Carlo](https://mlsanity.com/paper/2412.17136/) (2024)

## Abstract (arXiv)

A trivializing map is a field transformation whose Jacobian determinant exactly cancels the interaction terms in the action, providing a representation of the theory in terms of a deterministic transformation of a distribution from which sampling is trivial. Recently, a proof-of-principle study by Albergo, Kanwar and Shanahan [arXiv:1904.12072] demonstrated that approximations of trivializing maps can be `machine-learned' by a class of invertible, differentiable neural models called textitnormalizing flows. By ensuring that the Jacobian determinant can be computed efficiently, asymptotically exact sampling from the theory of interest can be performed by drawing samples from a simple distribution and passing them through the network. From a theoretical perspective, this approach has the potential to become more efficient than traditional Markov Chain Monte Carlo sampling techniques, where autocorrelations severely diminish the sampling efficiency as one approaches the continuum limit. A major caveat is that it is not yet understood how the size of models and the cost of training them is expected to scale. As a first step, we have conducted an exploratory scaling study using two-dimensional φ4 with up to 202 lattice sites. Although the scope of our study is limited to a particular model architecture and training algorithm, initial results paint an interesting picture in which training costs grow very quickly indeed. We describe a candidate explanation for the poor scaling, and outline our intentions to clarify the situation in future work.