# Neural Product Importance Sampling via Warp Composition

arXiv: https://arxiv.org/abs/2409.18974 · HTML: https://mlsanity.com/paper/2409.18974/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 44 references that are in the map.
- First posted on arXiv 12 Sep 2024.

## Builds on (most cited first, 16 of 16)

- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019)
- [Neural Importance Sampling](https://mlsanity.com/paper/1808.03856/) (2018)
- [Normalizing Flows on Tori and Spheres](https://mlsanity.com/paper/2002.02428/) (2020)
- [Neural Control Variates](https://mlsanity.com/paper/2006.01524/) (2020)
- [Neural BRDF Representation and Importance Sampling](https://mlsanity.com/paper/2102.05963/) (2021)
- [Random-Access Neural Compression of Material Textures](https://mlsanity.com/paper/2305.17105/) (2023)
- [Real-Time Neural Appearance Models](https://mlsanity.com/paper/2305.02678/) (2023)
- [Dr.Jit: A Just-In-Time Compiler for Differentiable Rendering](https://mlsanity.com/paper/2202.01284/) (2022)
- [NeuMIP: Multi-Resolution Neural Materials](https://mlsanity.com/paper/2104.02789/) (2021)
- [Learning to Importance Sample in Primary Sample Space](https://mlsanity.com/paper/1808.07840/) (2018)

## Built on by (most cited first, 3 of 3)

- [Neural BRDF Importance Sampling by Reparameterization](https://mlsanity.com/paper/2505.08998/) (2025)
- [Neural Path Guiding with Distribution Factorization](https://mlsanity.com/paper/2506.00839/) (2025)
- [8DNA: 8D Neural Asset Light Transport by Distribution Learning](https://mlsanity.com/paper/2604.25129/) (2026)

## Abstract (arXiv)

Achieving high efficiency in modern photorealistic rendering hinges on using Monte Carlo sampling distributions that closely approximate the illumination integral estimated for every pixel. Samples are typically generated from a set of simple distributions, each targeting a different factor in the integrand, which are combined via multiple importance sampling. The resulting mixture distribution can be far from the actual product of all factors, leading to sub-optimal variance even for direct-illumination estimation. We present a learning-based method that uses normalizing flows to efficiently importance sample illumination product integrals, e.g., the product of environment lighting and material terms. Our sampler composes a flow head warp with an emitter tail warp. The small conditional head warp is represented by a neural spline flow, while the large unconditional tail is discretized per environment map and its evaluation is instant. If the conditioning is low-dimensional, the head warp can be also discretized to achieve even better performance. We demonstrate variance reduction over prior methods on a range of applications comprising complex geometry, materials and illumination.