# Quasi Monte Carlo methods enable extremely low-dimensional deep generative models

arXiv: https://arxiv.org/abs/2601.18676 · HTML: https://mlsanity.com/paper/2601.18676/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 62 references that are in the map.
- First posted on arXiv 26 Jan 2026.

## Builds on (most cited first, 28 of 28)

- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [An Introduction to Variational Autoencoders](https://mlsanity.com/paper/1906.02691/) (2019)
- [NVAE: A Deep Hierarchical Variational Autoencoder](https://mlsanity.com/paper/2007.03898/) (2020)
- [Variational Autoencoders and Nonlinear ICA: A Unifying Framework](https://mlsanity.com/paper/1907.04809/) (2019)
- [Importance Weighted Autoencoders](https://mlsanity.com/paper/1509.00519/) (2015)
- [Tutorial on Variational Autoencoders](https://mlsanity.com/paper/1606.05908/) (2016)
- [Latent Space Oddity: on the Curvature of Deep Generative Models](https://mlsanity.com/paper/1710.11379/) (2017)
- [Understanding How Dimension Reduction Tools Work: An Empirical Approach to Deciphering t-SNE, UMAP, TriMAP, and PaCMAP for Data Visualization](https://mlsanity.com/paper/2012.04456/) (2020)
- [Lagging Inference Networks and Posterior Collapse in Variational Autoencoders](https://mlsanity.com/paper/1901.05534/) (2019)
- [Iterative Amortized Inference](https://mlsanity.com/paper/1807.09356/) (2018)
- [Semi-Amortized Variational Autoencoders](https://mlsanity.com/paper/1802.02550/) (2018)
- [Tighter Variational Bounds are Not Necessarily Better](https://mlsanity.com/paper/1802.04537/) (2018)
- [The Usual Suspects? Reassessing Blame for VAE Posterior Collapse](https://mlsanity.com/paper/1912.10702/) (2019)
- [Doubly Reparameterized Gradient Estimators for Monte Carlo Objectives](https://mlsanity.com/paper/1810.04152/) (2018)
- [Efficient Iterative Amortized Inference for Learning Symmetric and Disentangled Multi-Object Representations](https://mlsanity.com/paper/2106.03630/) (2021)
- [Monte Carlo Variational Auto-Encoders](https://mlsanity.com/paper/2106.15921/) (2021)
- [Variational Walkback: Learning a Transition Operator as a Stochastic Recurrent Net](https://mlsanity.com/paper/1711.02282/) (2017)
- [Eigen-Distortions of Hierarchical Representations](https://mlsanity.com/paper/1710.02266/) (2017)
- [Quasi-Monte Carlo Variational Inference](https://mlsanity.com/paper/1807.01604/) (2018)
- [Improving Explorability in Variational Inference with Annealed Variational Objectives](https://mlsanity.com/paper/1809.01818/) (2018)
- [Leveraging the Exact Likelihood of Deep Latent Variable Models](https://mlsanity.com/paper/1802.04826/) (2018)
- [Automatic Differentiation Variational Inference with Mixtures](https://mlsanity.com/paper/2003.01687/) (2020)
- [Multiple Importance Sampling ELBO and Deep Ensembles of Variational Approximations](https://mlsanity.com/paper/2202.10951/) (2022)
- [Controlling Posterior Collapse by an Inverse Lipschitz Constraint on the Decoder Network](https://mlsanity.com/paper/2304.12770/) (2023)
- [Bounding Evidence and Estimating Log-Likelihood in VAE](https://mlsanity.com/paper/2206.09453/) (2022)
- [Combining Normalizing Flows and Quasi-Monte Carlo](https://mlsanity.com/paper/2401.05934/) (2024)

## Abstract (arXiv)

This paper introduces quasi-Monte Carlo latent variable models (QLVMs): a class of deep generative models that are specialized for finding extremely low-dimensional and interpretable embeddings of high-dimensional datasets. Unlike standard approaches, which rely on a learned encoder and variational lower bounds, QLVMs directly approximate the marginal likelihood by randomized quasi-Monte Carlo integration. While this brute force approach has drawbacks in higher-dimensional spaces, we find that it excels in fitting one, two, and three dimensional deep latent variable models. Empirical results on a range of datasets show that QLVMs consistently outperform conventional variational autoencoders (VAEs) and importance weighted autoencoders (IWAEs) with matched latent dimensionality. The resulting embeddings enable transparent visualization and post hoc analyses such as nonparametric density estimation, clustering, and geodesic path computation, which are nontrivial to validate in higher-dimensional spaces. While our approach is compute-intensive and struggles to generate fine-scale details in complex datasets, it offers a compelling solution for applications prioritizing interpretability and latent space analysis.