# Cramer-Wold AutoEncoder

arXiv: https://arxiv.org/abs/1805.09235 · HTML: https://mlsanity.com/paper/1805.09235/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 May 2018.

## Built on by (most cited first, 10 of 10)

- [Generative Sliced MMD Flows with Riesz Kernels](https://mlsanity.com/paper/2305.11463/) (2023)
- [Points2NeRF: Generating Neural Radiance Fields from 3D point cloud](https://mlsanity.com/paper/2206.01290/) (2022)
- [Diversity in deep generative models and generative AI](https://mlsanity.com/paper/2202.09573/) (2022)
- [Data Interpolants -- That's What Discriminators in Higher-order Gradient-regularized GANs Are](https://mlsanity.com/paper/2306.00785/) (2023)
- [HyperColor: A HyperNetwork Approach for Synthesizing Auto-colored 3D Models for Game Scenes Population](https://mlsanity.com/paper/2108.01411/) (2021)
- [Momentum Contrastive Autoencoder: Using Contrastive Learning for Latent Space Distribution Matching in WAE](https://mlsanity.com/paper/2110.10303/) (2021)
- [Cramer Type Distances for Learning Gaussian Mixture Models by Gradient Descent](https://mlsanity.com/paper/2307.06753/) (2023)
- [Algorithms that get old : the case of generative deep neural networks](https://mlsanity.com/paper/2202.03008/) (2022)
- [Joint Distributional Learning via Cramer-Wold Distance](https://mlsanity.com/paper/2310.16374/) (2023)
- [Balanced Marginal and Joint Distributional Learning via Mixture Cramer-Wold Distance](https://mlsanity.com/paper/2312.03307/) (2023)

## Abstract (arXiv)

We propose a new generative model, Cramer-Wold Autoencoder (CWAE). Following WAE, we directly encourage normality of the latent space. Our paper uses also the recent idea from Sliced WAE (SWAE) model, which uses one-dimensional projections as a method of verifying closeness of two distributions. The crucial new ingredient is the introduction of a new (Cramer-Wold) metric in the space of densities, which replaces the Wasserstein metric used in SWAE. We show that the Cramer-Wold metric between Gaussian mixtures is given by a simple analytic formula, which results in the removal of sampling necessary to estimate the cost function in WAE and SWAE models. As a consequence, while drastically simplifying the optimization procedure, CWAE produces samples of a matching perceptual quality to other SOTA models.