# Invertible Gaussian Reparameterization: Revisiting the Gumbel-Softmax

arXiv: https://arxiv.org/abs/1912.09588 · HTML: https://mlsanity.com/paper/1912.09588/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Dec 2019.

## Built on by (most cited first, 18 of 18)

- [A Review of the Gumbel-max Trick and its Extensions for Discrete Stochasticity in Machine Learning](https://mlsanity.com/paper/2110.01515/) (2021)
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- [Identifying Semantic Component for Robust Molecular Property Prediction](https://mlsanity.com/paper/2311.04837/) (2023)
- [Generative Modeling of Discrete Joint Distributions by E-Geodesic Flow Matching on Assignment Manifolds](https://mlsanity.com/paper/2402.07846/) (2024)
- [MGF: Mixed Gaussian Flow for Diverse Trajectory Prediction](https://mlsanity.com/paper/2402.12238/) (2024)
- [Bayesian Nonparametrics for Offline Skill Discovery](https://mlsanity.com/paper/2202.04675/) (2022)
- [Understanding What Affects the Generalization Gap in Visual Reinforcement Learning: Theory and Empirical Evidence](https://mlsanity.com/paper/2402.02701/) (2024)
- [SketchDNN: Joint Continuous-Discrete Diffusion for CAD Sketch Generation](https://mlsanity.com/paper/2507.11579/) (2025)
- [Revisiting the Gumbel-Softmax in MADDPG](https://mlsanity.com/paper/2302.11793/) (2023)
- [Simplex-to-Euclidean Bijections for Categorical Flow Matching](https://mlsanity.com/paper/2510.27480/) (2025)
- [Multi-View Oriented GPLVM: Expressiveness and Efficiency](https://mlsanity.com/paper/2502.08253/) (2025)
- [Categorical Reparameterization with Denoising Diffusion models](https://mlsanity.com/paper/2601.00781/) (2026)
- [QWID: Quantized Weed Identification Deep neural network](https://mlsanity.com/paper/2310.18921/) (2023)
- [On The Statistical Representation Properties Of The Perturb-Softmax And The Perturb-Argmax Probability Distributions](https://mlsanity.com/paper/2406.02180/) (2024)
- [A Matrix Variational Auto-Encoder for Variant Effect Prediction in Pharmacogenes](https://mlsanity.com/paper/2507.02624/) (2025)

## Abstract (arXiv)

The Gumbel-Softmax is a continuous distribution over the simplex that is often used as a relaxation of discrete distributions. Because it can be readily interpreted and easily reparameterized, it enjoys widespread use. We propose a modular and more flexible family of reparameterizable distributions where Gaussian noise is transformed into a one-hot approximation through an invertible function. This invertible function is composed of a modified softmax and can incorporate diverse transformations that serve different specific purposes. For example, the stick-breaking procedure allows us to extend the reparameterization trick to distributions with countably infinite support, thus enabling the use of our distribution along nonparametric models, or normalizing flows let us increase the flexibility of the distribution. Our construction enjoys theoretical advantages over the Gumbel-Softmax, such as closed form KL, and significantly outperforms it in a variety of experiments. Our code is available at https://github.com/cunningham-lab/igr.