# Kernel Density Matrices for Probabilistic Deep Learning

arXiv: https://arxiv.org/abs/2305.18204 · HTML: https://mlsanity.com/paper/2305.18204/

## Key facts

- Cited by 1 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 26 May 2023.

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

- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [WaveNet: A Generative Model for Raw Audio](https://mlsanity.com/paper/1609.03499/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Variational Deep Embedding: An Unsupervised and Generative Approach to Clustering](https://mlsanity.com/paper/1611.05148/) (2016)
- [Challenges and Opportunities in Quantum Machine Learning](https://mlsanity.com/paper/2303.09491/) (2023)
- [Bayesian Neural Networks: An Introduction and Survey](https://mlsanity.com/paper/2006.12024/) (2020)
- [Neural Expectation Maximization](https://mlsanity.com/paper/1708.03498/) (2017)
- [Implicit Reparameterization Gradients](https://mlsanity.com/paper/1805.08498/) (2018)
- [The Promises and Pitfalls of Deep Kernel Learning](https://mlsanity.com/paper/2102.12108/) (2021)
- [ClusterGAN : Latent Space Clustering in Generative Adversarial Networks](https://mlsanity.com/paper/1809.03627/) (2018)
- [Learning from Label Proportions by Learning with Label Noise](https://mlsanity.com/paper/2203.02496/) (2022)
- [Learning with Density Matrices and Random Features](https://mlsanity.com/paper/2102.04394/) (2021)
- [Quantum-inspired Multimodal Fusion for Video Sentiment Analysis](https://mlsanity.com/paper/2103.10572/) (2021)
- [Pattern recognition on the quantum Bloch sphere](https://mlsanity.com/paper/1603.00173/) (2016)
- [Learning and Inference in Hilbert Space with Quantum Graphical Models](https://mlsanity.com/paper/1810.12369/) (2018)

## Built on by (most cited first, 1 of 1)

- [Scalable Machine Learning Analysis of Parker Solar Probe Solar Wind Data](https://mlsanity.com/paper/2510.21066/) (2025)

## Abstract (arXiv)

This paper introduces a novel approach to probabilistic deep learning, kernel density matrices, which provide a simpler yet effective mechanism for representing joint probability distributions of both continuous and discrete random variables. In quantum mechanics, a density matrix is the most general way to describe the state of a quantum system. This work extends the concept of density matrices by allowing them to be defined in a reproducing kernel Hilbert space. This abstraction allows the construction of differentiable models for density estimation, inference, and sampling, and enables their integration into end-to-end deep neural models. In doing so, we provide a versatile representation of marginal and joint probability distributions that allows us to develop a differentiable, compositional, and reversible inference procedure that covers a wide range of machine learning tasks, including density estimation, discriminative learning, and generative modeling. The broad applicability of the framework is illustrated by two examples: an image classification model that can be naturally transformed into a conditional generative model, and a model for learning with label proportions that demonstrates the framework's ability to deal with uncertainty in the training samples. The framework is implemented as a library and is available at: https://github.com/fagonzalezo/kdm.