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Kernel density matrices for probabilistic deep learning
Journal article   Open access   Peer reviewed

Kernel density matrices for probabilistic deep learning

Fabio A. Gonzalez, Raul Ramos-Pollan and Joseph Gallego
Quantum machine intelligence, v 7(2), 94
01 Dec 2025
DOI: https://doi.org/10.1007/s42484-025-00299-9
url
https://doi.org/10.1007/s42484-025-00299-9View
Published, Version of Record (VoR) Open

Abstract

Computer Science, Artificial Intelligence Quantum Science & Technology Science & Technology Computer Science Physical Sciences Physics Technology
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.

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