Kernel Density Matrices for Probabilistic Deep Learning

• URL: http://arxiv.org/abs/2305.18204v3
• Date: Tue, 30 Apr 2024 17:54:43 GMT
• Title: Kernel Density Matrices for Probabilistic Deep Learning
• Authors: Fabio A. González, Raúl Ramos-Pollán, Joseph A. Gallego-Mejia,
• Abstract summary: In quantum mechanics, a density matrix is the most general way to describe the state of a quantum system. This paper introduces a novel approach to probabilistic deep learning, kernel density matrices. It provides a simpler yet effective mechanism for representing joint probability distributions of both continuous and discrete random variables.
• Score: 8.486487001779416
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: 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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