This paper presents a novel approach to probabilistic deep learning (PDL),
quantum kernel mixtures, derived from the mathematical formalism of quantum
density matrices, which provides a simpler yet effective mechanism for
representing joint probability distributions of both continuous and discrete
random variables. The framework allows for the construction of differentiable
models for density estimation, inference, and sampling, enabling 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. We illustrate the
broad applicability of the framework with two examples: an image classification
model, which can be naturally transformed into a conditional generative model
thanks to the reversibility of our inference procedure; and a model for
learning with label proportions, which is a weakly supervised classification
task, demonstrating the framework's ability to deal with uncertainty in the
training samples.

该研究通过量子密度矩阵的数学形式导出了一种新颖的概率深度学习方法，即量子核混合模型，旨在建立不同 iable 的模型进行密度估计、推理、采样，实现端到端深度神经模型集成，其具有广泛的应用价值，特别是在密度估计、判别学习和生成建模中展现出了显著的优势 。