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Intell."],"published-print":{"date-parts":[[2025,12]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    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\u2019s ability to deal with uncertainty in the training samples. The framework is implemented as a library and is available at:\n                    <jats:ext-link xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"https:\/\/github.com\/fagonzalezo\/kdm\" ext-link-type=\"uri\">https:\/\/github.com\/fagonzalezo\/kdm<\/jats:ext-link>\n                    .\n                  <\/jats:p>","DOI":"10.1007\/s42484-025-00299-9","type":"journal-article","created":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T12:38:30Z","timestamp":1759840710000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Kernel density matrices for probabilistic deep learning"],"prefix":"10.1007","volume":"7","author":[{"given":"Fabio\u00a0A.","family":"Gonz\u00e1lez","sequence":"first","affiliation":[]},{"given":"Ra\u00fal","family":"Ramos-Poll\u00e1n","sequence":"additional","affiliation":[]},{"given":"Joseph","family":"Gallego","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,10,7]]},"reference":[{"key":"299_CR1","unstructured":"Bengio Y, Larochelle H, Vincent P (2005) Non-local manifold parzen windows. 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