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Partial Trace Regression and Low-Rank Kraus Decomposition

Abstract : The trace regression model, a direct extension of the well-studied linear regression model, allows one to map matrices to real-valued outputs. We here introduce an even more general model, namely the partial-trace regression model, a family of linear mappings from matrix-valued inputs to matrix-valued outputs; this model subsumes the trace regression model and thus the linear regression model. Borrowing tools from quantum information theory, where partial trace operators have been extensively studied, we propose a framework for learning partial trace regression models from data by taking advantage of the so-called low-rank Kraus representation of completely positive maps. We show the relevance of our framework with synthetic and real-world experiments conducted for both i) matrix-to-matrix regression and ii) positive semidefinite matrix completion, two tasks which can be formulated as partial trace regression problems.
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https://hal.archives-ouvertes.fr/hal-02885339
Contributor : Hachem Kadri <>
Submitted on : Wednesday, August 12, 2020 - 7:30:02 PM
Last modification on : Wednesday, August 26, 2020 - 3:30:02 AM

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  • HAL Id : hal-02885339, version 2
  • ARXIV : 2007.00935

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Hachem Kadri, Stéphane Ayache, Riikka Huusari, Alain Rakotomamonjy, Liva Ralaivola. Partial Trace Regression and Low-Rank Kraus Decomposition. International Conference on Machine Learning, Jul 2020, Vienne (Online), Austria. ⟨hal-02885339v2⟩

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