Partial Trace Regression and Low-Rank Kraus Decomposition

• URL: http://arxiv.org/abs/2007.00935v2
• Date: Tue, 25 Aug 2020 08:41:59 GMT
• Title: Partial Trace Regression and Low-Rank Kraus Decomposition
• Authors: Hachem Kadri (QARMA), St\'ephane Ayache (QARMA), Riikka Huusari, Alain Rakotomamonjy (DocApp - LITIS), Liva Ralaivola
• Abstract summary: We introduce the partial-trace regression model, a family of linear mappings from matrix-valued inputs to matrix-valued outputs. 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.
• Score: 9.292155894591874
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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