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.
Cite
@article{arxiv.2007.00935,
title = {Partial Trace Regression and Low-Rank Kraus Decomposition},
author = {Hachem Kadri and Stéphane Ayache and Riikka Huusari and Alain Rakotomamonjy and Liva Ralaivola},
journal= {arXiv preprint arXiv:2007.00935},
year = {2020}
}