Partial Trace Regression and Low-Rank Kraus Decomposition
Hachem Kadri, Stephane Ayache, Riikka Huusari, Alain Rakotomamonjy, Ralaivola Liva
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.
BibTeX
@InProceedings{pmlr-v119-kadri20a,
title = {Partial Trace Regression and Low-Rank Kraus Decomposition},
author = {Kadri, Hachem and Ayache, Stephane and Huusari, Riikka and Rakotomamonjy, Alain and Liva, Ralaivola},
booktitle = {Proceedings of the 37th International Conference on Machine Learning},
pages = {5031--5041},
year = {2020},
editor = {III, Hal Daumé and Singh, Aarti},
volume = {119},
series = {Proceedings of Machine Learning Research},
month = {13--18 Jul},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v119/kadri20a/kadri20a.pdf},
url = {https://proceedings.mlr.press/v119/kadri20a.html},
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.}
}