Partial Trace Regression and Low-Rank Kraus Decomposition
This provides a new mathematical framework for matrix-valued regression problems, though it appears incremental as an extension of existing trace regression models.
The authors introduced the partial-trace regression model, a generalization of trace regression that maps matrix-valued inputs to matrix-valued outputs, and developed a learning framework using low-rank Kraus representations from quantum information theory. They demonstrated its effectiveness on synthetic and real-world experiments for matrix-to-matrix regression and positive semidefinite matrix completion.
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.