  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.
