Partial Trace Regression and Low-Rank Kraus Decomposition

TL;DR

This paper introduces the partial-trace regression model, a generalization of trace regression, utilizing quantum information tools and low-rank Kraus decompositions to learn matrix-to-matrix mappings with applications in matrix regression and completion.

Contribution

It proposes a novel partial-trace regression framework that extends trace regression, leveraging quantum-inspired low-rank Kraus representations for improved learning of matrix-valued functions.

Findings

Effective in synthetic experiments

Successful application to real-world matrix completion

Outperforms existing methods in specified tasks

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.