Partial Trace Regression and Low-Rank Kraus Decomposition

Hachem Kadri (1) , Stéphane Ayache (1) , Riikka Huusari (2, 3) , Alain Rakotomamonjy (4, 5) , Liva Ralaivola (5)

1 QARMA - éQuipe d'AppRentissage de MArseille
2 HIIT - Helsinki Institute for Information Technology
3 Department of Computer Science [Aalto]
4 DocApp - LITIS - Equipe Apprentissage
5 Criteo AI Lab

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.

Mots clés: Partial Trace; Trace Regression; Quantum Machine Learning; Kraus Decomposition; Matrix Completion; Completely Positive Maps

Domaines: Apprentissage [cs.LG]; Machine Learning [stat.ML]; Physique Quantique [quant-ph]

Hachem Kadri, Stéphane Ayache, Riikka Huusari, Alain Rakotomamonjy, Liva Ralaivola. Partial Trace Regression and Low-Rank Kraus Decomposition. International Conference on Machine Learning, Jul 2020, Vienne (Online), Austria. ⟨hal-02885339v2⟩