# Partial Trace Regression and Low-Rank Kraus Decomposition

arXiv: https://arxiv.org/abs/2007.00935 · HTML: https://mlsanity.com/paper/2007.00935/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jul 2020.

## Built on by (most cited first, 5 of 5)

- [Gradient-descent quantum process tomography by learning Kraus operators](https://mlsanity.com/paper/2208.00812/) (2022)
- [Gradient Clipping Beyond Vector Norms: A Spectral Approach for Matrix-Valued Parameters](https://mlsanity.com/paper/2605.11838/) (2026)
- [Lower and Upper Bounds on the VC-Dimension of Tensor Network Models](https://mlsanity.com/paper/2106.11827/) (2021)
- [Fast Quantum Process Tomography via Riemannian Gradient Descent](https://mlsanity.com/paper/2404.18840/) (2024)
- [Position: Quantum Kernel Machines Should Move Beyond Scalar-Valued Kernels to Realize Their Potential](https://mlsanity.com/paper/2506.03779/) (2025)

## Abstract (arXiv)

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.