The Normal Procrustes Problem: A Riemannian Optimization Approach
Abstract
For given m × n data matrices $X, Y$, we investigate the Normal Procrustes Problem---the least squares optimization problem that aims to minimize |AX−Y|F2, where A is constrained to be a normal m × m matrix. As far as the author of this article is aware, no other method that attempts to solve the Normal Procrustes Problem exists in the literature; we thus propose what is, to our knowledge, the first such method. We, furthermore, adapt our approach to address the Real Normal Procrustes Problem, where A must be real. In our treatment of these problems, we first reduce our complex and real objective functions to be purely optimizable over the Riemannian manifolds of the unitary and real orthogonal matrices, respectively. This reduction enables us to apply techniques in Riemannian manifold optimization to approximate solutions to both. The Closest Normal Matrix and Real Closest Normal Matrix Problems are both special cases of their respective Procrustes Problems (by setting m = n and X = Im) and have been previously studied in the literature. Our approach thus recovers a novel Riemannian optimization method for approximating solutions to both these problems. We further numerically test the performance of our method across all such problems (including against previously developed algorithms on the Closest Normal Matrix Problems) and obtain competitive residuals and favorable scaling in wall-clock time.
Related
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BibTeX
@misc{bierly2026normal,
title = {{The Normal Procrustes Problem: A Riemannian Optimization Approach}},
author = {Kyle Bierly},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.19513},
eprint = {2608.19513},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.19513},
}
RIS
TY - UNPB
TI - The Normal Procrustes Problem: A Riemannian Optimization Approach
AU - Kyle Bierly
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2608.19513
UR - https://arxiv.org/abs/2608.19513
AB - For given m × n data matrices $X, Y$, we investigate the Normal Procrustes Problem---the least squares optimization problem that aims to minimize |AX−Y|_F^2, where A is constrained to be a normal m × m matrix. As far as the author of this article is aware, no other method that attempts to solve the Normal Procrustes Problem exists in the literature; we thus propose what is, to our knowledge, the first such method. We, furthermore, adapt our approach to address the Real Normal Procrustes Problem, where A must be real. In our treatment of these problems, we first reduce our complex and real objective functions to be purely optimizable over the Riemannian manifolds of the unitary and real orthogonal matrices, respectively. This reduction enables us to apply techniques in Riemannian manifold optimization to approximate solutions to both. The Closest Normal Matrix and Real Closest Normal Matrix Problems are both special cases of their respective Procrustes Problems (by setting m = n and X = I_m) and have been previously studied in the literature. Our approach thus recovers a novel Riemannian optimization method for approximating solutions to both these problems. We further numerically test the performance of our method across all such problems (including against previously developed algorithms on the Closest Normal Matrix Problems) and obtain competitive residuals and favorable scaling in wall-clock time.
ER -