The geometry of monarch matrix approximation
Abstract
Helmke and Shayman showed that, for generic targets, least-squares distance to the manifold of rank-r matrices is a nondegenerate Morse function whose indices are governed by Bruhat-cell codimensions. We develop the analogous theory for Monarch matrices, products of a fixed permutation matrix with two block-diagonal factors. The parametrization's gauge dimension, as well as the codimensions and closure relations of the associated Bruhat-type cells, is determined by a common transfer-matrix invariant, although these quantities are generally distinct. In the equal-block case, we prove the Monarch-Schur theorem, characterizing the permutations that minimize gauge freedom and maximize manifold dimension, and derive a general lower bound for arbitrary partitions. We further establish generic second-order nondegeneracy, a Morse-index formula, and an explicit focal set of measure zero. Finally, we show that the unregularized gradient flow lacks coercivity because of unbounded flat directions. A quartic regularizer restores global convergence, selects a canonical minimum-norm exact solution when one exists, and otherwise yields a well-defined best-approximation image. The regularizer preserves the Monarch architecture and provides a geometric interpretation of weight decay in structured factorized layers.
Related
- A deterministic (1+ε)n approximation for the permanent of a nonnegative matrix
- LoCCA: Localized Chebyshev Cross Approximation for Kernel Matrix Factorization via Nodal Perturbation Stability
- Generic factorizations and alternating direction method for fixed-rank approximation of matrix polynomials
- On the Approximation of the Unitary Operator Group Associated with a Rotation Matrix and Its Applications to Abstract Hyperbolic Equations
- Matrix algebras with orthogonal Cauchy-like transforms and Toeplitz approximation: a negative result
BibTeX
@misc{yoshizawa2026geometry,
title = {{The geometry of monarch matrix approximation}},
author = {Shintaro Yoshizawa},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02226},
eprint = {2610.02226},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02226},
}
RIS
TY - UNPB
TI - The geometry of monarch matrix approximation
AU - Shintaro Yoshizawa
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02226
UR - https://arxiv.org/abs/2610.02226
AB - Helmke and Shayman showed that, for generic targets, least-squares distance to the manifold of rank-r matrices is a nondegenerate Morse function whose indices are governed by Bruhat-cell codimensions. We develop the analogous theory for Monarch matrices, products of a fixed permutation matrix with two block-diagonal factors. The parametrization's gauge dimension, as well as the codimensions and closure relations of the associated Bruhat-type cells, is determined by a common transfer-matrix invariant, although these quantities are generally distinct. In the equal-block case, we prove the Monarch-Schur theorem, characterizing the permutations that minimize gauge freedom and maximize manifold dimension, and derive a general lower bound for arbitrary partitions. We further establish generic second-order nondegeneracy, a Morse-index formula, and an explicit focal set of measure zero. Finally, we show that the unregularized gradient flow lacks coercivity because of unbounded flat directions. A quartic regularizer restores global convergence, selects a canonical minimum-norm exact solution when one exists, and otherwise yields a well-defined best-approximation image. The regularizer preserves the Monarch architecture and provides a geometric interpretation of weight decay in structured factorized layers.
ER -