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Sparsity-preserving means on a homogeneous convex cone
Abstract
We study two means on the five-dimensional Vinberg cone Ω, viewed as a sparse cone of positive definite 3×3 matrices. The Cholesky-Vinberg mean is obtained from the simply transitive solvable group H associated with Ω. It is H-equivariant, and its interpolation curves are geodesics for two flat metric connections with torsion. One of the corresponding metrics is the canonical Hessian metric of the cone. The Log-Vinberg mean is defined by affine interpolation in the global logarithmic coordinates of the associated Vinberg algebra, or clan. The resulting Riemannian metric is complete and flat, and its weighted Fréchet means have explicit formulas. Both constructions preserve the zero pattern defining Ω.
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BibTeX
@misc{koufany2026sparsitypreserving,
title = {{Sparsity-preserving means on a homogeneous convex cone}},
author = {Khalid Koufany},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.07555},
eprint = {2609.07555},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05792754v1},
}
RIS
TY - UNPB
TI - Sparsity-preserving means on a homogeneous convex cone
AU - Khalid Koufany
PY - 2026
DA - 2026/10/04
PB - HAL
DO - 10.48550/arxiv.2609.07555
UR - https://hal.science/hal-05792754v1
AB - We study two means on the five-dimensional Vinberg cone Ω, viewed as a sparse cone of positive definite 3×3 matrices. The Cholesky-Vinberg mean is obtained from the simply transitive solvable group H associated with Ω. It is H-equivariant, and its interpolation curves are geodesics for two flat metric connections with torsion. One of the corresponding metrics is the canonical Hessian metric of the cone. The Log-Vinberg mean is defined by affine interpolation in the global logarithmic coordinates of the associated Vinberg algebra, or clan. The resulting Riemannian metric is complete and flat, and its weighted Fréchet means have explicit formulas. Both constructions preserve the zero pattern defining Ω.
ER -