Universal optimality of the double-centred matrix under unitarily invariant norms for dissimilarity data
Abstract
The classical identity relating squared distances to inner products determines every off-diagonal entry of a symmetric matrix associated with a set of pairwise dissimilarities, while leaving its diagonal free. This paper studies the resulting affine family of symmetric matrices and asks which representative is selected by minimising a matrix norm. We prove that the same representative, namely the double-centred matrix used in classical multidimensional scaling and principal coordinate analysis, minimises every unitarily invariant norm on the family. The result includes the Frobenius, spectral, nuclear, Schatten, and Ky Fan norms, and requires no assumption that the dissimilarities are Euclidean distances. The proof rests on a compression-reduction principle: the centring projection maps the entire compatible set to a single element of the family, and orthogonal compression is contractive for unitarily invariant norms; the conclusion follows without solving a separate minimisation problem for each norm. We then determine how many minimisers there are: the minimiser is unique for the Frobenius norm, for the nuclear norm and for every strictly convex unitarily invariant norm, while for the spectral norm we describe the whole minimising set, a compact convex set that reduces to a single point precisely when the double-centred matrix vanishes.
Related
- Clarkson--McCarthy type inequalities, part II: refinements, unitary orbits, unitarily invariant norms and ℓs--ℓs estimates
- Measure Convergence under Weak Domination for Unitarily Invariant Norms on Finite von Neumann Algebras
- Short curves in homogeneous spaces of the unitary group for unitarily invariant norms
- Unitarily invariant norms
- A Sharp Unitarily Invariant Norm Bound for the Off-Diagonal Block Perturbation of a Hermitian Matrix
BibTeX
@misc{santos2026universal,
title = {{Universal optimality of the double-centred matrix under unitarily invariant norms for dissimilarity data}},
author = {M. Nuria de las Heras Santos and Antonio Falcó and Francisco Javier Muñoz Almaraz},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.09670},
eprint = {2607.09670},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.09670},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Universal optimality of the double-centred matrix under unitarily invariant norms for dissimilarity data
AU - M. Nuria de las Heras Santos
AU - Antonio Falcó
AU - Francisco Javier Muñoz Almaraz
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2607.09670
UR - https://arxiv.org/abs/2607.09670
AB - The classical identity relating squared distances to inner products determines every off-diagonal entry of a symmetric matrix associated with a set of pairwise dissimilarities, while leaving its diagonal free. This paper studies the resulting affine family of symmetric matrices and asks which representative is selected by minimising a matrix norm. We prove that the same representative, namely the double-centred matrix used in classical multidimensional scaling and principal coordinate analysis, minimises every unitarily invariant norm on the family. The result includes the Frobenius, spectral, nuclear, Schatten, and Ky Fan norms, and requires no assumption that the dissimilarities are Euclidean distances. The proof rests on a compression-reduction principle: the centring projection maps the entire compatible set to a single element of the family, and orthogonal compression is contractive for unitarily invariant norms; the conclusion follows without solving a separate minimisation problem for each norm. We then determine how many minimisers there are: the minimiser is unique for the Frobenius norm, for the nuclear norm and for every strictly convex unitarily invariant norm, while for the spectral norm we describe the whole minimising set, a compact convex set that reduces to a single point precisely when the double-centred matrix vanishes.
ER -