Universal optimality of the double-centred matrix under unitarily invariant norms for dissimilarity data

Abstract

Let $D=(D_{ij})_{i,j=1}^{n}$ be a symmetric dissimilarity matrix and let $D^{(2)}=(D_{ij}^{2})_{i,j=1}^{n}$. We study the affine family of real symmetric matrices \[ A(\mathbf{g})=\tfrac{1}{2}\!\left(\mathbf{1}\mathbf{g}^{\top} +\mathbf{g}\mathbf{1}^{\top}-D^{(2)}\right), \qquad \mathbf{g}\in\mathbb{R}^{n}, \] parametrised by a free diagonal vector $\mathbf{g}\in\mathbb{R}^n$, whose off-diagonal entries satisfy \[ D_{ij}^{2}=a_{ii}+a_{jj}-2a_{ij}, \qquad i\neq j. \] We prove that the double-centred matrix \[ A(\mathbf{g}^F)=-\tfrac{1}{2}J D^{(2)} J, \qquad J=I-\tfrac{1}{n}\mathbf{1}\mathbf{1}^{\top}, \] is the unique minimiser of the Frobenius norm over this family, with minimiser $\mathbf{g}^F$ given explicitly by \[ g^{F}_{k} = \frac{1}{n}\sum_{i=1}^{n}D_{ik}^{2} - \frac{1}{2n^{2}}\sum_{i,j=1}^{n}D_{ij}^{2}, \qquad k=1,\dots,n, \] and that this same representative simultaneously minimises every unitarily invariant norm \[ \min_{\mathbf{g}\in\mathbb{R}^n}\left|\!\left|\!\left|A(\mathbf{g})\right|\!\right|\!\right|, \] including the spectral norm, the nuclear norm, and all Schatten $p$-norms and Ky Fan $k$-norms. Thus the double-centred matrix, central to classical multidimensional scaling, admits a purely variational characterisation that does not depend on the choice of norm and requires no Euclidean realisability assumption on $D$.