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<p>Multidimensional scaling (MDS) serves as a widely adopted methodology for projecting data on a finite metric space onto a lower-dimensional Euclidean space, with the goal of preserving pairwise distances as accurately as possible. While the classical metric MDS model typically involves fitting double-centered squared dissimilarity, Browne (1987) advocated for the direct fitting of original squared dissimilarity, suggesting potentially superior outcomes. This study redefines and investigates the problem of directly fitting the metric MDS model to the data as a matrix optimization problem over the product of the Stiefel sub-manifold of matrices with zero column sums and the linear sub-space comprising all <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10543_2025_1066_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(r\times r\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>×</mo> <mi>r</mi> </mrow> </math></EquationSource> </InlineEquation> diagonal matrices. We introduce an effective algorithm, drawing upon the generic Riemannian trust-region method developed by Absil et al., to address this problem, offering both global convergence and local superlinear convergence rates. Numerical experiments demonstrate that direct fitting yields a Euclidean distance matrix with reduced error. Additionally, we provide numerical comparisons with the projected gradient flow method, as well as several existing Riemannian first and second-order algorithms in the Riemannian optimization toolbox Manopt, to highlight the merits of our proposed approach.</p>

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A trust-region approach for iteration solution of the direct fitting metric MDS

  • 期刊论文

  • 发表日期：

    2025年5月19日

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  • Jiao-fen Li,
  • Jing Zhou,
  • Xue-lin Zhou,
  • Chao-qian Li,
  • Jia-shuo Song

摘要

Multidimensional scaling (MDS) serves as a widely adopted methodology for projecting data on a finite metric space onto a lower-dimensional Euclidean space, with the goal of preserving pairwise distances as accurately as possible. While the classical metric MDS model typically involves fitting double-centered squared dissimilarity, Browne (1987) advocated for the direct fitting of original squared dissimilarity, suggesting potentially superior outcomes. This study redefines and investigates the problem of directly fitting the metric MDS model to the data as a matrix optimization problem over the product of the Stiefel sub-manifold of matrices with zero column sums and the linear sub-space comprising all \(r\times r\) r × r diagonal matrices. We introduce an effective algorithm, drawing upon the generic Riemannian trust-region method developed by Absil et al., to address this problem, offering both global convergence and local superlinear convergence rates. Numerical experiments demonstrate that direct fitting yields a Euclidean distance matrix with reduced error. Additionally, we provide numerical comparisons with the projected gradient flow method, as well as several existing Riemannian first and second-order algorithms in the Riemannian optimization toolbox Manopt, to highlight the merits of our proposed approach.