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<p>The Generalized Inner Product SCALing (GIPSCAL) model is a specialized tool designed for the analysis of square asymmetric tables, with wide applications across disciplines such as sociology for social mobility tables and marketing for brand switching data. A significant problem in practical applications of GIPSCAL is the presence of missing data, which can compromise the accuracy and reliability of the results. This paper addresses the problem of fitting the three-way GIPSCAL model in the presence of missing values by reformulating it as a matrix optimization problem on the product manifold of orthonormal, diagonal, and skew-symmetric matrices. We introduce an efficient algorithm based on the Riemannian trust-region method, as proposed by Absil <i>et al.</i>, which guarantees global convergence and a locally superlinear convergence rate. Through numerical experiments, we demonstrate the effectiveness of the proposed method, providing a comparison with existing approaches such as the projected gradient method with necessary adjustments and several classical methods available in the Riemannian optimization toolbox Manopt. Additionally, we report comparisons with two state-of-the-art algorithms—the original alternating least-squares algorithm and its minimal polynomial extrapolation accelerated variant—applied to the standard three-way GIPSCAL fitting problem without missing values, further highlighting the advantages of our approach.</p>

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An efficient algorithm for fitting the three-way GIPSCAL problem with missing values from asymmetric multidimensional scaling

  • 期刊论文

  • 发表日期：

    2025年6月28日

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  • Xue-lin Zhou,
  • Jiao-fen Li,
  • Chao-qian Li

摘要

The Generalized Inner Product SCALing (GIPSCAL) model is a specialized tool designed for the analysis of square asymmetric tables, with wide applications across disciplines such as sociology for social mobility tables and marketing for brand switching data. A significant problem in practical applications of GIPSCAL is the presence of missing data, which can compromise the accuracy and reliability of the results. This paper addresses the problem of fitting the three-way GIPSCAL model in the presence of missing values by reformulating it as a matrix optimization problem on the product manifold of orthonormal, diagonal, and skew-symmetric matrices. We introduce an efficient algorithm based on the Riemannian trust-region method, as proposed by Absil et al., which guarantees global convergence and a locally superlinear convergence rate. Through numerical experiments, we demonstrate the effectiveness of the proposed method, providing a comparison with existing approaches such as the projected gradient method with necessary adjustments and several classical methods available in the Riemannian optimization toolbox Manopt. Additionally, we report comparisons with two state-of-the-art algorithms—the original alternating least-squares algorithm and its minimal polynomial extrapolation accelerated variant—applied to the standard three-way GIPSCAL fitting problem without missing values, further highlighting the advantages of our approach.