A Globally Convergent Inertial First-Order Optimization Method for Multidimensional Scaling

Noga Ram () and Shoham Sabach ()
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Noga Ram: Technion—Israel Institute of Technology
Shoham Sabach: Technion—Israel Institute of Technology

Journal of Optimization Theory and Applications, 2024, vol. 202, issue 2, No 17, 949-974

Abstract: Abstract Multidimensional scaling (MDS) is a popular tool for dimensionality reduction and data visualization. Given distances between data points and a target low-dimension, the MDS problem seeks to find a configuration of these points in the low-dimensional space, such that the inter-point distances are preserved as well as possible. We focus on the most common approach to formulate the MDS problem, known as stress minimization, which results in a challenging non-smooth and non-convex optimization problem. In this paper, we propose an inertial version of the well-known SMACOF Algorithm, which we call AI-SMACOF. This algorithm is proven to be globally convergent, and to the best of our knowledge this is the first result of this kind for algorithms aiming at solving the stress MDS minimization. In addition to the theoretical findings, numerical experiments provide another evidence for the superiority of the proposed algorithm.

Keywords: Multidimensional scaling; Nonconvex and nonsmooth optimization; First-order methods; Global convergence; Majorization minimization; 90C26; 90C30; 49M37; 65K10 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10957-024-02486-3

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