Chen Xin, Qin Yuefeng, Zhou Xuelin, Li Jiaofen
Multidimensional scaling (MDS) is a technique used in multidimensional data analysis that depicts the similarities or relationships between observed objects as distances between points in a lower-dimensional space. By representing high-dimensional data within a lowdimensional framework, MDS preserves the relative distances between data points. This study focuses on developing an efficient numerical algorithm for a specific type of individual differences scaling model, known as O-INDSCAL, within symmetric multidimensional scaling, which accounts for individual differences among observed objects. Initially, leveraging the concept of the alternating least squares algorithm, the multivariable constrained matrix optimization model associated with the O-INDSCAL model is transformed into a fixed-point iteration problem. By thoroughly examining the acceleration principles and implementation processes of various polynomial extrapolation and Anderson acceleration methods in vector sequence acceleration, we have designed a matrix sequence acceleration algorithm tailored to this problem model. Numerical experiments indicate that the proposed acceleration algorithms significantly enhance the convergence speed of sequences generated by fixed-point iterations. Furthermore, when compared with existing continuous-time projection gradient flow algorithms and the first-order and second-order Riemannian algorithms available in the Manopt toolbox for manifold optimization, the proposed algorithms demonstrate notable improvements in iterative efficiency.
