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Published in:
2020 | OriginalPaper | Chapter
A Novel Geometric Approach to the Problem of Multidimensional Scaling
Authors : Gintautas Dzemyda, Martynas Sabaliauskas
Published in: Numerical Computations: Theory and Algorithms
Publisher: Springer International Publishing
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Abstract
Multidimensional scaling (MDS) is one of the most popular methods for a visual representation of multidimensional data. A novel geometric interpretation of the stress function and multidimensional scaling in general (Geometric MDS) has been proposed. Following this interpretation, the step size and direction forward the minimum of the stress function are found analytically for a separate point without reference to the analytical expression of the stress function, numerical evaluation of its derivatives and the linear search. It is proved theoretically that the direction coincides with the steepest descent direction, and the analytically found step size guarantees the decrease of stress in this direction. A strategy of application of the discovered option to minimize the stress function is presented and examined. It is compared with SMACOF version of MDS. The novel geometric approach will allow developing a new class of algorithms to minimize MDS stress, including global optimization and high-performance computing.