Functional Multidimensional Scaling

Abstract

This article introduces a functional method for lower-dimensional smooth representations in terms of time-varying dissimilarities. The method incorporates dissimilarity representation in multidimensional scaling and smoothness approach of functional data analysis by using cubic B-spline basis functions. The model is designed to arrive at optimal representations with an iterative procedure such that dissimilarities evaluated by estimated representations are almost the same as original dissimilarities of objects in a low dimension which is easier for people to recognize. To solve expensive computation in optimization, we propose a computationally efficient method by taking gradient steps with respect to individual sub-functions of target functions using a Stochastic Gradient Descent algorithm. Keywords: Multidimensional Scaling, Functional Data Analysis, Statistical Modeling, Quasi-Newton Method, Stochastic Gradient Descent