Global Mapping Analysis: Stochastic Gradient Algorithm in Multidimensional Scaling

摘要

In order to implement multidimensional scaling (MDS) efficiently, we propose a new method named "global mapping analysis" (GMA), which applies stochastic approximation to minimizing MDS criteria. GMA can solve MDS more efficiently in both the linear case (classical MDS) and non-linear one (e.g., ALSCAL) if only the MDS criteria are polynomial. GMA separates the polynomial criteria into the local factors and the global ones. Because the global factors need to be calculated only once in each iteration, GMA is of linear order in the number of objects. Numerical experiments on artificial data verify the efficiency of GMA. It is also shown that GMA can find out various interesting structures from massive document collections.