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.

In this paper, a new approach is proposed for extracting the spatio-temporal patterns from a location-based social networking system (SNS) such as Foursquare. The proposed approach consists of the following procedures. First, the spatio-temporal behaviors of users in SNS are approximated as a probabilistic distribution by using a diffusion-type formula. Since the SNS datasets generally consist of sparse check-in's of users at some time points and locations, it is difficult to investigate the spatio-temporal patterns on a wide range of time and space scales. The proposed method can estimate such wide range patterns by smoothing the sparse datasets by a diffusion-type formula. It is crucial in this method to estimate robustly the scale parameter by giving a prior generative model on check-in's of users. The robust estimation enables the method to extract appropriate patterns even in small local areas. Next, the covariance matrix among the time points is calculated from the estimated distribution. Then, the principal eigenfunctions are approximately extracted as the spatio-temporal patterns by principal component analysis (PCA). The distribution is a mixture of various patterns, some of which are regular ones with a periodic cycle and some of which are irregular ones corresponding to transient events. Though it is generally difficult to separate such complicated mixtures, the experiments on an actual Foursquare dataset showed that the proposed method can extract many plausible and interesting spatio-temporal patterns.