Nonlinear Dimensionality Reduction on Graphs

In this era of data deluge, many signal processing and machine learning tasks are faced with high-dimensional datasets, including images, videos, as well as time series generated from social, commercial and brain network interactions. Their efficient processing calls for dimensionality reduction techniques capable of properly compressing the data while preserving task-related characteristics, going beyond pairwise data correlations. The present paper puts forth a nonlinear dimensionality reduction framework that accounts for data lying on known graphs. The novel framework turns out to encompass most of the existing dimensionality reduction methods as special cases, and it is capable of capturing and preserving possibly nonlinear correlations that are ignored by linear methods, as well as taking into account information from multiple graphs. An efficient algorithm admitting closed-form solution is developed and tested on synthetic datasets to corroborate its effectiveness.