Randomized approximations of operators and their spectral decomposition for diffusion based embeddings of heterogeneous data

Modern sensing platforms acquire heterogeneous data collected from multiple sensing modalities. We are then faced with the challenge of fusing the already collected heterogeneous data for effective use in such applications as data classification, segmentation, or target detection. Data fusion is a non-trivial problem and many existing approaches are either highly dependent on the types of data or fuse the information post classification. When the data is high dimensional and possesses a large number of data points (pixels), such as is the case for Hyperspectral Satellite Imagery, it can be burdensome on even the most efficient algorithms. Previous techniques of data integration have relied harmonic analysis methods to fuse data and produce low dimensional representations of large heterogeneous datasets. These techniques construct data-driven operators whose spectral decomposition creates a representation of the data. The bottlenecks of this approach are in the construction of the operator and its spectral decomposition, which may require a significant computational cost. Thus, we introduce randomized approximations at the most computationally intensive steps to significantly reduce the computational cost with minimal loss of fidelity, thus enabling the practical use of these highly sophisticated and geometrically reliable methods.