  Dealing with land cover classification of the new image sources has also
turned to be a complex problem requiring large amount of memory and processing
time. In order to cope with these problems, statistical learning has greatly
helped in the last years to develop statistical retrieval and classification
models that can ingest large amounts of Earth observation data. Kernel methods
constitute a family of powerful machine learning algorithms, which have found
wide use in remote sensing and geosciences. However, kernel methods are still
not widely adopted because of the high computational cost when dealing with
large scale problems, such as the inversion of radiative transfer models or the
classification of high spatial-spectral-temporal resolution data. This paper
introduces an efficient kernel method for fast statistical retrieval of
bio-geo-physical parameters and image classification problems. The method
allows to approximate a kernel matrix with a set of projections on random bases
sampled from the Fourier domain. The method is simple, computationally very
efficient in both memory and processing costs, and easily parallelizable. We
show that kernel regression and classification is now possible for datasets
with millions of examples and high dimensionality. Examples on atmospheric
parameter retrieval from hyperspectral infrared sounders like IASI/Metop; large
scale emulation and inversion of the familiar PROSAIL radiative transfer model
on Sentinel-2 data; and the identification of clouds over landmarks in time
series of MSG/Seviri images show the efficiency and effectiveness of the
proposed technique.
