  We extend the diffusion-map formalism to data sets that are induced by
asymmetric kernels. Analytical convergence results of the resulting expansion
are proved, and an algorithm is proposed to perform the dimensional reduction.
In this work we study data sets in which its geometry structure is induced by
an asymmetric kernel. We use a priori coordinate system to represent this
geometry and, thus, be able to improve the computational complexity of reducing
the dimensionality of data sets. A coordinate system connected to the tensor
product of Fourier basis is used to represent the underlying geometric
structure obtained by the diffusion-map, thus reducing the dimensionality of
the data set and making use of the speedup provided by the two-dimensional Fast
Fourier Transform algorithm (2-D FFT). We compare our results with those
obtained by other eigenvalue expansions, and verify the efficiency of the
algorithms with synthetic data, as well as with real data from applications
including climate change studies.
