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.

扩展了扩散映射形式化方法，提出了一种使用不对称核的数据集进行维度规约的算法，通过使用与Fourier基底张量积相关的坐标系统来表示数据集的几何结构，从而减少数据集的维度并利用两维快速Fourier变换算法（2-D FFT）提供的加速优势验证了其效率。

非对称核的扩散表示