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.

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.

我们将扩散图形式主义扩展到由非对称核导出的数据集。证明了由此产生的展开的解析收敛结果，并提出了一种进行降维的算法。

连接到傅立叶张量积的坐标系用于表示通过扩散图获得的基础几何结构，从而减小了数据集的维数，并利用了二维快速傅立叶变换算法提供的加速（2-D FFT）。

我们将我们的结果与其他特征值扩展获得的结果进行比较，并使用合成数据以及包括气候变化研究在内的实际数据验证算法的效率。