Colloquium

High dimensional data become ubiquitous in modern science and engineering. Some data are known or considered to have nonlinear structures and linear methods usually do not work well. We developed a series of methods that can detect nonlinear structures or nonlinear dependence underlying the data. These methods involve the use of reproducing kernel Hilbert spaces and l₁ minimization technique. The reproducing kernel Hilbert space enables nonlinear solutions to describe nonlinear structures by tuning the kernels and is computationally efficient due to the reproducing property. The l₁ minimization allows sparsity to eliminate irrelevant information. The utility of our methods is validated in various simulations and applications such as gene expression data analysis, hand written digits recognition, biomedical time series, and stylometry analysis.