Flatten a Curved Space by Kernel [Applications Corner]

Due to the recent explosion of data from all fields of science, there is an increasing need for pattern analysis tools, which are capable of analyzing data patterns in a non-Euclidean (curved) space. Because linear approaches are not directly applicable to handle data in a curved space, nonlinear approaches are to be used. Early-day nonlinear approaches were usually based on gradient descent or greedy heuristics, and these approaches suffered from local minima and overfitting [1]. In contrast, kernel methods provide a powerful means for transforming data in a non-Euclidean curved space into points in a high-dimensional Euclidean flat space, so that linear approaches can be applied to the transformed points in the high-dimensional Euclidean space. With this flattening capability, kernel methods combine the best features of linear approaches and nonlinear approaches, i.e., kernel methods are capable of dealing with nonlinear structures while enjoying a low computational complexity. In this column, we provide insights on and illustrate the power of kernel methods in two important pattern analysis problems: feature extraction and clustering.