Flatten a Curved Space by Kernel: From Einstein to Euclid

Einstein’s general theory of relativity fundamentally changed our view about the physical world. Different from Newton’s theory, Einstein’s space and time are not flat but can be warped by matter. For a curved space such as Einstein’s space, Euclidean geometry is no longer suitable, and Riemannian geometry is usually used instead. In parallel with physics, due to an explosion of data from all fields of science, there is an increasing need for pattern analysis tools, which are capable of analyzing patterns of data in a non-Euclidean (curved) space. To handle data in a curved space, linear approaches are not directly applicable, and instead nonlinear approaches are the right weapon. However, early-day nonlinear approaches were usually based on gradient descent or greedy heuristics, and suffered from local minima and overfitting [1]. In contrast, kernel methods provide a powerful means for transforming data in a non-Euclidean curved space (such as Einstein space) into points in a highdimensional 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 like linear approaches. In this column, we provide important insights into kernel methods and illustrate the power of kernel methods in two important pattern analysis problems: feature extraction and clustering.