Can you tell me what the Kernel trick is, in the context of SVMs?
The kernel trick is a technique used to map data from one feature space (e.g., low-dimensional) into another feature space (e.g., higher-dimensional or more appropriate for machine learning algorithms). In the case of support vector machines (SVM), it allows us to solve non-linearly separable problems in Hilbert spaces (feature spaces) of arbitrary dimension, with only minimal changes made to the algorithm itself.

In layman's terms, the kernel trick involves transforming each instance X using a function K(X,.) instead of working directly with the dot product, as would be done in a linear SVM. The kernel function is capable of mapping instances from the original feature space into a higher-dimensional space, where the instances become linearly separable. This transformation process enables the kernelized version of the linear classifier (SVM) to solve classification problems in much higher dimensions without running into computational complexity issues during training and prediction stages, unlike plain vanilla perceptron or linear classifiers.

The evaluation of K(x1, x2), which involves some complex math, is not as computationally expensive as calculating the dot product x1^T * x2, which requires 2N complexity (where ^ signifies tensor multiplication for the dot product). This trade-off between mathematical representation and complexity at prediction makes the kernel trick an indispensable part of state-of-the-art models like SVM and many other flavors of machine learning classifiers built upon SVM for big data applications.