I first encountered this in my machine learning class in the context of solving ridge regression or SVM. Simply put, when performing SVM you want to find a pla

I first encountered this in my machine learning class in the context of solving ridge regression or SVM. Simply put, when performing SVM you want to find a plane that minimizes the sum of its distance from coordinates belonging to two classes. The coordinates can be very high dimensional, which makes the minimization problem hard (primal formulation: $y = w^Tx + b$, w being the weights you need to estimate, which is equal to a very high dimension). With help of the Lagrangian of the primal formulation, you can show that $x' ~ w^Tx + b ~ ayx^Tx + b$. So instead of having to estimate a very high number of w's, you can do away with only estimating the dot product x^Tx, which is limited by the number of inputs. This is the kernel trick you can use to avoid the curse of dimensionality.