TITLE: What are some lesser-known examples where increasing the dimensionality makes the problem easier to solve?
QUESTION [13 upvotes]: I feel like there is a common pattern in mathematics where increasing the dimensionality makes the problem easier to solve or provides a solution where otherwise one would not exist. Some examples:

Going from real numbers to complex numbers
Working with quaternions (4D) instead of (roll, pitch, yaw) or rotation matrices which have singularities
Laplace transform which lets us solve differential equations with algebra (ok this isn't really increasing the dimensionality, more like working in a different dimension)

I'm not a mathematician but I imagine this appears in other areas. What else is there? Is there a "method" or systematic way for increasing the dimensionality of a problem to make it easier to solve? Bonus if there is something in the field of optimization and/or linear algebra, which I am most interested in at the moment.

REPLY [2 votes]: Any nonlinear decision problem can be solved in higher dimensions as a linear decision problem. The XOR decision function, for example, is nonlinear in two dimensions, but becomes linearly separable when mapping to an appropriate third dimension. This is essentially the goal of supervised machine learning methods - find a way to take some set of input features, and construct a new output feature that maps to the target variable. With the correct mapping to an additional dimension, any classification problem is linearly separable.