The Munificence of High Dimensionality

Many methods of mathematical analysis that are of practical value in low-dimensional Euclidean spaces have computational requirements that grow exponentially as the dimensionality of the space increases to tens, hundreds, or thousands. Mathematician Richard Bellman termed this explosion of difficulty “the curse of dimensionality.” The point of this talk is that this curse is not universal. In fact, there are some wonderfully beneficial properties of Euclidean spaces that improve with increasing dimension. These recently discovered and/or appreciated properties bear directly on studies of neural networks and related subjects.