Abstract
This talk is about a new direction in my research that arose from the exposure to a broad set of topics during the big data program. Given a real-valued, differentiable multivariate function, an active subspace is a linear combination of the function inputs where the function exhibits large changes in its values. Although this sounds similar to principal components, the idea is distinct. Checking if a function has an active subspace involves computing eigenvalues of a matrix that results from a high-dimensional integration problem. This integral is typically intractable in the cases where active subspaces are used in science and engineering applications. We discuss a randomized procedure to accomplish this check using the randomized matrix methods discussed during the program. In terms of big data, this problem illustrates a case when only a small amount of data seems to be necessary for a useful approximation whereas a tremendously large amount of data is necessary for an exact computation.
For more technical detail, see:
Computing Active Subspaces
Paul Constantine, David Gleich
http://arxiv.org/abs/1408.0545