Lecture : Some Statistical Inference Issues ( 2 of 3 )

Last time, we talked about whether the Laplacian constructed from point clouds converged to the Laplace-Beltrami operator on the manifold from which the data were drawn, under the assumption that the unseen hypothesized data points are drawn from a probability distribution that is supported on a low-dimensional Riemannian manifold. While potentially interesting, that result is a little unsatisfactory for a number of reasons, basically since one typically does not test the hypothesis that the underlying manifold even exists, and since the result doesn’t imply anything statistical about cluster quality or prediction quality or some other inferential goal. For example, if one is going to use the Laplacian for spectral clustering, then probably a more interesting question is to ask whether the actual clusters that are identified make any sense, e.g., do they converge, are they consistent, etc. So, let’s consider these questions. Today and next time, we will do this in two different ways.