Nick Whiteley

On the Statistical Origin of Manifold Structure in Data

The `manifold hypothesis' is a collective label for assumptions that data are concentrated near topological or geometric structures embedded in high-dimensional spaces. For data such as natural images, global weather recordings and genomic sequences, there is well-established empirical evidence that manifold structure is present. In some cases, such structure can be explained intuitively albeit heuristically in terms of the physical mechanism that generated the data, in other cases manifold structure is less expected, but startling and sometimes mysterious. This talk will describe recent research which gives a new, unifying and essentially statistical explanation for the presence of manifold structure in data in terms of mathematical interactions between latent variables and correlations. This includes a refreshed view on Principal Component Analysis and connections to the methodology and theory of clustering, topological data analysis and nonlinear dimension reduction.
Joint work with Annie Gray (Compass CDT) and Prof. Patrick Rubin-Delanchy (Institute for Statistical Science, University of Bristol).