Multi-Resolution Geometric Analysis for Data in High Dimensions

Large data sets arise in a wide variety of applications and are often modeled as samples from a probability distribution in high-dimensional space. It is sometimes assumed that the support of such probability distribution is well approximated by a set of low intrinsic dimension, perhaps even a lowdimensional smooth manifold. Samples are often corrupted by high-dimensional noise. We are interested in developing tools for studying the geometry of such high-dimensional data sets. In particular, we present here a multiscale transform that maps high-dimensional data as above to a set of multiscale coefficients that are compressible/sparse under suitable assumptions on the data. We think of this as a geometric counterpart to multi-resolution analysis in wavelet theory: whereas wavelets map a signal (typically low dimensional, such as a one-dimensional time series or a two-dimensional image) to a set of multiscale coefficients, the geometric wavelets discussed here map points in a high-dimensional point cloud to a multiscale set of coefficients. The geometric multi-resolution analysis (GMRA) we construct depends on the support of the probability distribution, and in this sense it fits with the paradigm of dictionary learning or data-adaptive representations, albeit the type of representation we construct is in fact mildly nonlinear, as opposed to standard linear representations. Finally, we apply the transform to a set of synthetic and real-world data sets.