Time: Tuesday, Oct 17, 2017, 2:00pm @ MTH1311

Speaker: Stefano Vigogna (JHU)

Title: Multiscale analysis of data and functions in high dimension

Abstract: In contemporary science and technology we constantly collect data which, albeit described by a large number of coordinates, feature an underlying low-dimensional structure. Common examples of intrinsic parameters include pose variations of a scene, facial expressions of a person, and position of lighting sources in high resolution images. Due to the well known curse of dimensionality, exploiting such structure turns out to be crucial in order to perform learning tasks with good accuracy while maintaining low sample complexity and affordable computational cost. The high nonlinearity of the data makes global, uniform, uniscale approaches generally ineffective, and calls for a finer multiscale analysis, capable of resolving local singularities and heterogeneous behaviors. In my talk I will present a universal algorithm to regress response variables on high-dimensional predictors concentrating near sets of lower dimension. Exploring the empirical distribution of the data, we construct piecewise polynomial approximations along a tree of multiscale partitions. Our estimators automatically adapt to the intrinsic geometry of the data, unveiling local features at different levels of detail. We show convergence both in probability and in expectation on uniform and nonuniform smoothness spaces, providing explicit rates for the finite-sample performance. The rates achieved are optimal (up to logarithmic factors,) and are independent of the ambient dimension. Moreover, the computational cost is quasilinear, and depends only linearly on the ambient dimension. Back to seminar