Seminar by Dr. Chang Jun Im (Hybrid)

For regression problems in which the response is a distribution, covariance matrix, network, point on a manifold, or another complex object, Fréchet regression provides a framework for defining regression targets while preserving the geometry of the response space. One natural way to handle non-Euclidean predictors is to introduce a finite-dimensional coordinate representation. In global Fréchet regression, however, the population regression target depends on the coordinate representation of the predictor. Thus, the choice of chart is not merely a computational convenience but may change the regression target itself.

In this work, we formulate chart-induced global Fréchet regression based on a general finite-dimensional coordinate representation. First, for a coordinate representation fixed in advance, we establish uniform consistency and convergence rates for the estimator. We then analyze the case in which the anchor of the chart is estimated from the data and show that, when the anchor is estimated at the usual parametric rate, the same convergence rate is obtained as when the anchor is fixed in advance. Finally, for logarithmic and stereographic charts on spheres and Cartesian products of spheres, we verify the conditions required to apply the general theory.