Beyond Euclidean responses: Fréchet regression for complex data objects

Fréchet regression provides a framework for relating complex response objects that reside in non-Euclidean metric spaces with Euclidean vector predictors. Its foundation is the extension of the classical Fréchet mean (Fréchet 1948) to the conditional Fréchet mean, introduced in 2019. Initial implementations included global Fréchet regression, based on a generalization of least squares, and local Fréchet regression, extending locally weighted smoothing methods to object-valued responses. Early methodological developments further incorporated total variation regularization to accommodate discontinuities and trend-filtering type behavior, as well as single-index Fréchet regression for multivariate predictors.

Recent advances include statistical inference and predictor selection for global Fréchet regression and a theoretically challenging additive Fréchet regression model for multivariate predictors. For high-dimensional predictors and large-scale data, several deep learning approaches have recently emerged. These include deep Fréchet regression methods that combine neural networks with low-dimensional object representations obtained through Isomap embeddings, as well as a version with a fully end-to-end architecture that achieves state-of-the-art performance.

Adapted versions of Fréchet regression have proven instrumental for interpolation and extrapolation of distributions, sliced Wasserstein regression, network regression, global Fréchet manifold learning, and causal inference for random object outcomes. The talk will survey the core methodology, theoretical developments, and emerging machine learning connections, illustrated with various data applications.