Nonparametric Feature Screening for Non-Euclidean Data

We first revisit the concept of the Fréchet R⊕2 and prove that it possesses important properties analogous to those of the classical R2, including existence, boundedness, goodness of fit, monotonicity, and scale invariance. Building on these properties, we develop a novel feature screening procedure, termed local Fréchet sure independence screening, which identifies important covariates by evaluating the empirical Fréchet R⊕2 from corresponding univariate local Fréchet regressions.

Under mild conditions, we establish the uniform consistency, sure screening, and sure ranking properties of the proposed procedure. Extensive numerical studies provide strong empirical support for the correctness and effectiveness of our methodology. Two representative non-Euclidean datasets, mortality data and FC Barcelona match data, with distribution-valued and matrix-valued responses, respectively, are used to illustrate the practical performance of our method.</p>