Semi-Supervised Kernel Ridge Fr\'echet Regression

Abstract

Fr\'echet regression provides a framework for predicting random object-valued responses by minimizing the conditional expected squared distances. We propose semi-supervised kernel ridge Fr\'echet regression (SS-KRFR) to exploit unlabeled covariates when labeled samples are scarce. The method learns kernel spectral features from both labeled and unlabeled covariates, then uses labeled responses to estimate the conditional Fr\'echet function by ridge regression. We derive prediction error bounds that distinguish the roles of labeled and unlabeled samples, and establish polynomial convergence rates under suitable conditions. Under additional kernel assumptions, a sharper analysis relaxes the sample-size requirements for learning the features. We also develop hybrid methods that combine features, predictions, or both from models fitted with and without unlabeled covariates, and establish hybrid stability uniformly over mixing weights in Hadamard spaces at common tuning, with a separate rate guarantee for prediction mixing. Simulations and an EEG application show semi-supervised gain and improvements over competing Fr\'echet regression methods in several settings.