Cluster-Based Dimensionality Reduction by Nonparametric Distributional Screening
We consider dimensionality reduction for high-dimensional observations accompanied by a supplied partition into two or more clusters. The objective is not to construct a low-rank projection, but to retain an interpretable subset of the original coordinates that preserves the distributional information distinguishing the clusters. For each coordinate, the proposed procedure compares the cluster-specific empirical distribution functions through a several-sample Kolmogorov-Smirnov separation statistic. We formalize the resulting marginal cluster support and establish simultaneous finite-sample concentration over all coordinates, explicit bounds for false inclusions and omissions, and exact support recovery when the minimum distributional separation dominates the high-dimensional stochastic error. We also quantify the dimension inflation induced by using an unadjusted testing level and give a familywise-error-controlled version. Under a conditional sufficiency condition, sure screening preserves the full-data posterior cluster probabilities, mutual information, and Bayes risk; an additional result characterizes robustness to imperfectly estimated cluster labels. The procedure is invariant to strictly increasing coordinate transformations and can retain low-variance cluster signals that principal components may discard. We further develop average dual information, a criterion combining partition agreement after transformation with structural coverage of cluster-relevant coordinates, and derive its basic properties and consistency. Simulations illustrate the theory, the interpretability of the selected coordinates, and the distinction between cluster-directed screening and variance-directed projection.